#set document(title: "6.7 Integer Exponents and Scientific Notation", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 6.7#h(0.6em)Integer Exponents and Scientific Notation #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. What is the place value of the #math.equation(block: false, alt: "6")[$6$] in the number #math.equation(block: false, alt: "64,891")[$64,891$]? #linebreak() If you missed this problem, review . #solutionbox[ Ten thousands ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Name the decimal: #math.equation(block: false, alt: "0.0012.")[$0.0012 .$] #linebreak() If you missed this problem, review . #solutionbox[ Twelve ten thousandths ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Subtract: #math.equation(block: false, alt: "5 minus open parenthesis −3 close parenthesis .")[$5 − ( −3 ) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "8")[$8$] ] ] === Use the Definition of a Negative Exponent We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Quotient Property for Exponents] If #math.equation(block: false, alt: "a")[$a$] is a real number, #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$], and #math.equation(block: false, alt: "m and n")[$m #h(0.2em) "and" #h(0.2em) n$] are whole numbers, then #math.equation(block: true, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n , m greater than n and the fraction a to the power m over a to the power n equals the fraction 1 over a to the power n minus m , n greater than m")[$frac(a^(m), a^(n)) = a^(m − n) , m > n #h(1em) "and" #h(1em) frac(a^(m), a^(n)) = frac(1, a^(n − m)) , n > m$] ] What if we just subtract exponents regardless of which is larger? Let’s consider #math.equation(block: false, alt: "the fraction x squared over x to the power 5")[$frac(x^(2), x^(5))$]. We subtract the exponent in the denominator from the exponent in the numerator. #math.equation(block: true, alt: "the fraction x squared over x to the power 5; x to the power 2 minus 5; x to the power −3")[$frac(x^(2), x^(5)) \ x^(2 − 5) \ x^(−3)$]We can also simplify #math.equation(block: false, alt: "the fraction x squared over x to the power 5")[$frac(x^(2), x^(5))$] by dividing out common factors: #figure(figph[Illustrated in this figure is x times x divided by x times x times x times x times x. Two xes cancel out in the numerator and denominator. Below this is the simplified term: 1 divided by x cubed.], alt: "Illustrated in this figure is x times x divided by x times x times x times x times x. Two xes cancel out in the numerator and denominator. Below this is the simplified term: 1 divided by x cubed.", caption: none) This implies that #math.equation(block: false, alt: "x to the power −3 equals the fraction 1 over x cubed")[$x^(−3) = frac(1, x^(3))$] and it leads us to the definition of a #emph[negative exponent]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Negative Exponent] If #math.equation(block: false, alt: "n")[$n$] is an integer and #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$], then #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$]. ] The #strong[negative exponent] tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a #strong[negative exponent] and other properties of exponents to write the expression with only positive exponents. For example, if after simplifying an expression we end up with the expression #math.equation(block: false, alt: "x to the power −3")[$x^(−3)$], we will take one more step and write #math.equation(block: false, alt: "the fraction 1 over x cubed")[$frac(1, x^(3))$]. The answer is considered to be in simplest form when it has only positive exponents. #examplebox("Example 1")[][ Simplify: ⓐ #math.equation(block: false, alt: "4 to the power −2")[$4^(−2)$] ⓑ #math.equation(block: false, alt: "10 to the power −3.")[$10^(−3) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], [#math.equation(block: false, alt: "4 to the power −2")[$4^(−2)$]]), [Use the definition of a negative exponent, #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$].], [#math.equation(block: false, alt: "the fraction 1 over 4 squared")[$frac(1, 4^(2))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 16")[$frac(1, 16)$]], [ⓑ], [#math.equation(block: false, alt: "10 to the power −3")[$10^(−3)$]], [Use the definition of a negative exponent, #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$].], [#math.equation(block: false, alt: "the fraction 1 over 10 cubed")[$frac(1, 10^(3))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 1000")[$frac(1, 1000)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "2 to the power −3")[$2^(−3)$] ⓑ #math.equation(block: false, alt: "10 to the power −7.")[$10^(−7) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 10 to the power 7")[$frac(1, 10^(7))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "3 to the power −2")[$3^(−2)$] ⓑ #math.equation(block: false, alt: "10 to the power −4.")[$10^(−4) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 9")[$frac(1, 9)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 10,000")[$frac(1, 10","000)$] ] ] In we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 1 over a to the power − n")[$frac(1, a^("−" n))$]]), [Use the definition of a negative exponent, #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$].], [#math.equation(block: false, alt: "the fraction 1 over the fraction 1 over a to the power n")[$frac(1, frac(1, a^(n)))$]], [Simplify the complex fraction.], [#math.equation(block: false, alt: "1 times the fraction a to the power n over 1")[$1 · frac(a^(n), 1)$]], [Multiply.], [#math.equation(block: false, alt: "a to the power n")[$a^(n)$]], )) This leads to the Property of Negative Exponents. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Property of Negative Exponents] If #math.equation(block: false, alt: "n")[$n$] is an integer and #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$], then #math.equation(block: false, alt: "the fraction 1 over a to the power − n equals a to the power n")[$frac(1, a^("−" n)) = a^(n)$]. ] #examplebox("Example 2")[][ Simplify: ⓐ #math.equation(block: false, alt: "the fraction 1 over y to the power −4")[$frac(1, y^(−4))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 3 to the power −2.")[$frac(1, 3^(−2)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], [#math.equation(block: false, alt: "the fraction 1 over y to the power −4")[$frac(1, y^(−4))$]]), [Use the property of a negative exponent, #math.equation(block: false, alt: "the fraction 1 over a to the power − n equals a to the power n")[$frac(1, a^("−" n)) = a^(n)$].], [#math.equation(block: false, alt: "y to the power 4")[$y^(4)$]], [ⓑ], [#math.equation(block: false, alt: "the fraction 1 over 3 to the power −2")[$frac(1, 3^(−2))$]], [Use the property of a negative exponent, #math.equation(block: false, alt: "the fraction 1 over a to the power − n equals a to the power n")[$frac(1, a^("−" n)) = a^(n)$].], [#math.equation(block: false, alt: "3 squared")[$3^(2)$]], [Simplify.], [#math.equation(block: false, alt: "9")[$9$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction 1 over p to the power −8")[$frac(1, p^(−8))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 4 to the power −3.")[$frac(1, 4^(−3)) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "p to the power 8")[$p^(8)$] ⓑ #math.equation(block: false, alt: "64")[$64$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction 1 over q to the power −7")[$frac(1, q^(−7))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 2 to the power −4.")[$frac(1, 2^(−4)) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "q to the power 7")[$q^(7)$] ⓑ #math.equation(block: false, alt: "16")[$16$] ] ] Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction 3 over 4 close parenthesis to the power −2")[$attach(( frac(3, 4) ), t: −2)$]]), [Use the definition of a negative exponent, #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$].], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis the fraction 3 over 4 close parenthesis squared")[$frac(1, attach(( frac(3, 4) ), t: 2))$]], [Simplify the denominator.], [#math.equation(block: false, alt: "the fraction 1 over the fraction 9 over 16")[$frac(1, frac(9, 16))$]], [Simplify the complex fraction.], [#math.equation(block: false, alt: "the fraction 16 over 9")[$frac(16, 9)$]], [But we know that #math.equation(block: false, alt: "the fraction 16 over 9")[$frac(16, 9)$] is #math.equation(block: false, alt: "open parenthesis the fraction 4 over 3 close parenthesis squared")[$attach(( frac(4, 3) ), t: 2)$].], [], [This tells us that:], [#math.equation(block: false, alt: "open parenthesis the fraction 3 over 4 close parenthesis to the power −2 equals open parenthesis the fraction 4 over 3 close parenthesis squared")[$attach(( frac(3, 4) ), t: −2) = attach(( frac(4, 3) ), t: 2)$]], )) To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent. This leads us to the #emph[Quotient to a Negative Exponent Property]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Quotient to a Negative Exponent Property] If #math.equation(block: false, alt: "a and b")[$a #h(0.2em) "and" #h(0.2em) b$] are real numbers, #math.equation(block: false, alt: "a not equal to 0 , b not equal to 0 ,")[$a ≠ 0 , b ≠ 0 ,$] and #math.equation(block: false, alt: "n")[$n$] is an integer, then #math.equation(block: false, alt: "open parenthesis the fraction a over b close parenthesis to the power − n equals open parenthesis the fraction b over a close parenthesis to the power n")[$attach(( frac(a, b) ), t: "−" n) = attach(( frac(b, a) ), t: n)$]. ] #examplebox("Example 3")[][ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 5 over 7 close parenthesis to the power −2")[$attach(( frac(5, 7) ), t: −2)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 2 x over y close parenthesis to the power −3.")[$attach(( − frac(2 x, y) ), t: −3) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], [#math.equation(block: false, alt: "open parenthesis the fraction 5 over 7 close parenthesis to the power −2")[$attach(( frac(5, 7) ), t: −2)$]]), [Use the Quotient to a Negative Exponent Property, #math.equation(block: false, alt: "open parenthesis the fraction a over b close parenthesis to the power − n equals open parenthesis the fraction b over a close parenthesis to the power n")[$attach(( frac(a, b) ), t: "−" n) = attach(( frac(b, a) ), t: n)$].], [], [Take the reciprocal of the fraction and change the sign of the exponent.], [#math.equation(block: false, alt: "open parenthesis the fraction 7 over 5 close parenthesis squared")[$attach(( frac(7, 5) ), t: 2)$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 49 over 25")[$frac(49, 25)$]], [ⓑ], [#math.equation(block: false, alt: "open parenthesis minus the fraction 2 x over y close parenthesis to the power −3")[$attach(( − frac(2 x, y) ), t: −3)$]], [Use the Quotient to a Negative Exponent Property, #math.equation(block: false, alt: "open parenthesis the fraction a over b close parenthesis to the power − n equals open parenthesis the fraction b over a close parenthesis to the power n")[$attach(( frac(a, b) ), t: "−" n) = attach(( frac(b, a) ), t: n)$].], [], [Take the reciprocal of the fraction and change the sign of the exponent.], [#math.equation(block: false, alt: "open parenthesis minus the fraction y over 2 x close parenthesis cubed")[$attach(( − frac(y, 2 x) ), t: 3)$]], [Simplify.], [#math.equation(block: false, alt: "minus the fraction y cubed over 8 x cubed")[$− frac(y^(3), 8 x^(3))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 2 over 3 close parenthesis to the power −4")[$attach(( frac(2, 3) ), t: −4)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 6 m over n close parenthesis to the power −2.")[$attach(( − frac(6 m, n) ), t: −2) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 81 over 16")[$frac(81, 16)$] ⓑ #math.equation(block: false, alt: "the fraction n squared over 36 m squared")[$frac(n^(2), 36 m^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 3 over 5 close parenthesis to the power −3")[$attach(( frac(3, 5) ), t: −3)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction a over 2 b close parenthesis to the power −4.")[$attach(( − frac(a, 2 b) ), t: −4) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 125 over 27")[$frac(125, 27)$] ⓑ #math.equation(block: false, alt: "the fraction 16 b to the power 4 over a to the power 4")[$frac(16 b^(4), a^(4))$] ] ] When simplifying an expression with exponents, we must be careful to correctly identify the base. #examplebox("Example 4")[][ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power −2")[$attach(( −3 ), t: −2)$] ⓑ #math.equation(block: false, alt: "− 3 to the power −2")[$"−" 3^(−2)$] ⓒ #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 3 close parenthesis to the power −2")[$attach(( − frac(1, 3) ), t: −2)$] ⓓ #math.equation(block: false, alt: "− open parenthesis the fraction 1 over 3 close parenthesis to the power −2.")[$"−" attach(( frac(1, 3) ), t: −2) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ Here the exponent applies to the base #math.equation(block: false, alt: "−3")[$−3$].], [#math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power −2")[$attach(( −3 ), t: −2)$]]), [Take the reciprocal of the base and change the sign of the exponent.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis −3 close parenthesis squared")[$frac(1, attach(( −3 ), t: 2))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 9")[$frac(1, 9)$]], [ⓑ The expression #math.equation(block: false, alt: "− 3 to the power −2")[$"−" 3^(−2)$] means "find the opposite of #math.equation(block: false, alt: "3 to the power −2")[$3^(−2)$]." Here the exponent applies to the base #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 3 close parenthesis")[$attach(( − frac(1, 3) ))$].], [#math.equation(block: false, alt: "− 3 to the power −2")[$"−" 3^(−2)$]], [Rewrite as a product with #math.equation(block: false, alt: "-1")[$-1$].], [#math.equation(block: false, alt: "−1 times 3 to the power −2")[$−1 · 3^(−2)$]], [Take the reciprocal of the base and change the sign of the exponent.], [#math.equation(block: false, alt: "−1 times the fraction 1 over 3 squared")[$−1 · frac(1, 3^(2))$]], [Simplify.], [#math.equation(block: false, alt: "minus the fraction 1 over 9")[$− frac(1, 9)$]], [ⓒ Here the exponent applies to the base #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 3 close parenthesis")[$attach(( − frac(1, 3) ))$].], [#math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 3 close parenthesis to the power −2")[$attach(( − frac(1, 3) ), t: −2)$]], [Take the reciprocal of the base and change the sign of the exponent.], [#math.equation(block: false, alt: "open parenthesis minus the fraction 3 over 1 close parenthesis squared")[$attach(( − frac(3, 1) ), t: 2)$]], [Simplify.], [#math.equation(block: false, alt: "9")[$9$]], [ⓓ The expression #math.equation(block: false, alt: "− open parenthesis the fraction 1 over 3 close parenthesis to the power −2")[$"−" attach(( frac(1, 3) ), t: −2)$] means "find the opposite of #math.equation(block: false, alt: "open parenthesis the fraction 1 over 3 close parenthesis to the power −2")[$attach(( frac(1, 3) ), t: −2)$]." Here the exponent applies to the base #math.equation(block: false, alt: "open parenthesis the fraction 1 over 3 close parenthesis")[$( frac(1, 3) )$].], [$$], [Rewrite as a product with #math.equation(block: false, alt: "-1")[$-1$].], [#math.equation(block: false, alt: "−1 times open parenthesis the fraction 1 over 3 close parenthesis to the power −2")[$−1 · attach(( frac(1, 3) ), t: −2)$]], [Take the reciprocal of the base and change the sign of the exponent.], [#math.equation(block: false, alt: "−1 times open parenthesis the fraction 3 over 1 close parenthesis squared")[$−1 · attach(( frac(3, 1) ), t: 2)$]], [Simplify.], [#math.equation(block: false, alt: "−9")[$−9$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −5 close parenthesis to the power −2")[$attach(( −5 ), t: −2)$] ⓑ #math.equation(block: false, alt: "− 5 to the power −2")[$"−" 5^(−2)$] ⓒ #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 5 close parenthesis to the power −2")[$attach(( − frac(1, 5) ), t: −2)$] ⓓ #math.equation(block: false, alt: "− open parenthesis the fraction 1 over 5 close parenthesis to the power −2.")[$"−" attach(( frac(1, 5) ), t: −2) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 25")[$frac(1, 25)$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 25")[$− frac(1, 25)$] ⓒ 25 ⓓ #math.equation(block: false, alt: "−25")[$−25$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −7 close parenthesis to the power −2")[$attach(( −7 ), t: −2)$] ⓑ #math.equation(block: false, alt: "− 7 to the power −2")[$"−" 7^(−2)$], ⓒ #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 7 close parenthesis to the power −2")[$attach(( − frac(1, 7) ), t: −2)$] ⓓ #math.equation(block: false, alt: "− open parenthesis the fraction 1 over 7 close parenthesis to the power −2.")[$"−" attach(( frac(1, 7) ), t: −2) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 49")[$frac(1, 49)$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 49")[$− frac(1, 49)$] ⓒ 49 ⓓ #math.equation(block: false, alt: "−49")[$−49$] ] ] We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different. #examplebox("Example 5")[][ Simplify: ⓐ #math.equation(block: false, alt: "4 times 2 to the power −1")[$4 · 2^(−1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 times 2 close parenthesis to the power −1.")[$attach(( 4 · 2 ), t: −1) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ #linebreak() Do exponents before multiplication.], [#math.equation(block: false, alt: "4 times 2 to the power −1")[$4 · 2^(−1)$]]), [Use #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$].], [#math.equation(block: false, alt: "4 times the fraction 1 over 2 to the power 1")[$4 · frac(1, 2^(1))$]], [Simplify.], [#math.equation(block: false, alt: "2")[$2$]], [ⓑ], [#math.equation(block: false, alt: "open parenthesis 4 times 2 close parenthesis to the power −1")[$attach(( 4 · 2 ), t: −1)$]], [Simplify inside the parentheses first.], [#math.equation(block: false, alt: "open parenthesis 8 close parenthesis to the power −1")[$attach(( 8 ), t: −1)$]], [Use #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$].], [#math.equation(block: false, alt: "the fraction 1 over 8 to the power 1")[$frac(1, 8^(1))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "6 times 3 to the power −1")[$6 · 3^(−1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 6 times 3 close parenthesis to the power −1.")[$attach(( 6 · 3 ), t: −1) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "2")[$2$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 18")[$frac(1, 18)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "8 times 2 to the power −2")[$8 · 2^(−2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 times 2 close parenthesis to the power −2.")[$attach(( 8 · 2 ), t: −2) .$] #solutionbox[ ⓐ 2 ⓑ #math.equation(block: false, alt: "the fraction 1 over 256")[$frac(1, 256)$] ] ] When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero. #examplebox("Example 6")[][ Simplify: ⓐ #math.equation(block: false, alt: "x to the power −6")[$x^(−6)$] ⓑ #math.equation(block: false, alt: "open parenthesis u to the power 4 close parenthesis to the power −3.")[$attach(( u^(4) ), t: −3) .$] #solutionbox[ + ⓐ #linebreak() #math.equation(block: false, alt: "x to the power −6; Use the definition of a negative exponent, a to the power − n equals the fraction 1 over a to the power n ., the fraction 1 over x to the power 6")[$& & & #h(4em) x^(−6) \ "Use the definition of a negative exponent," #h(0.2em) a^("−" n) = frac(1, a^(n)) . & & & #h(4em) frac(1, x^(6))$] #linebreak() #linebreak() + ⓑ #linebreak() #math.equation(block: false, alt: "open parenthesis u to the power 4 close parenthesis to the power −3; Use the definition of a negative exponent, a to the power − n equals the fraction 1 over a to the power n ., the fraction 1 over open parenthesis u to the power 4 close parenthesis cubed; Simplify., the fraction 1 over u to the power 12")[$& & & #h(4em) attach(( u^(4) ), t: −3) \ "Use the definition of a negative exponent," #h(0.2em) a^("−" n) = frac(1, a^(n)) . & & & #h(4em) frac(1, attach(( u^(4) ), t: 3)) \ "Simplify." & & & #h(4em) frac(1, u^(12))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "y to the power −7")[$y^(−7)$] ⓑ #math.equation(block: false, alt: "open parenthesis z cubed close parenthesis to the power −5.")[$attach(( z^(3) ), t: −5) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over y to the power 7")[$frac(1, y^(7))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over z to the power 15")[$frac(1, z^(15))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "p to the power −9")[$p^(−9)$] ⓑ #math.equation(block: false, alt: "open parenthesis q to the power 4 close parenthesis to the power −6.")[$attach(( q^(4) ), t: −6) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over p to the power 9")[$frac(1, p^(9))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over q to the power 24")[$frac(1, q^(24))$] ] ] When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example. #examplebox("Example 7")[][ Simplify: ⓐ #math.equation(block: false, alt: "5 y to the power −1")[$5 y^(−1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 5 y close parenthesis to the power −1")[$attach(( 5 y ), t: −1)$] ⓒ #math.equation(block: false, alt: "open parenthesis −5 y close parenthesis to the power −1.")[$attach(( −5 y ), t: −1) .$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "5 y to the power −1")[$5 y^(−1)$]]), [Notice the exponent applies to just the base #emph[y]. #linebreak() Take the reciprocal of #emph[y] and change the sign of the exponent.], [#math.equation(block: false, alt: "5 times the fraction 1 over y to the power 1")[$5 · frac(1, y^(1))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 5 over y")[$frac(5, y)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis 5 y close parenthesis to the power −1")[$attach(( 5 y ), t: −1)$]]), [Here the parentheses make the exponent apply to the base 5#emph[y]. #linebreak() Take the reciprocal of 5#emph[y] and change the sign of the exponent.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis 5 y close parenthesis to the power 1")[$frac(1, attach(( 5 y ), t: 1))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 5 y")[$frac(1, 5 y)$]], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis −5 y close parenthesis to the power −1")[$attach(( −5 y ), t: −1)$]]), [The base here is −5#emph[y]. #linebreak() Take the reciprocal of −5#emph[y] and change the sign of the exponent.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis −5 y close parenthesis to the power 1")[$frac(1, attach(( −5 y ), t: 1))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over −5 y")[$frac(1, −5 y)$]], [Use #math.equation(block: false, alt: "the fraction a over − b equals minus the fraction a over b .")[$frac(a, "−" b) = − frac(a, b) .$]], [#math.equation(block: false, alt: "minus the fraction 1 over 5 y")[$− frac(1, 5 y)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "8 p to the power −1")[$8 p^(−1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 p close parenthesis to the power −1")[$attach(( 8 p ), t: −1)$] ⓒ #math.equation(block: false, alt: "open parenthesis −8 p close parenthesis to the power −1.")[$attach(( −8 p ), t: −1) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 8 over p")[$frac(8, p)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 8 p")[$frac(1, 8 p)$] ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 8 p")[$− frac(1, 8 p)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "11 q to the power −1")[$attach(11 q, t: −1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 11 q close parenthesis to the power −1")[$attach(( 11 q ), t: −1)$]#math.equation(block: false, alt: "− open parenthesis 11 q close parenthesis to the power −1")[$"−" attach(( 11 q ), t: −1)$] ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 11 q")[$− frac(1, 11 q)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 11 q")[$frac(1, 11 q)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 11 q")[$frac(1, 11 q)$]#math.equation(block: false, alt: "minus the fraction 1 over 11 q")[$− frac(1, 11 q)$] ⓒ #math.equation(block: false, alt: "minus the fraction 11 over q")[$− frac(11, q)$] ] ] With negative exponents, the Quotient Rule needs only one form #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n")[$frac(a^(m), a^(n)) = a^(m − n)$], for #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$]. When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. === Simplify Expressions with Integer Exponents All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Summary of Exponent Properties] If #math.equation(block: false, alt: "a and b")[$a #h(0.2em) "and" #h(0.2em) b$] are real numbers, and #math.equation(block: false, alt: "m and n")[$m #h(0.2em) "and" #h(0.2em) n$] are integers, then #math.equation(block: true, alt: "Product Property, a to the power m times a to the power n, equals, a to the power m plus n; Power Property, open parenthesis a to the power m close parenthesis to the power n, equals, a to the power m times n; Product to a Power, open parenthesis a b close parenthesis to the power m, equals, a to the power m b to the power m; Quotient Property, the fraction a to the power m over a to the power n, equals, a to the power m minus n , a not equal to 0; Zero Exponent Property, a to the power 0, equals, 1 , a not equal to 0; Quotient to a Power Property, open parenthesis the fraction a over b close parenthesis to the power m, equals, the fraction a to the power m over b to the power m , b not equal to 0; Properties of Negative Exponents, a to the power − n, equals, the fraction 1 over a to the p")[$"Product Property" & & & a^(m) · a^(n) & = & a^(m + n) \ "Power Property" & & & attach(( a^(m) ), t: n) & = & a^(m · n) \ "Product to a Power" & & & attach(( a b ), t: m) & = & a^(m) b^(m) \ "Quotient Property" & & & frac(a^(m), a^(n)) & = & a^(m − n) , a ≠ 0 \ "Zero Exponent Property" & & & a^(0) & = & 1 , a ≠ 0 \ "Quotient to a Power Property" & & & attach(( frac(a, b) ), t: m) & = & frac(a^(m), b^(m)) , #h(0.2em) #h(0.2em) b ≠ 0 \ "Properties of Negative Exponents" & & & a^("−" n) & = & frac(1, a^(n)) #h(0.5em) "and" #h(0.5em) frac(1, a^("−" n)) = a^(n) \ "Quotient to a Negative Exponent" & & & attach(( frac(a, b) ), t: "−" n) & = & attach(( frac(b, a) ), t: n)$] ] #examplebox("Example 8")[][ Simplify: ⓐ #math.equation(block: false, alt: "x to the power −4 times x to the power 6")[$x^(−4) · x^(6)$] ⓑ #math.equation(block: false, alt: "y to the power −6 times y to the power 4")[$y^(−6) · y^(4)$] ⓒ #math.equation(block: false, alt: "z to the power −5 times z to the power −3.")[$z^(−5) · z^(−3) .$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "x to the power −4 times x to the power 6")[$x^(−4) · x^(6)$]]), [Use the Product Property, #math.equation(block: false, alt: "a to the power m times a to the power n equals a to the power m plus n .")[$a^(m) · a^(n) = a^(m + n) .$]], [#math.equation(block: false, alt: "x to the power −4 plus 6")[$x^(−4 + 6)$]], [Simplify.], [#math.equation(block: false, alt: "x squared")[$x^(2)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "y to the power −6 times y to the power 4")[$y^(−6) · y^(4)$]]), [Notice the same bases, so add the exponents.], [#math.equation(block: false, alt: "y to the power −6 plus 4")[$y^(−6 + 4)$]], [Simplify.], [#math.equation(block: false, alt: "y to the power −2")[$y^(−2)$]], [Use the definition of a negative exponent, #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n .")[$a^("−" n) = frac(1, a^(n)) .$]], [#math.equation(block: false, alt: "the fraction 1 over y squared")[$frac(1, y^(2))$]], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "z to the power −5 times z to the power −3")[$z^(−5) · z^(−3)$]]), [Add the exponents, since the bases are the same.], [#math.equation(block: false, alt: "z to the power −5 minus 3")[$z^(−5 − 3)$]], [Simplify.], [#math.equation(block: false, alt: "z to the power −8")[$z^(−8)$]], [Take the reciprocal and change the sign of the exponent, #linebreak() using the definition of a negative exponent.], [#math.equation(block: false, alt: "the fraction 1 over z to the power 8")[$frac(1, z^(8))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "x to the power −3 times x to the power 7")[$x^(−3) · x^(7)$] ⓑ #math.equation(block: false, alt: "y to the power −7 times y squared")[$y^(−7) · y^(2)$] ⓒ #math.equation(block: false, alt: "z to the power −4 times z to the power −5.")[$z^(−4) · z^(−5) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x to the power 4")[$x^(4)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over y to the power 5")[$frac(1, y^(5))$] ⓒ #math.equation(block: false, alt: "the fraction 1 over z to the power 9")[$frac(1, z^(9))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "a to the power −1 times a to the power 6")[$a^(−1) · a^(6)$] ⓑ #math.equation(block: false, alt: "b to the power −8 times b to the power 4")[$b^(−8) · b^(4)$] ⓒ #math.equation(block: false, alt: "c to the power −8 times c to the power −7.")[$c^(−8) · c^(−7) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "a to the power 5")[$a^(5)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over b to the power 4")[$frac(1, b^(4))$] ⓒ #math.equation(block: false, alt: "the fraction 1 over c to the power 15")[$frac(1, c^(15))$] ] ] In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property. #examplebox("Example 9")[][ Simplify: #math.equation(block: false, alt: "open parenthesis m to the power 4 n to the power −3 close parenthesis open parenthesis m to the power −5 n to the power −2 close parenthesis .")[$( m^(4) n^(−3) ) ( m^(−5) n^(−2) ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis m to the power 4 n to the power −3 close parenthesis open parenthesis m to the power −5 n to the power −2 close parenthesis")[$( m^(4) n^(−3) ) ( m^(−5) n^(−2) )$]]), [Use the Commutative Property to get like bases together.], [#math.equation(block: false, alt: "m to the power 4 m to the power −5 times n to the power −2 n to the power −3")[$m^(4) m^(−5) · n^(−2) n^(−3)$]], [Add the exponents for each base.], [#math.equation(block: false, alt: "m to the power −1 times n to the power −5")[$m^(−1) · n^(−5)$]], [Take reciprocals and change the signs of the exponents.], [#math.equation(block: false, alt: "the fraction 1 over m to the power 1 times the fraction 1 over n to the power 5")[$frac(1, m^(1)) · frac(1, n^(5))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over m n to the power 5")[$frac(1, m n^(5))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis p to the power 6 q to the power −2 close parenthesis open parenthesis p to the power −9 q to the power −1 close parenthesis .")[$( p^(6) q^(−2) ) ( p^(−9) q^(−1) ) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over p cubed q cubed")[$frac(1, p^(3) q^(3))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis r to the power 5 s to the power −3 close parenthesis open parenthesis r to the power −7 s to the power −5 close parenthesis .")[$( r^(5) s^(−3) ) ( r^(−7) s^(−5) ) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over r squared s to the power 8")[$frac(1, r^(2) s^(8))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis −6 c to the power −6 d to the power 4 close parenthesis open parenthesis −5 c to the power −2 d to the power −1 close parenthesis .")[$( −6 c^(−6) d^(4) ) ( −5 c^(−2) d^(−1) ) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 30 d cubed over c to the power 8")[$frac(30 d^(3), c^(8))$] ] ] In the next two examples, we’ll use the Power Property and the Product to a Power Property. #examplebox("Example 10")[][ Simplify: #math.equation(block: false, alt: "open parenthesis 6 k cubed close parenthesis to the power −2.")[$attach(( 6 k^(3) ), t: −2) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis 6 k cubed close parenthesis to the power −2")[$attach(( 6 k^(3) ), t: −2)$]]), [Use the Product to a Power Property, #math.equation(block: false, alt: "open parenthesis a b close parenthesis to the power m equals a to the power m b to the power m .")[$attach(( a b ), t: m) = a^(m) b^(m) .$]], [#math.equation(block: false, alt: "open parenthesis 6 close parenthesis to the power −2 open parenthesis k cubed close parenthesis to the power −2")[$attach(( 6 ), t: −2) attach(( k^(3) ), t: −2)$]], [Use the Power Property, #math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n .")[$attach(( a^(m) ), t: n) = a^(m · n) .$]], [#math.equation(block: false, alt: "6 to the power −2 k to the power −6")[$6^(−2) k^(−6)$]], [Use the Definition of a Negative Exponent, #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n .")[$a^("−" n) = frac(1, a^(n)) .$]], [#math.equation(block: false, alt: "the fraction 1 over 6 squared times the fraction 1 over k to the power 6")[$frac(1, 6^(2)) · frac(1, k^(6))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 36 k to the power 6")[$frac(1, 36 k^(6))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis −4 x to the power 4 close parenthesis to the power −2.")[$attach(( −4 x^(4) ), t: −2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 16 x to the power 8")[$frac(1, 16 x^(8))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis 2 b cubed close parenthesis to the power −4.")[$attach(( 2 b^(3) ), t: −4) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 16 b to the power 12")[$frac(1, 16 b^(12))$] ] ] #examplebox("Example 11")[][ Simplify: #math.equation(block: false, alt: "open parenthesis 5 x to the power −3 close parenthesis squared .")[$attach(( 5 x^(−3) ), t: 2) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis 5 x to the power −3 close parenthesis squared")[$attach(( 5 x^(−3) ), t: 2)$]]), [Use the Product to a Power Property, #math.equation(block: false, alt: "open parenthesis a b close parenthesis to the power m equals a to the power m b to the power m .")[$attach(( a b ), t: m) = a^(m) b^(m) .$]], [#math.equation(block: false, alt: "5 squared open parenthesis x to the power −3 close parenthesis squared")[$5^(2) attach(( x^(−3) ), t: 2)$]], [Simplify 5#super[2] and multiply the exponents of #emph[x] using the Power #linebreak() Property, #math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n .")[$attach(( a^(m) ), t: n) = a^(m · n) .$]], [#math.equation(block: false, alt: "25 times x to the power −6")[$25 · x^(−6)$]], [Rewrite #emph[x]#super[−6] by using the Definition of a Negative Exponent, #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n .")[$a^("−" n) = frac(1, a^(n)) .$]], [#math.equation(block: false, alt: "25 times the fraction 1 over x to the power 6")[$25 · frac(1, x^(6))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 25 over x to the power 6")[$frac(25, x^(6))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis 8 a to the power −4 close parenthesis squared .")[$attach(( 8 a^(−4) ), t: 2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 64 over a to the power 8")[$frac(64, a^(8))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis 2 c to the power −4 close parenthesis cubed .")[$attach(( 2 c^(−4) ), t: 3) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 8 over c to the power 12")[$frac(8, c^(12))$] ] ] To simplify a fraction, we use the Quotient Property and subtract the exponents. #examplebox("Example 12")[][ Simplify: #math.equation(block: false, alt: "the fraction r to the power 5 over r to the power −4.")[$frac(r^(5), r^(−4)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction r to the power 5 over r to the power −4")[$frac(r^(5), r^(−4))$]]), [Use the Quotient Property, #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n .")[$frac(a^(m), a^(n)) = a^(m − n) .$]], [#math.equation(block: false, alt: "r to the power 5 minus open parenthesis −4 close parenthesis")[$r^(5 − ( −4 ))$]], [Simplify.], [#math.equation(block: false, alt: "r to the power 9")[$r^(9)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x to the power 8 over x to the power −3.")[$frac(x^(8), x^(−3)) .$] #solutionbox[ #math.equation(block: true, alt: "x to the power 11")[$x^(11)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction y to the power 8 over y to the power −6.")[$frac(y^(8), y^(−6)) .$] #solutionbox[ #math.equation(block: true, alt: "y to the power 14")[$y^(14)$] ] ] === Convert from Decimal Notation to Scientific Notation Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and #math.equation(block: false, alt: "0.004")[$0.004$]. We know that 4,000 means #math.equation(block: false, alt: "4 times 1,000")[$4 #h(0.2em) × #h(0.2em) 1,000$] and 0.004 means #math.equation(block: false, alt: "4 times the fraction 1 over 1,000")[$4 #h(0.2em) × #h(0.2em) frac(1, 1","000)$]. If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way: #math.equation(block: true, alt: "4,000, 0.004; 4 times 1,000, 4 times the fraction 1 over 1,000; 4 times 10 cubed, 4 times the fraction 1 over 10 cubed; 4 times 10 to the power −3")[$4,000 & & & #h(4em) 0.004 \ 4 #h(0.2em) × #h(0.2em) 1,000 & & & #h(4em) 4 #h(0.2em) × #h(0.2em) frac(1, 1","000) \ 4 #h(0.2em) × #h(0.2em) 10^(3) & & & #h(4em) 4 #h(0.2em) × #h(0.2em) frac(1, 10^(3)) \ & & & #h(4em) 4 #h(0.2em) × #h(0.2em) 10^(−3)$]When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in #emph[scientific notation.] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Scientific Notation] A number is expressed in #strong[scientific notation] when it is of the form #math.equation(block: true, alt: "a times 10 to the power n where 1 less than or equal to │ a │ less than 10 and n is an integer")[$& & & a #h(0.2em) × #h(0.2em) 10^(n) #h(0.2em) "where" #h(0.2em) 1 ≤ │ a │ < 10 #h(0.2em) "and" #h(0.2em) n #h(0.2em) "is an integer"$] ] It is customary in scientific notation to use #math.equation(block: false, alt: "times")[$#h(0.2em) × #h(0.2em)$] as the multiplication sign, even though we avoid using this sign elsewhere in algebra. If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to #strong[scientific notation]. #figure(figph[This figure illustrates how to convert a number to scientific notation. It has two columns. In the first column is 4000 equals 4 times 10 to the third power. Below this, the equation is repeated, with an arrow demonstrating that the decimal point at the end of 4000 has moved three places to the left, so that 4000 becomes 4.000. The second column has 0.004 equals 4 times 10 to the negative third power. Below this, the equation is repeated, with an arrow demonstrating how the decimal point in 0.004 is moved three places to the right to produce 4.], alt: "This figure illustrates how to convert a number to scientific notation. It has two columns. In the first column is 4000 equals 4 times 10 to the third power. Below this, the equation is repeated, with an arrow demonstrating that the decimal point at the end of 4000 has moved three places to the left, so that 4000 becomes 4.000. The second column has 0.004 equals 4 times 10 to the negative third power. Below this, the equation is repeated, with an arrow demonstrating how the decimal point in 0.004 is moved three places to the right to produce 4.", caption: none) In both cases, the decimal was moved 3 places to get the first factor between 1 and 10. #math.equation(block: true, alt: "The power of 10 is positive when the number is larger than 1:, 4,000 equals 4 times 10 cubed; The power of 10 is negative when the number is between 0 and 1:, 0.004 equals 4 times 10 to the power −3")[$"The power of 10 is positive when the number is larger than 1:" & & & 4,000 = 4 #h(0.2em) × #h(0.2em) 10^(3) \ "The power of 10 is negative when the number is between 0 and 1:" & & & 0.004 = 4 #h(0.2em) × #h(0.2em) 10^(−3)$] #examplebox("Example 13")[How to Convert from Decimal Notation to Scientific Notation][ Write in scientific notation: 37,000. #solutionbox[ #figure(figph[This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.” The second cell reads “Remember, there is a decimal at the end of 37,000.” The third cell contains 37,000. One line down, the second cell reads “Move the decimal after the 3. 3.7000 is between 1 and 10.”], alt: "This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.” The second cell reads “Remember, there is a decimal at the end of 37,000.” The third cell contains 37,000. One line down, the second cell reads “Move the decimal after the 3. 3.7000 is between 1 and 10.”", caption: none) #figure(figph[In the second row, the first cell reads “Step 2. Count the number of decimal places, n, that the decimal place was moved. The second cell reads “The decimal point was moved 4 places to the left.” The third cell contains 370000 again, with an arrow showing the decimal point jumping places to the left from the end of the number until it ends up between the 3 and the 7.], alt: "In the second row, the first cell reads “Step 2. Count the number of decimal places, n, that the decimal place was moved. The second cell reads “The decimal point was moved 4 places to the left.” The third cell contains 370000 again, with an arrow showing the decimal point jumping places to the left from the end of the number until it ends up between the 3 and the 7.", caption: none) #figure(figph[In the third row, the first cell reads “Step 3. Write the number as a product with a power of 10. If the original number is greater than 1, the power of 10 will be 10 to the n power. If it’s between 0 and 1, the power of 10 will be 10 to the negative n power.” The second cell reads “37,000 is greater than 1, so the power of 10 will have exponent 4.” The third cell contains 3.7 times 10 to the fourth power.], alt: "In the third row, the first cell reads “Step 3. Write the number as a product with a power of 10. If the original number is greater than 1, the power of 10 will be 10 to the n power. If it’s between 0 and 1, the power of 10 will be 10 to the negative n power.” The second cell reads “37,000 is greater than 1, so the power of 10 will have exponent 4.” The third cell contains 3.7 times 10 to the fourth power.", caption: none) #figure(figph[In the fourth row, the first cell reads “Step 4. Check.” The second cell reads “Check to see if your answer makes sense.” The third cell reads “10 to the fourth power is 10,000 and 10,000 times 3.7 will be 37,000.” Below this is 37,000 equals 3.7 times 10 to the fourth power.], alt: "In the fourth row, the first cell reads “Step 4. Check.” The second cell reads “Check to see if your answer makes sense.” The third cell reads “10 to the fourth power is 10,000 and 10,000 times 3.7 will be 37,000.” Below this is 37,000 equals 3.7 times 10 to the fourth power.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in scientific notation: #math.equation(block: false, alt: "96,000.")[$96,000 .$] #solutionbox[ #math.equation(block: true, alt: "9.6 times 10 to the power 4")[$9.6 #h(0.2em) × #h(0.2em) 10^(4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in scientific notation: #math.equation(block: false, alt: "48,300.")[$48,300 .$] #solutionbox[ #math.equation(block: true, alt: "4.83 times 10 to the power 4")[$4.83 #h(0.2em) × #h(0.2em) 10^(4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Convert from decimal notation to scientific notation] + Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. + Count the number of decimal places, #emph[n], that the decimal point was moved. + Write the number as a product with a power of 10. #linebreak() If the original number is: - greater than 1, the power of 10 will be 10#super[#emph[n]]. - between 0 and 1, the power of 10 will be 10#super[#emph[−n]]. + Check. ] #examplebox("Example 14")[][ Write in scientific notation: #math.equation(block: false, alt: "0.0052.")[$0.0052 .$] #solutionbox[ The original number, #math.equation(block: false, alt: "0.0052")[$0.0052$], is between 0 and 1 so we will have a negative power of 10. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[The image displays the numerical value 0.0052 in a simple, clear font against a white background.], alt: "The image displays the numerical value 0.0052 in a simple, clear font against a white background.", caption: none)]), [Move the decimal point to get 5.2, a number between 1 and 10.], [#figure(figph[0.0052, with an arrow showing the decimal point jumping three places to the right until it ends up between the 5 and 2.], alt: "0.0052, with an arrow showing the decimal point jumping three places to the right until it ends up between the 5 and 2.", caption: none)], [Count the number of decimal places the point was moved.], [#figure(figph[The text '3 places' is displayed in a simple, gray font against a plain white background.], alt: "The text '3 places' is displayed in a simple, gray font against a plain white background.", caption: none)], [Write as a product with a power of 10.], [#figure(figph[5.2 times 10 to the power of negative 3.], alt: "5.2 times 10 to the power of negative 3.", caption: none)], [Check.], [], [#math.equation(block: false, alt: "5.2 times 10 to the power −3; 5.2 times the fraction 1 over 10 cubed; 5.2 times the fraction 1 over 1000; 5.2 times 0.001")[$#h(3em) 5.2 #h(0.2em) × #h(0.2em) 10^(−3) \ #h(3em) 5.2 #h(0.2em) × #h(0.2em) frac(1, 10^(3)) \ #h(3em) 5.2 #h(0.2em) × #h(0.2em) frac(1, 1000) \ #h(3em) 5.2 #h(0.2em) × #h(0.2em) 0.001$]], [], [#math.equation(block: false, alt: "0.0052")[$#h(2em) 0.0052$]], [#figure(figph[0.0052 equals 5.2 times 10 to the power of negative 3.], alt: "0.0052 equals 5.2 times 10 to the power of negative 3.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in scientific notation: #math.equation(block: false, alt: "0.0078.")[$0.0078 .$] #solutionbox[ #math.equation(block: true, alt: "7.8 times 10 to the power −3")[$7.8 #h(0.2em) × #h(0.2em) 10^(−3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in scientific notation: #math.equation(block: false, alt: "0.0129.")[$0.0129 .$] #solutionbox[ #math.equation(block: true, alt: "1.29 times 10 to the power −2")[$1.29 #h(0.2em) × #h(0.2em) 10^(−2)$] ] ] === Convert Scientific Notation to Decimal Form How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see. #math.equation(block: true, alt: "9.12 times 10 to the power 4, 9.12 times 10 to the power −4; 9.12 times 10,000, 9.12 times 0.0001; 91,200, 0.000912")[$9.12 #h(0.2em) × #h(0.2em) 10^(4) & & & #h(4em) 9.12 #h(0.2em) × #h(0.2em) 10^(−4) \ 9.12 #h(0.2em) × #h(0.2em) 10,000 & & & #h(4em) 9.12 #h(0.2em) × #h(0.2em) 0.0001 \ 91,200 & & & #h(4em) 0.000912$]If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form. #math.equation(block: true, alt: "9.12 times 10 to the power 4 equals 91,200, 9.12 times 10 to the power −4 equals 0.000912")[$9.12 #h(0.2em) × #h(0.2em) 10^(4) = 91,200 & & & #h(4em) 9.12 #h(0.2em) × #h(0.2em) 10^(−4) = 0.000912$]#figure(figph[This figure has two columns. In the left column is 9.12 times 10 to the fourth power equals 91,200. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the right. Because there are no digits after 2, the final two places are represented by blank spaces. Below this is the text “Move the decimal point four places to the right.” In the right column is 9.12 times 10 to the negative fourth power equals 0.000912. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the left. Because there are no digits before 9, the remaining three places are represented by spaces. Below this is the text “Move the decimal point 4 places to the left.”], alt: "This figure has two columns. In the left column is 9.12 times 10 to the fourth power equals 91,200. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the right. Because there are no digits after 2, the final two places are represented by blank spaces. Below this is the text “Move the decimal point four places to the right.” In the right column is 9.12 times 10 to the negative fourth power equals 0.000912. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the left. Because there are no digits before 9, the remaining three places are represented by spaces. Below this is the text “Move the decimal point 4 places to the left.”", caption: none) In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left. #examplebox("Example 15")[How to Convert Scientific Notation to Decimal Form][ Convert to decimal form: #math.equation(block: false, alt: "6.2 times 10 cubed .")[$6.2 #h(0.2em) × #h(0.2em) 10^(3) .$] #solutionbox[ #figure(figph[This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Determine the exponent, n, on the factor 10.” The second cell reads “The exponent is 3.” The third cell contains 6.2 times 10 cubed.], alt: "This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Determine the exponent, n, on the factor 10.” The second cell reads “The exponent is 3.” The third cell contains 6.2 times 10 cubed.", caption: none) #figure(figph[In the second row, the first cell reads “Step 2. Move the decimal n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point absolute value of n places to the left.” The second cell reads “The exponent is positive so move the decimal point 3 places to the right. We need to add two zeros as placeholders.” The third cell contains 6.200, with an arrow showing the decimal point jumping places to the right, from between the 6 and 2 to after the second 00 in 6.200. Below this is the number 6,200.], alt: "In the second row, the first cell reads “Step 2. Move the decimal n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point absolute value of n places to the left.” The second cell reads “The exponent is positive so move the decimal point 3 places to the right. We need to add two zeros as placeholders.” The third cell contains 6.200, with an arrow showing the decimal point jumping places to the right, from between the 6 and 2 to after the second 00 in 6.200. Below this is the number 6,200.", caption: none) #figure(figph[In the third row, the first cell reads “Step 3. Check to see if your answer makes sense.” The second cell is blank. The third reads “10 cubed is 1000 and 1000 times 6.2 will be 6,200.” Beneath this is 6.2 times 10 cubed equals 6,200.], alt: "In the third row, the first cell reads “Step 3. Check to see if your answer makes sense.” The second cell is blank. The third reads “10 cubed is 1000 and 1000 times 6.2 will be 6,200.” Beneath this is 6.2 times 10 cubed equals 6,200.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Convert to decimal form: #math.equation(block: false, alt: "1.3 times 10 cubed .")[$1.3 #h(0.2em) × #h(0.2em) 10^(3) .$] #solutionbox[ 1,300 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Convert to decimal form: #math.equation(block: false, alt: "9.25 times 10 to the power 4.")[$9.25 #h(0.2em) × #h(0.2em) 10^(4) .$] #solutionbox[ 92,500 ] ] The steps are summarized below. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Convert scientific notation to decimal form.] To convert scientific notation to decimal form: + Determine the exponent, #math.equation(block: false, alt: "n")[$n$], on the factor 10. + Move the decimal #math.equation(block: false, alt: "n")[$n$] places, adding zeros if needed. - If the exponent is positive, move the decimal point #math.equation(block: false, alt: "n")[$n$] places to the right. - If the exponent is negative, move the decimal point #math.equation(block: false, alt: "vertical bar n vertical bar")[$| n |$] places to the left. + Check. ] #examplebox("Example 16")[][ Convert to decimal form: #math.equation(block: false, alt: "8.9 times 10 to the power −2.")[$8.9 #h(0.2em) × #h(0.2em) 10^(−2) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[8.9 times 10 to the power of negative 2.], alt: "8.9 times 10 to the power of negative 2.", caption: none)]), [Determine the exponent, #emph[n], on the factor 10.], [#figure(figph[The exponent is negative 2.], alt: "The exponent is negative 2.", caption: none)], [Since the exponent is negative, move the decimal point 2 places to the left.], [#figure(figph[8.9, with an arrow the decimal place showing the decimal point being moved two places to the left.], alt: "8.9, with an arrow the decimal place showing the decimal point being moved two places to the left.", caption: none)], [Add zeros as needed for placeholders.], [#figure(figph[8.9 times 10 to the power of negative 2 equals 0.089.], alt: "8.9 times 10 to the power of negative 2 equals 0.089.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Convert to decimal form: #math.equation(block: false, alt: "1.2 times 10 to the power −4.")[$1.2 #h(0.2em) × #h(0.2em) 10^(−4) .$] #solutionbox[ 0.00012 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Convert to decimal form: #math.equation(block: false, alt: "7.5 times 10 to the power −2.")[$7.5 #h(0.2em) × #h(0.2em) 10^(−2) .$] #solutionbox[ 0.075 ] ] === Multiply and Divide Using Scientific Notation Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation. #examplebox("Example 17")[][ Multiply. Write answers in decimal form: #math.equation(block: false, alt: "open parenthesis 4 times 10 to the power 5 close parenthesis open parenthesis 2 times 10 to the power −7 close parenthesis .")[$( 4 #h(0.2em) × #h(0.2em) 10^(5) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−7) ) .$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis 4 times 10 to the power 5 close parenthesis open parenthesis 2 times 10 to the power −7 close parenthesis; Use the Commutative Property to rearrange the factors., 4 times 2 times 10 to the power 5 times 10 to the power −7; Multiply., 8 times 10 to the power −2; Change to decimal form by moving the decimal two places left., 0.08")[$& & & ( 4 #h(0.2em) × #h(0.2em) 10^(5) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−7) ) \ \ \ "Use the Commutative Property to rearrange the factors." & & & 4 · 2 · 10^(5) · 10^(−7) \ \ \ "Multiply." & & & 8 #h(0.2em) × #h(0.2em) 10^(−2) \ \ \ "Change to decimal form by moving the decimal two places left." & & & 0.08$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Multiply #math.equation(block: false, alt: "open parenthesis 3 times 10 to the power 6 close parenthesis open parenthesis 2 times 10 to the power −8 close parenthesis")[$( 3 #h(0.2em) × #h(0.2em) 10^(6) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−8) )$]. Write answers in decimal form. #solutionbox[ 0.06 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Multiply #math.equation(block: false, alt: "open parenthesis 3 times 10 to the power −2 close parenthesis open parenthesis 3 times 10 to the power −1 close parenthesis")[$( 3 #h(0.2em) × #h(0.2em) 10^(−2) ) ( 3 #h(0.2em) × #h(0.2em) 10^(−1) )$]. Write answers in decimal form. #solutionbox[ 0.009 ] ] #examplebox("Example 18")[][ Divide. Write answers in decimal form: #math.equation(block: false, alt: "the fraction 9 times 10 cubed over 3 times 10 to the power −2.")[$frac(9 #h(0.2em) × #h(0.2em) 10^(3), 3 #h(0.2em) × #h(0.2em) 10^(−2)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 9 times 10 cubed over 3 times 10 to the power −2")[$frac(9 #h(0.2em) × #h(0.2em) 10^(3), 3 #h(0.2em) × #h(0.2em) 10^(−2))$]]), [Separate the factors, rewriting as the product of two fractions.], [#math.equation(block: false, alt: "the fraction 9 over 3 times the fraction 10 cubed over 10 to the power −2")[$frac(9, 3) #h(0.2em) × #h(0.2em) frac(10^(3), 10^(−2))$]], [Divide.], [#math.equation(block: false, alt: "3 times 10 to the power 5")[$3 #h(0.2em) × #h(0.2em) 10^(5)$]], [Change to decimal form by moving the decimal five places right.], [300,000], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Divide #math.equation(block: false, alt: "the fraction 8 times 10 to the power 4 over 2 times 10 to the power −1")[$frac(8 #h(0.2em) × #h(0.2em) 10^(4), 2 #h(0.2em) × #h(0.2em) 10^(−1))$]. Write answers in decimal form. #solutionbox[ 400,000 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Divide #math.equation(block: false, alt: "the fraction 8 times 10 squared over 4 times 10 to the power −2")[$frac(8 #h(0.2em) × #h(0.2em) 10^(2), 4 #h(0.2em) × #h(0.2em) 10^(−2))$]. Write answers in decimal form. #solutionbox[ 20,000 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access these online resources for additional instruction and practice with integer exponents and scientific notation: - #link("https://openstax.org/l/25Negexponents")[Negative Exponents] - #link("https://openstax.org/l/25Scientnot1")[Scientific Notation] - #link("https://openstax.org/l/25Scientnot2")[Scientific Notation 2] ] === Key Concepts - #strong[Property of Negative Exponents] - If #math.equation(block: false, alt: "n")[$n$] is a positive integer and #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$], then #math.equation(block: false, alt: "the fraction 1 over a to the power − n equals a to the power n")[$frac(1, a^("−" n)) = a^(n)$] - #strong[Quotient to a Negative Exponent] - If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers, #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$] and #math.equation(block: false, alt: "n")[$n$] is an integer , then #math.equation(block: false, alt: "open parenthesis the fraction a over b close parenthesis to the power − n equals open parenthesis the fraction b over a close parenthesis to the power n")[$attach(( frac(a, b) ), t: "−" n) = attach(( frac(b, a) ), t: n)$] - #strong[To convert a decimal to scientific notation:] + Move the decimal point so that the first factor is greater than or equal to 1 but less than 10. + Count the number of decimal places, #math.equation(block: false, alt: "n")[$n$], that the decimal point was moved. + Write the number as a product with a power of 10. If the original number is: - greater than 1, the power of 10 will be #math.equation(block: false, alt: "10 to the power n")[$10^(n)$] - between 0 and 1, the power of 10 will be #math.equation(block: false, alt: "10 to the power − n")[$10^("−" n)$] + Check. #linebreak() #linebreak() - #strong[To convert scientific notation to decimal form:] + Determine the exponent, #math.equation(block: false, alt: "n")[$n$], on the factor 10. + Move the decimal #math.equation(block: false, alt: "n")[$n$]places, adding zeros if needed. - If the exponent is positive, move the decimal point #math.equation(block: false, alt: "n")[$n$] places to the right. - If the exponent is negative, move the decimal point #math.equation(block: false, alt: "vertical bar n vertical bar")[$| n |$] places to the left. + Check. === Section Exercises ==== Practice Makes Perfect #strong[Use the Definition of a Negative Exponent] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "4 to the power −2")[$4^(−2)$] ⓑ #math.equation(block: false, alt: "10 to the power −3")[$10^(−3)$] ⓐ #math.equation(block: false, alt: "3 to the power −4")[$3^(−4)$] ⓑ #math.equation(block: false, alt: "10 to the power −2")[$10^(−2)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 81")[$frac(1, 81)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 100")[$frac(1, 100)$] ] ⓐ #math.equation(block: false, alt: "5 to the power minus 3")[$5^(− 3)$] ⓑ #math.equation(block: false, alt: "10 to the power −5")[$10^(−5)$] ⓐ #math.equation(block: false, alt: "2 to the power −8")[$2^(−8)$] ⓑ #math.equation(block: false, alt: "10 to the power −2")[$10^(−2)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 256")[$frac(1, 256)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 100")[$frac(1, 100)$] ] ⓐ #math.equation(block: false, alt: "the fraction 1 over c to the power −5")[$frac(1, c^(−5))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 3 to the power −2")[$frac(1, 3^(−2))$] ⓐ #math.equation(block: false, alt: "the fraction 1 over c to the power −5")[$frac(1, c^(−5))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 5 to the power −2")[$frac(1, 5^(−2))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "c to the power 5")[$c^(5)$] ⓑ 25 ] ⓐ #math.equation(block: false, alt: "the fraction 1 over q to the power −10")[$frac(1, q^(−10))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 10 to the power −3")[$frac(1, 10^(−3))$] ⓐ #math.equation(block: false, alt: "the fraction 1 over t to the power −9")[$frac(1, t^(−9))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 10 to the power −4")[$frac(1, 10^(−4))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "t to the power 9")[$t^(9)$] ⓑ 10000 ] ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 5 over 8 close parenthesis to the power −2")[$attach(( frac(5, 8) ), t: −2)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 3 m over n close parenthesis to the power −2")[$attach(( − frac(3 m, n) ), t: −2)$] ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 3 over 10 close parenthesis to the power −2")[$attach(( frac(3, 10) ), t: −2)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 2 over c d close parenthesis to the power −3")[$attach(( − frac(2, c d) ), t: −3)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 100 over 9")[$frac(100, 9)$] ⓑ #math.equation(block: false, alt: "minus the fraction c cubed d cubed over 8")[$− frac(c^(3) d^(3), 8)$] ] ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 4 over 9 close parenthesis to the power −3")[$attach(( frac(4, 9) ), t: −3)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction u squared over 2 v close parenthesis to the power −5")[$attach(( − frac(u^(2), 2 v) ), t: −5)$] ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 7 over 2 close parenthesis to the power −3")[$attach(( frac(7, 2) ), t: −3)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 3 over x y squared close parenthesis to the power −3")[$attach(( − frac(3, x y^(2)) ), t: −3)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 8 over 343")[$frac(8, 343)$] ⓑ #math.equation(block: false, alt: "minus the fraction x cubed y to the power 6 over 27")[$− frac(x^(3) y^(6), 27)$] ] ⓐ #math.equation(block: false, alt: "open parenthesis minus 5 close parenthesis to the power minus 2")[$attach(( − 5 ), t: − 2)$] ⓑ #math.equation(block: false, alt: "− 5 to the power −2")[$"−" 5^(−2)$] ⓒ #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 5 close parenthesis to the power −2")[$attach(( − frac(1, 5) ), t: −2)$] ⓓ #math.equation(block: false, alt: "− open parenthesis the fraction 1 over 5 close parenthesis to the power −2")[$"−" attach(( frac(1, 5) ), t: −2)$] ⓐ #math.equation(block: false, alt: "open parenthesis −7 close parenthesis to the power −2")[$attach(( −7 ), t: −2)$] ⓑ #math.equation(block: false, alt: "minus 7 to the power minus 2")[$− 7^(− 2)$] ⓒ #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 7 close parenthesis to the power −2")[$attach(( − frac(1, 7) ), t: −2)$] ⓓ #math.equation(block: false, alt: "− open parenthesis the fraction 1 over 7 close parenthesis to the power −2")[$"−" attach(( frac(1, 7) ), t: −2)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 49")[$frac(1, 49)$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 49")[$− frac(1, 49)$] ⓒ 49 ⓓ #math.equation(block: false, alt: "−49")[$−49$] ] ⓐ #math.equation(block: false, alt: "− 3 to the power −3")[$"−" 3^(−3)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 3 close parenthesis to the power −3")[$attach(( − frac(1, 3) ), t: −3)$] ⓒ #math.equation(block: false, alt: "− open parenthesis the fraction 1 over 3 close parenthesis to the power −3")[$"−" attach(( frac(1, 3) ), t: −3)$] ⓓ #math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power −3")[$attach(( −3 ), t: −3)$] ⓐ #math.equation(block: false, alt: "− 5 to the power −3")[$"−" 5^(−3)$] ⓑ #math.equation(block: false, alt: "open parenthesis minus the fraction 1 over 5 close parenthesis to the power −3")[$attach(( − frac(1, 5) ), t: −3)$] ⓒ #math.equation(block: false, alt: "− open parenthesis the fraction 1 over 5 close parenthesis to the power −3")[$"−" attach(( frac(1, 5) ), t: −3)$] ⓓ #math.equation(block: false, alt: "open parenthesis −5 close parenthesis to the power −3")[$attach(( −5 ), t: −3)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "minus the fraction 1 over 125")[$− frac(1, 125)$] ⓑ #math.equation(block: false, alt: "−125")[$−125$] ⓒ #math.equation(block: false, alt: "−125")[$−125$] ⓓ #math.equation(block: false, alt: "minus the fraction 1 over 125")[$− frac(1, 125)$] ] ⓐ #math.equation(block: false, alt: "3 times 5 to the power −1")[$3 · 5^(−1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 times 5 close parenthesis to the power −1")[$attach(( 3 · 5 ), t: −1)$] ⓐ #math.equation(block: false, alt: "2 times 5 to the power −1")[$2 · 5^(−1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 times 5 close parenthesis to the power −1")[$attach(( 2 · 5 ), t: −1)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 2 over 5")[$frac(2, 5)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 10")[$frac(1, 10)$] ] ⓐ #math.equation(block: false, alt: "4 times 5 to the power −2")[$4 · 5^(−2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 times 5 close parenthesis to the power −2")[$attach(( 4 · 5 ), t: −2)$] ⓐ #math.equation(block: false, alt: "3 times 4 to the power −2")[$3 · 4^(−2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 times 4 close parenthesis to the power −2")[$attach(( 3 · 4 ), t: −2)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 3 over 16")[$frac(3, 16)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 144")[$frac(1, 144)$] ] ⓐ #math.equation(block: false, alt: "m to the power −4")[$m^(−4)$] ⓑ #math.equation(block: false, alt: "open parenthesis x cubed close parenthesis to the power −4")[$attach(( x^(3) ), t: −4)$] ⓐ #math.equation(block: false, alt: "b to the power −5")[$b^(−5)$] ⓑ #math.equation(block: false, alt: "open parenthesis k squared close parenthesis to the power −5")[$attach(( k^(2) ), t: −5)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over b to the power 5")[$frac(1, b^(5))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over k to the power 10")[$frac(1, k^(10))$] ] ⓐ #math.equation(block: false, alt: "p to the power −10")[$p^(−10)$] ⓑ #math.equation(block: false, alt: "open parenthesis q to the power 6 close parenthesis to the power −8")[$attach(( q^(6) ), t: −8)$] ⓐ #math.equation(block: false, alt: "s to the power −8")[$s^(−8)$] ⓑ #math.equation(block: false, alt: "open parenthesis a to the power 9 close parenthesis to the power −10")[$attach(( a^(9) ), t: −10)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over s to the power 8")[$frac(1, s^(8))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over a to the power 90")[$frac(1, a^(90))$] ] ⓐ #math.equation(block: false, alt: "7 n to the power −1")[$7 n^(−1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 7 n close parenthesis to the power −1")[$attach(( 7 n ), t: −1)$] ⓒ #math.equation(block: false, alt: "open parenthesis −7 n close parenthesis to the power −1")[$attach(( −7 n ), t: −1)$] ⓐ #math.equation(block: false, alt: "6 r to the power −1")[$6 r^(−1)$] ⓑ #math.equation(block: false, alt: "open parenthesis 6 r close parenthesis to the power −1")[$attach(( 6 r ), t: −1)$] ⓒ #math.equation(block: false, alt: "open parenthesis −6 r close parenthesis to the power −1")[$attach(( −6 r ), t: −1)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 6 over r")[$frac(6, r)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 6 r")[$frac(1, 6 r)$] ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 6 r")[$− frac(1, 6 r)$] ] ⓐ #math.equation(block: false, alt: "open parenthesis 3 p close parenthesis to the power −2")[$attach(( 3 p ), t: −2)$] ⓑ #math.equation(block: false, alt: "3 p to the power −2")[$3 p^(−2)$] ⓒ #math.equation(block: false, alt: "−3 p to the power −2")[$−3 p^(−2)$] ⓐ #math.equation(block: false, alt: "open parenthesis 2 q close parenthesis to the power −4")[$attach(( 2 q ), t: −4)$] ⓑ #math.equation(block: false, alt: "2 q to the power −4")[$2 q^(−4)$] ⓒ #math.equation(block: false, alt: "−2 q to the power −4")[$−2 q^(−4)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 16 q to the power 4")[$frac(1, 16 q^(4))$] ⓑ #math.equation(block: false, alt: "the fraction 2 over q to the power 4")[$frac(2, q^(4))$] ⓒ #math.equation(block: false, alt: "minus the fraction 2 over q to the power 4")[$− frac(2, q^(4))$] ] #strong[Simplify Expressions with Integer Exponents] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "b to the power 4 b to the power −8")[$b^(4) b^(−8)$] ⓑ #math.equation(block: false, alt: "r to the power −2 r to the power 5")[$r^(−2) r^(5)$] ⓒ #math.equation(block: false, alt: "x to the power −7 x to the power −3")[$x^(−7) x^(−3)$] ⓐ #math.equation(block: false, alt: "s cubed times s to the power −7")[$s^(3) · s^(−7)$] ⓑ #math.equation(block: false, alt: "q to the power −8 times q cubed")[$q^(−8) · q^(3)$] ⓒ #math.equation(block: false, alt: "y to the power −2 times y to the power −5")[$y^(−2) · y^(−5)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over s to the power 4")[$frac(1, s^(4))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over q to the power 5")[$frac(1, q^(5))$] ⓒ #math.equation(block: false, alt: "the fraction 1 over y to the power 7")[$frac(1, y^(7))$] ] ⓐ #math.equation(block: false, alt: "a cubed times a to the power −3")[$a^(3) · a^(−3)$] ⓑ #math.equation(block: false, alt: "a times a cubed")[$a · a^(3)$] ⓒ #math.equation(block: false, alt: "a times a to the power −3")[$a · a^(−3)$] ⓐ #math.equation(block: false, alt: "y to the power 5 times y to the power −5")[$y^(5) · y^(−5)$] ⓑ #math.equation(block: false, alt: "y times y to the power 5")[$y · y^(5)$] ⓒ #math.equation(block: false, alt: "y times y to the power −5")[$y · y^(−5)$] #solutionbox[ ⓐ 1 ⓑ #math.equation(block: false, alt: "y to the power 6")[$y^(6)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over y to the power 4")[$frac(1, y^(4))$] ] #math.equation(block: true, alt: "p to the power 5 times p to the power −2 times p to the power −4")[$p^(5) · p^(−2) · p^(−4)$] #math.equation(block: true, alt: "x to the power 4 times x to the power −2 times x to the power −3")[$x^(4) · x^(−2) · x^(−3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over x")[$frac(1, x)$] ] #math.equation(block: true, alt: "open parenthesis w to the power 4 x to the power −5 close parenthesis open parenthesis w to the power −2 x to the power −4 close parenthesis")[$( w^(4) x^(−5) ) ( w^(−2) x^(−4) )$] #math.equation(block: true, alt: "open parenthesis m cubed n to the power −3 close parenthesis open parenthesis m to the power −5 n to the power −1 close parenthesis")[$( m^(3) n^(−3) ) ( m^(−5) n^(−1) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over m squared n to the power 4")[$frac(1, m^(2) n^(4))$] ] #math.equation(block: true, alt: "open parenthesis u v to the power −2 close parenthesis open parenthesis u to the power −5 v to the power −3 close parenthesis")[$( u v^(−2) ) ( u^(−5) v^(−3) )$] #math.equation(block: true, alt: "open parenthesis p q to the power −4 close parenthesis open parenthesis p to the power −6 q to the power −3 close parenthesis")[$( p q^(−4) ) ( p^(−6) q^(−3) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over p to the power 5 q to the power 7")[$frac(1, p^(5) q^(7))$] ] #math.equation(block: true, alt: "open parenthesis −6 c to the power −3 d to the power 9 close parenthesis open parenthesis 2 c to the power 4 d to the power −5 close parenthesis")[$( −6 c^(−3) d^(9) ) ( 2 c^(4) d^(−5) )$] #math.equation(block: true, alt: "open parenthesis −2 j to the power −5 k to the power 8 close parenthesis open parenthesis 7 j squared k to the power −3 close parenthesis")[$( −2 j^(−5) k^(8) ) ( 7 j^(2) k^(−3) )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 14 k to the power 5 over j cubed")[$− frac(14 k^(5), j^(3))$] ] #math.equation(block: true, alt: "open parenthesis −4 r to the power −2 s to the power −8 close parenthesis open parenthesis 9 r to the power 4 s cubed close parenthesis")[$( −4 r^(−2) s^(−8) ) ( 9 r^(4) s^(3) )$] #math.equation(block: true, alt: "open parenthesis −5 m to the power 4 n to the power 6 close parenthesis open parenthesis 8 m to the power −5 n to the power −3 close parenthesis")[$( −5 m^(4) n^(6) ) ( 8 m^(−5) n^(−3) )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 40 n cubed over m")[$− frac(40 n^(3), m)$] ] #math.equation(block: true, alt: "open parenthesis 5 x squared close parenthesis to the power −2")[$attach(( 5 x^(2) ), t: −2)$] #math.equation(block: true, alt: "open parenthesis 4 y cubed close parenthesis to the power −3")[$attach(( 4 y^(3) ), t: −3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 64 y to the power 9")[$frac(1, 64 y^(9))$] ] #math.equation(block: true, alt: "open parenthesis 3 z to the power −3 close parenthesis squared")[$attach(( 3 z^(−3) ), t: 2)$] #math.equation(block: true, alt: "open parenthesis 2 p to the power −5 close parenthesis squared")[$attach(( 2 p^(−5) ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over p to the power 10")[$frac(4, p^(10))$] ] #math.equation(block: true, alt: "the fraction t to the power 9 over t to the power −3")[$frac(t^(9), t^(−3))$] #math.equation(block: true, alt: "the fraction n to the power 5 over n to the power −2")[$frac(n^(5), n^(−2))$] #solutionbox[ #math.equation(block: true, alt: "n to the power 7")[$n^(7)$] ] #math.equation(block: true, alt: "the fraction x to the power −7 over x to the power −3")[$frac(x^(−7), x^(−3))$] #math.equation(block: true, alt: "the fraction y to the power −5 over y to the power −10")[$frac(y^(−5), y^(−10))$] #solutionbox[ #math.equation(block: true, alt: "y to the power 5")[$y^(5)$] ] #strong[Convert from Decimal Notation to Scientific Notation] In the following exercises, write each number in scientific notation. 57,000 340,000 #solutionbox[ #math.equation(block: true, alt: "3.4 times 10 to the power 5")[$3.4 #h(0.2em) × #h(0.2em) 10^(5)$] ] 8,750,000 1,290,000 #solutionbox[ #math.equation(block: true, alt: "1.29 times 10 to the power 6")[$1.29 #h(0.2em) × #h(0.2em) 10^(6)$] ] 0.026 0.041 #solutionbox[ #math.equation(block: true, alt: "4.1 times 10 to the power −2")[$4.1 #h(0.2em) × #h(0.2em) 10^(−2)$] ] 0.00000871 0.00000103 #solutionbox[ #math.equation(block: true, alt: "1.03 times 10 to the power −6")[$1.03 #h(0.2em) × #h(0.2em) 10^(−6)$] ] #strong[Convert Scientific Notation to Decimal Form] In the following exercises, convert each number to decimal form. #math.equation(block: true, alt: "5.2 times 10 squared")[$5.2 #h(0.2em) × #h(0.2em) 10^(2)$] #math.equation(block: true, alt: "8.3 times 10 squared")[$8.3 #h(0.2em) × #h(0.2em) 10^(2)$] #solutionbox[ 830 ] #math.equation(block: true, alt: "7.5 times 10 to the power 6")[$7.5 #h(0.2em) × #h(0.2em) 10^(6)$] #math.equation(block: true, alt: "1.6 times 10 to the power 10")[$1.6 #h(0.2em) × #h(0.2em) 10^(10)$] #solutionbox[ 16,000,000,000 ] #math.equation(block: true, alt: "2.5 times 10 to the power −2")[$2.5 #h(0.2em) × #h(0.2em) 10^(−2)$] #math.equation(block: true, alt: "3.8 times 10 to the power −2")[$3.8 #h(0.2em) × #h(0.2em) 10^(−2)$] #solutionbox[ 0.038 ] #math.equation(block: true, alt: "4.13 times 10 to the power −5")[$4.13 #h(0.2em) × #h(0.2em) 10^(−5)$] #math.equation(block: true, alt: "1.93 times 10 to the power −5")[$1.93 #h(0.2em) × #h(0.2em) 10^(−5)$] #solutionbox[ 0.0000193 ] #strong[Multiply and Divide Using Scientific Notation] In the following exercises, multiply. Write your answer in decimal form. #math.equation(block: true, alt: "open parenthesis 3 times 10 to the power −5 close parenthesis open parenthesis 3 times 10 to the power 9 close parenthesis")[$( 3 #h(0.2em) × #h(0.2em) 10^(−5) ) ( 3 #h(0.2em) × #h(0.2em) 10^(9) )$] #math.equation(block: true, alt: "open parenthesis 2 times 10 squared close parenthesis open parenthesis 1 times 10 to the power −4 close parenthesis")[$( 2 #h(0.2em) × #h(0.2em) 10^(2) ) ( 1 #h(0.2em) × #h(0.2em) 10^(−4) )$] #solutionbox[ 0.02 ] #math.equation(block: true, alt: "open parenthesis 7.1 times 10 to the power −2 close parenthesis open parenthesis 2.4 times 10 to the power −4 close parenthesis")[$( 7.1 #h(0.2em) × #h(0.2em) 10^(−2) ) ( 2.4 #h(0.2em) × #h(0.2em) 10^(−4) )$] #math.equation(block: true, alt: "open parenthesis 3.5 times 10 to the power −4 close parenthesis open parenthesis 1.6 times 10 to the power −2 close parenthesis")[$( 3.5 #h(0.2em) × #h(0.2em) 10^(−4) ) ( 1.6 #h(0.2em) × #h(0.2em) 10^(−2) )$] #solutionbox[ #math.equation(block: true, alt: "0.0000056")[$0.0000056$] ] In the following exercises, divide. Write your answer in decimal form. #math.equation(block: true, alt: "the fraction 7 times 10 to the power −3 over 1 times 10 to the power −7")[$frac(7 #h(0.2em) × #h(0.2em) 10^(−3), 1 #h(0.2em) × #h(0.2em) 10^(−7))$] #math.equation(block: true, alt: "the fraction 5 times 10 to the power −2 over 1 times 10 to the power −10")[$frac(5 #h(0.2em) × #h(0.2em) 10^(−2), 1 #h(0.2em) × #h(0.2em) 10^(−10))$] #solutionbox[ 500,000,000 ] #math.equation(block: true, alt: "the fraction 6 times 10 to the power 4 over 3 times 10 to the power −2")[$frac(6 #h(0.2em) × #h(0.2em) 10^(4), 3 #h(0.2em) × #h(0.2em) 10^(−2))$] #math.equation(block: true, alt: "the fraction 8 times 10 to the power 6 over 4 times 10 to the power −1")[$frac(8 #h(0.2em) × #h(0.2em) 10^(6), 4 #h(0.2em) × #h(0.2em) 10^(−1))$] #solutionbox[ 20,000,000 ] ==== Everyday Math The population of the United States on July 4, 2010 was almost 310,000,000. Write the number in scientific notation. The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation #solutionbox[ #math.equation(block: true, alt: "6.85 times 10 to the power 9")[$6.85 #h(0.2em) × #h(0.2em) 10^(9)$]. ] The average width of a human hair is 0.0018 centimeters. Write the number in scientific notation. The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation. #solutionbox[ #math.equation(block: true, alt: "5.7 times 10 to the power −9")[$5.7 #h(0.2em) × #h(0.2em) 10^(−9)$] ] In 2010, the number of Facebook users each day who changed their status to ‘engaged’ was #math.equation(block: false, alt: "2 times 10 to the power 4")[$2 #h(0.2em) × #h(0.2em) 10^(4)$]. Convert this number to decimal form. At the start of 2012, the US national debt was more than #math.equation(block: false, alt: "$ 1.5 times 10 to the power 13")[$"$" 1.5 #h(0.2em) × #h(0.2em) 10^(13)$]. Convert this number to decimal form. #solutionbox[ 15,000,000,000,000 ] The concentration of carbon dioxide in the atmosphere is #math.equation(block: false, alt: "3.9 times 10 to the power −4")[$3.9 #h(0.2em) × #h(0.2em) 10^(−4)$]. Convert this number to decimal form. The width of a proton is #math.equation(block: false, alt: "1 times 10 to the power −5")[$1 #h(0.2em) × #h(0.2em) 10^(−5)$] of the width of an atom. Convert this number to decimal form. #solutionbox[ 0.00001 ] #strong[Health care costs] The Centers for Medicare and Medicaid projects that consumers will spend more than \$4 trillion on health care by 2017. + ⓐ Write 4 trillion in decimal notation. + ⓑ Write 4 trillion in scientific notation. #strong[Coin production] In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation. #solutionbox[ #math.equation(block: true, alt: "1.545 times 10 to the power 8")[$1.545 #h(0.2em) × #h(0.2em) 10^(8)$] ] #strong[Distance] The distance between Earth and one of the brightest stars in the night sky is 33.7 light years. One light year is about 6,000,000,000,000 (6 trillion) miles. + ⓐ Write the number of miles in one light year in scientific notation. + ⓑ Use scientific notation to find the distance between Earth and the star in miles. Write the answer in scientific notation. #strong[Debt] At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately \$18,600,000,000,000 (\$18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015. + ⓐ Write the debt in scientific notation. + ⓑ Write the population in scientific notation. + ⓒ Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation. #solutionbox[ ⓐ #math.equation(block: false, alt: "1.86 times 10 to the power 13")[$1.86 #h(0.2em) × #h(0.2em) 10^(13)$] ⓑ #math.equation(block: false, alt: "3 times 10 to the power 8")[$3 #h(0.2em) × #h(0.2em) 10^(8)$] ⓒ #math.equation(block: false, alt: "6.2 times 10 to the power 4")[$6.2 #h(0.2em) × #h(0.2em) 10^(4)$] ] ==== Writing Exercises + ⓐ Explain the meaning of the exponent in the expression #math.equation(block: false, alt: "2 cubed")[$2^(3)$]. + ⓑ Explain the meaning of the exponent in the expression #math.equation(block: false, alt: "2 to the power −3")[$2^(−3)$]. When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative? #solutionbox[ answers will vary ] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “use the definition of a negative exponent,” “simplify expressions with integer exponents,” “convert from decimal notation to scientific notation,” “convert scientific notation to decimal form,” and “multiply and divide using scientific notation.” The rest of the cells are blank.], alt: "This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “use the definition of a negative exponent,” “simplify expressions with integer exponents,” “convert from decimal notation to scientific notation,” “convert scientific notation to decimal form,” and “multiply and divide using scientific notation.” The rest of the cells are blank.", caption: none) ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not? === Chapter 6 Review Exercises ==== Add and Subtract Polynomials #strong[Identify Polynomials, Monomials, Binomials and Trinomials] In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial. ⓐ #math.equation(block: false, alt: "11 c to the power 4 minus 23 c squared plus 1")[$11 c^(4) − 23 c^(2) + 1$] #linebreak() ⓑ #math.equation(block: false, alt: "9 p cubed plus 6 p squared minus p minus 5")[$9 p^(3) + 6 p^(2) − p − 5$] #linebreak() ⓒ #math.equation(block: false, alt: "the fraction 3 over 7 x plus the fraction 5 over 14")[$frac(3, 7) x + frac(5, 14)$] #linebreak() ⓓ 10 #linebreak() ⓔ #math.equation(block: false, alt: "2 y minus 12")[$2 y − 12$] ⓐ #math.equation(block: false, alt: "a squared minus b squared")[$a^(2) − b^(2)$] #linebreak() ⓑ #math.equation(block: false, alt: "24 d cubed")[$24 d^(3)$] #linebreak() ⓒ #math.equation(block: false, alt: "x squared plus 8 x minus 10")[$x^(2) + 8 x − 10$] #linebreak() ⓓ #math.equation(block: false, alt: "m squared n squared minus 2 m n plus 6")[$m^(2) n^(2) − 2 m n + 6$] #linebreak() ⓔ #math.equation(block: false, alt: "7 y cubed plus y squared minus 2 y minus 4")[$7 y^(3) + y^(2) − 2 y − 4$] #solutionbox[ ⓐ binomial ⓑ monomial ⓒ trinomial ⓓ trinomial ⓔ other polynomial ] #strong[Determine the Degree of Polynomials] In the following exercises, determine the degree of each polynomial. + ⓐ #math.equation(block: false, alt: "3 x squared plus 9 x plus 10")[$3 x^(2) + 9 x + 10$] + ⓑ #math.equation(block: false, alt: "14 a squared b c")[$14 a^(2) b c$] + ⓒ #math.equation(block: false, alt: "6 y plus 1")[$6 y + 1$] + ⓓ #math.equation(block: false, alt: "n cubed minus 4 n squared plus 2 n minus 8")[$n^(3) − 4 n^(2) + 2 n − 8$] + ⓔ #math.equation(block: false, alt: "−19")[$−19$] + ⓐ #math.equation(block: false, alt: "5 p cubed minus 8 p squared plus 10 p minus 4")[$5 p^(3) − 8 p^(2) + 10 p − 4$] + ⓑ #math.equation(block: false, alt: "−20 q to the power 4")[$−20 q^(4)$] + ⓒ #math.equation(block: false, alt: "x squared plus 6 x plus 12")[$x^(2) + 6 x + 12$] + ⓓ #math.equation(block: false, alt: "23 r squared s squared minus 4 r s plus 5")[$23 r^(2) s^(2) − 4 r s + 5$] + ⓔ 100 #solutionbox[ ⓐ 3 ⓑ 4 ⓒ 2 ⓓ 4 ⓔ 0 ] #strong[Add and Subtract Monomials] In the following exercises, add or subtract the monomials. #math.equation(block: true, alt: "5y cubed plus 8 y cubed")[$attach(#h(0.2em) "5y", t: "3") + 8 y^(3)$] #math.equation(block: true, alt: "−14 k plus 19 k")[$−14 k + 19 k$] #solutionbox[ #math.equation(block: true, alt: "5 k")[$5 k$] ] #math.equation(block: true, alt: "12 q minus open parenthesis −6 q close parenthesis")[$12 q − ( −6 q )$] #math.equation(block: true, alt: "−9 c minus 18 c")[$−9 c − 18 c$] #solutionbox[ #math.equation(block: true, alt: "−27 c")[$−27 c$] ] #math.equation(block: true, alt: "12x minus 4 y minus 9 x")[$"12x" − 4 y − 9 x$] #math.equation(block: true, alt: "3 m squared plus 7 n squared minus 3 m squared")[$3 m^(2) + 7 n^(2) − 3 m^(2)$] #solutionbox[ #math.equation(block: true, alt: "7 n squared")[$7 n^(2)$] ] #math.equation(block: true, alt: "6 x squared y minus 4 x plus 8 x y squared")[$6 x^(2) y − 4 x + 8 x y^(2)$] #math.equation(block: true, alt: "13a plus b")[$"13a" + b$] #solutionbox[ #math.equation(block: true, alt: "13a plus b")[$"13a" + b$] ] #strong[Add and Subtract Polynomials] In the following exercises, add or subtract the polynomials. #math.equation(block: true, alt: "open parenthesis 5 x squared plus 12 x plus 1 close parenthesis plus open parenthesis 6 x squared minus 8 x plus 3 close parenthesis")[$( 5 x^(2) + 12 x + 1 ) + ( 6 x^(2) − 8 x + 3 )$] #math.equation(block: true, alt: "open parenthesis 9 p squared minus 5 p plus 3 close parenthesis plus open parenthesis 4 p squared minus 4 close parenthesis")[$( 9 p^(2) − 5 p + 3 ) + ( 4 p^(2) − 4 )$] #solutionbox[ #math.equation(block: true, alt: "13 p squared minus 5 p minus 1")[$13 p^(2) − 5 p − 1$] ] #math.equation(block: true, alt: "open parenthesis 10 m squared minus 8 m minus 1 close parenthesis minus open parenthesis 5 m squared plus m minus 2 close parenthesis")[$( 10 m^(2) − 8 m − 1 ) − ( 5 m^(2) + m − 2 )$] #math.equation(block: true, alt: "open parenthesis 7 y squared minus 8 y close parenthesis minus open parenthesis y minus 4 close parenthesis")[$( 7 y^(2) − 8 y ) − ( y − 4 )$] #solutionbox[ #math.equation(block: true, alt: "7 y squared minus 9 y plus 4")[$7 y^(2) − 9 y + 4$] ] Subtract #linebreak() #math.equation(block: false, alt: "open parenthesis 3 s squared plus 10 close parenthesis from open parenthesis 15 s squared minus 2 s plus 8 close parenthesis")[$( 3 s^(2) + 10 ) #h(0.2em) "from" #h(0.2em) ( 15 s^(2) − 2 s + 8 )$] Find the sum of #math.equation(block: false, alt: "open parenthesis a squared plus 6 a plus 9 close parenthesis and open parenthesis 5 a cubed minus 7 close parenthesis")[$( a^(2) + 6 a + 9 ) #h(0.2em) "and" #h(0.2em) ( 5 a^(3) − 7 )$] #solutionbox[ #math.equation(block: true, alt: "5 a cubed plus a squared plus 6 a plus 2")[$5 a^(3) + a^(2) + 6 a + 2$] ] #strong[Evaluate a Polynomial for a Given Value of the Variable] In the following exercises, evaluate each polynomial for the given value. Evaluate #math.equation(block: false, alt: "3 y squared minus y plus 1")[$3 y^(2) − y + 1$] when: + ⓐ #math.equation(block: false, alt: "y equals 5")[$y = 5$] + ⓑ #math.equation(block: false, alt: "y equals −1")[$y = −1$] + ⓒ #math.equation(block: false, alt: "y equals 0")[$y = 0$] Evaluate #math.equation(block: false, alt: "10 minus 12 x")[$10 − 12 x$] when: + ⓐ #math.equation(block: false, alt: "x equals 3")[$x = 3$] + ⓑ #math.equation(block: false, alt: "x equals 0")[$x = 0$] + ⓒ #math.equation(block: false, alt: "x equals −1")[$x = −1$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−26")[$−26$] ⓑ 10 ⓒ 22 ] Randee drops a stone off the 200 foot high cliff into the ocean. The polynomial #math.equation(block: false, alt: "−16 t squared plus 200")[$−16 t^(2) + 200$] gives the height of a stone #math.equation(block: false, alt: "t")[$t$] seconds after it is dropped from the cliff. Find the height after #math.equation(block: false, alt: "t equals 3")[$t = 3$] seconds. A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of #emph[p] dollars each is given by the polynomial #math.equation(block: false, alt: "−4 p squared plus 460 p .")[$−4 p^(2) + 460 p .$] Find the revenue received when #math.equation(block: false, alt: "p equals 75")[$p = 75$] dollars. #solutionbox[ 12,000 ] ==== Use Multiplication Properties of Exponents #strong[Simplify Expressions with Exponents] In the following exercises, simplify. #math.equation(block: true, alt: "10 to the power 4")[$10^(4)$] #math.equation(block: true, alt: "17 to the power 1")[$17^(1)$] #solutionbox[ 17 ] #math.equation(block: true, alt: "open parenthesis the fraction 2 over 9 close parenthesis squared")[$attach(( frac(2, 9) ), t: 2)$] #math.equation(block: true, alt: "open parenthesis 0.5 close parenthesis cubed")[$attach(( 0.5 ), t: 3)$] #solutionbox[ 0.125 ] #math.equation(block: true, alt: "open parenthesis −2 close parenthesis to the power 6")[$attach(( −2 ), t: 6)$] #math.equation(block: true, alt: "− 2 to the power 6")[$"−" 2^(6)$] #solutionbox[ #math.equation(block: true, alt: "−64")[$−64$] ] #strong[Simplify Expressions Using the Product Property for Exponents] In the following exercises, simplify each expression. #math.equation(block: true, alt: "x to the power 4 times x cubed")[$x^(4) · x^(3)$] #math.equation(block: true, alt: "p to the power 15 times p to the power 16")[$p^(15) · p^(16)$] #solutionbox[ #math.equation(block: true, alt: "p to the power 31")[$p^(31)$] ] #math.equation(block: true, alt: "4 to the power 10 times 4 to the power 6")[$4^(10) · 4^(6)$] #math.equation(block: true, alt: "8 times 8 to the power 5")[$8 · 8^(5)$] #solutionbox[ #math.equation(block: true, alt: "8 to the power 6")[$8^(6)$] ] #math.equation(block: true, alt: "n times n squared times n to the power 4")[$n · n^(2) · n^(4)$] #math.equation(block: true, alt: "y to the power c times y cubed")[$y^(c) · y^(3)$] #solutionbox[ #math.equation(block: true, alt: "y to the power c plus 3")[$y^(c + 3)$] ] #strong[Simplify Expressions Using the Power Property for Exponents] In the following exercises, simplify each expression. #math.equation(block: true, alt: "open parenthesis m cubed close parenthesis to the power 5")[$attach(( m^(3) ), t: 5)$] #math.equation(block: true, alt: "open parenthesis 5 cubed close parenthesis squared")[$attach(( 5^(3) ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "5 to the power 6")[$5^(6)$] ] #math.equation(block: true, alt: "open parenthesis y to the power 4 close parenthesis to the power x")[$attach(( y^(4) ), t: x)$] #math.equation(block: true, alt: "open parenthesis 3 to the power r close parenthesis to the power s")[$attach(( 3^(r) ), t: s)$] #solutionbox[ #math.equation(block: true, alt: "3 to the power r s")[$3^(r s)$] ] #strong[Simplify Expressions Using the Product to a Power Property] In the following exercises, simplify each expression. #math.equation(block: true, alt: "open parenthesis 4 a close parenthesis squared")[$attach(( 4 a ), t: 2)$] #math.equation(block: true, alt: "open parenthesis −5 y close parenthesis cubed")[$attach(( −5 y ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "−125 y cubed")[$−125 y^(3)$] ] #math.equation(block: true, alt: "open parenthesis 2 m n close parenthesis to the power 5")[$attach(( 2 m n ), t: 5)$] #math.equation(block: true, alt: "open parenthesis 10 x y z close parenthesis cubed")[$attach(( 10 x y z ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "1000 x cubed y cubed z cubed")[$1000 x^(3) y^(3) z^(3)$] ] #strong[Simplify Expressions by Applying Several Properties] In the following exercises, simplify each expression. #math.equation(block: true, alt: "open parenthesis p squared close parenthesis to the power 5 times open parenthesis p cubed close parenthesis to the power 6")[$attach(( p^(2) ), t: 5) · attach(( p^(3) ), t: 6)$] #math.equation(block: true, alt: "open parenthesis 4 a cubed b squared close parenthesis cubed")[$attach(( 4 a^(3) b^(2) ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "64 a to the power 9 b to the power 6")[$64 a^(9) b^(6)$] ] #math.equation(block: true, alt: "open parenthesis 5 x close parenthesis squared open parenthesis 7 x close parenthesis")[$attach(( 5 x ), t: 2) ( 7 x )$] #math.equation(block: true, alt: "open parenthesis 2 q cubed close parenthesis to the power 4 open parenthesis 3 q close parenthesis squared")[$attach(( 2 q^(3) ), t: 4) attach(( 3 q ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "144 q to the power 14")[$144 q^(14)$] ] #math.equation(block: true, alt: "open parenthesis the fraction 1 over 3 x squared close parenthesis squared open parenthesis the fraction 1 over 2 x close parenthesis cubed")[$attach(( frac(1, 3) x^(2) ), t: 2) attach(( frac(1, 2) x ), t: 3)$] #math.equation(block: true, alt: "open parenthesis the fraction 2 over 5 m squared n close parenthesis cubed")[$attach(( frac(2, 5) m^(2) n ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 8 over 125 m to the power 6 n cubed")[$frac(8, 125) m^(6) n^(3)$] ] #strong[Multiply Monomials] In the following exercises, multiply the monomials. #math.equation(block: true, alt: "open parenthesis −15 x squared close parenthesis open parenthesis 6 x to the power 4 close parenthesis")[$( −15 x^(2) ) ( 6 x^(4) )$] #math.equation(block: true, alt: "open parenthesis −9 n to the power 7 close parenthesis open parenthesis −16 n close parenthesis")[$( −9 n^(7) ) ( −16 n )$] #solutionbox[ #math.equation(block: true, alt: "144 n to the power 8")[$144 n^(8)$] ] #math.equation(block: true, alt: "open parenthesis 7 p to the power 5 q cubed close parenthesis open parenthesis 8 p q to the power 9 close parenthesis")[$( 7 p^(5) q^(3) ) ( 8 p q^(9) )$] #math.equation(block: true, alt: "open parenthesis the fraction 5 over 9 a b squared close parenthesis open parenthesis 27 a b cubed close parenthesis")[$( frac(5, 9) a b^(2) ) ( 27 a b^(3) )$] #solutionbox[ #math.equation(block: true, alt: "15 a squared b to the power 5")[$15 a^(2) b^(5)$] ] ==== Multiply Polynomials #strong[Multiply a Polynomial by a Monomial] In the following exercises, multiply. #math.equation(block: true, alt: "7 open parenthesis a plus 9 close parenthesis")[$7 ( a + 9 )$] #math.equation(block: true, alt: "−4 open parenthesis y plus 13 close parenthesis")[$−4 ( y + 13 )$] #solutionbox[ #math.equation(block: true, alt: "−4 y minus 52")[$−4 y − 52$] ] #math.equation(block: true, alt: "−5 open parenthesis r minus 2 close parenthesis")[$−5 ( r − 2 )$] #math.equation(block: true, alt: "p open parenthesis p plus 3 close parenthesis")[$p ( p + 3 )$] #solutionbox[ #math.equation(block: true, alt: "p squared plus 3 p")[$p^(2) + 3 p$] ] #math.equation(block: true, alt: "− m open parenthesis m plus 15 close parenthesis")[$"−" m ( m + 15 )$] #math.equation(block: true, alt: "−6 u open parenthesis 2 u plus 7 close parenthesis")[$−6 u ( 2 u + 7 )$] #solutionbox[ #math.equation(block: true, alt: "−12 u squared minus 42 u")[$−12 u^(2) − 42 u$] ] #math.equation(block: true, alt: "9 open parenthesis b squared plus 6 b plus 8 close parenthesis")[$9 ( b^(2) + 6 b + 8 )$] #math.equation(block: false, alt: "3 q squared open parenthesis q squared minus 7 q plus 6 close parenthesis")[$3 q^(2) ( q^(2) − 7 q + 6 )$] 3 #solutionbox[ #math.equation(block: true, alt: "9 q to the power 4 minus 63 q cubed plus 54 q squared")[$9 q^(4) − 63 q^(3) + 54 q^(2)$] ] #math.equation(block: true, alt: "open parenthesis 5 z minus 1 close parenthesis z")[$( 5 z − 1 ) z$] #math.equation(block: true, alt: "open parenthesis b minus 4 close parenthesis times 11")[$( b − 4 ) · 11$] #solutionbox[ #math.equation(block: true, alt: "11 b minus 44")[$11 b − 44$] ] #strong[Multiply a Binomial by a Binomial] In the following exercises, multiply the binomials using: ⓐ the Distributive Property, ⓑ the FOIL method, ⓒ the Vertical Method. #math.equation(block: true, alt: "open parenthesis x minus 4 close parenthesis open parenthesis x plus 10 close parenthesis")[$( x − 4 ) ( x + 10 )$] #math.equation(block: true, alt: "open parenthesis 6 y minus 7 close parenthesis open parenthesis 2 y minus 5 close parenthesis")[$( 6 y − 7 ) ( 2 y − 5 )$] #solutionbox[ ⓐ #math.equation(block: false, alt: "12 y squared minus 44 y plus 35")[$12 y^(2) − 44 y + 35$] ⓑ #math.equation(block: false, alt: "12 y squared minus 44 y plus 35")[$12 y^(2) − 44 y + 35$] ⓒ #math.equation(block: false, alt: "12 y squared minus 44 y plus 35")[$12 y^(2) − 44 y + 35$] ] In the following exercises, multiply the binomials. Use any method. #math.equation(block: true, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x plus 9 close parenthesis")[$( x + 3 ) ( x + 9 )$] #math.equation(block: true, alt: "open parenthesis y minus 4 close parenthesis open parenthesis y minus 8 close parenthesis")[$( y − 4 ) ( y − 8 )$] #solutionbox[ #math.equation(block: true, alt: "y squared minus 12 y plus 32")[$y^(2) − 12 y + 32$] ] #math.equation(block: true, alt: "open parenthesis p minus 7 close parenthesis open parenthesis p plus 4 close parenthesis")[$( p − 7 ) ( p + 4 )$] #math.equation(block: true, alt: "open parenthesis q plus 16 close parenthesis open parenthesis q minus 3 close parenthesis")[$( q + 16 ) ( q − 3 )$] #solutionbox[ #math.equation(block: true, alt: "q squared plus 13 q minus 48")[$q^(2) + 13 q − 48$] ] #math.equation(block: true, alt: "open parenthesis 5 m minus 8 close parenthesis open parenthesis 12 m plus 1 close parenthesis")[$( 5 m − 8 ) ( 12 m + 1 )$] #math.equation(block: true, alt: "open parenthesis u squared plus 6 close parenthesis open parenthesis u squared minus 5 close parenthesis")[$( u^(2) + 6 ) ( u^(2) − 5 )$] #solutionbox[ #math.equation(block: true, alt: "u to the power 4 plus u squared minus 30")[$u^(4) + u^(2) − 30$] ] #math.equation(block: true, alt: "open parenthesis 9 x minus y close parenthesis open parenthesis 6 x minus 5 close parenthesis")[$( 9 x − y ) ( 6 x − 5 )$] #math.equation(block: true, alt: "open parenthesis 8 m n plus 3 close parenthesis open parenthesis 2 m n minus 1 close parenthesis")[$( 8 m n + 3 ) ( 2 m n − 1 )$] #solutionbox[ #math.equation(block: true, alt: "16 m squared n squared minus 2 m n minus 3")[$16 m^(2) n^(2) − 2 m n − 3$] ] #strong[Multiply a Trinomial by a Binomial] In the following exercises, multiply using ⓐ the Distributive Property, ⓑ the Vertical Method. #math.equation(block: true, alt: "open parenthesis n plus 1 close parenthesis open parenthesis n squared plus 5 n minus 2 close parenthesis")[$( n + 1 ) ( n^(2) + 5 n − 2 )$] #math.equation(block: true, alt: "open parenthesis 3 x minus 4 close parenthesis open parenthesis 6 x squared plus x minus 10 close parenthesis")[$( 3 x − 4 ) ( 6 x^(2) + x − 10 )$] #solutionbox[ ⓐ #math.equation(block: false, alt: "18 x cubed minus 21 x squared minus 34 x plus 40")[$18 x^(3) − 21 x^(2) − 34 x + 40$] ⓑ #math.equation(block: false, alt: "18 x cubed minus 21 x squared minus 34 x plus 40")[$18 x^(3) − 21 x^(2) − 34 x + 40$] ] In the following exercises, multiply. Use either method. #math.equation(block: true, alt: "open parenthesis y minus 2 close parenthesis open parenthesis y squared minus 8 y plus 9 close parenthesis")[$( y − 2 ) ( y^(2) − 8 y + 9 )$] #math.equation(block: true, alt: "open parenthesis 7 m plus 1 close parenthesis open parenthesis m squared minus 10 m minus 3 close parenthesis")[$( 7 m + 1 ) ( m^(2) − 10 m − 3 )$] #solutionbox[ #math.equation(block: true, alt: "7 m cubed minus 69 m squared minus 31 m minus 3")[$7 m^(3) − 69 m^(2) − 31 m − 3$] ] ==== Special Products #strong[Square a Binomial Using the Binomial Squares Pattern] In the following exercises, square each binomial using the Binomial Squares Pattern. #math.equation(block: true, alt: "open parenthesis c plus 11 close parenthesis squared")[$attach(( c + 11 ), t: 2)$] #math.equation(block: true, alt: "open parenthesis q minus 15 close parenthesis squared")[$attach(( q − 15 ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "q squared minus 30 q plus 225")[$q^(2) − 30 q + 225$] ] #math.equation(block: true, alt: "open parenthesis x plus the fraction 1 over 3 close parenthesis squared")[$attach(( x + frac(1, 3) ), t: 2)$] #math.equation(block: true, alt: "open parenthesis 8 u plus 1 close parenthesis squared")[$attach(( 8 u + 1 ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "64 u squared plus 16 u plus 1")[$64 u^(2) + 16 u + 1$] ] #math.equation(block: true, alt: "open parenthesis 3 n cubed minus 2 close parenthesis squared")[$attach(( 3 n^(3) − 2 ), t: 2)$] #math.equation(block: true, alt: "open parenthesis 4 a minus 3 b close parenthesis squared")[$attach(( 4 a − 3 b ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "16 a squared minus 24 a b plus 9 b squared")[$16 a^(2) − 24 a b + 9 b^(2)$] ] #strong[Multiply Conjugates Using the Product of Conjugates Pattern] In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern. #math.equation(block: true, alt: "open parenthesis s minus 7 close parenthesis open parenthesis s plus 7 close parenthesis")[$( s − 7 ) ( s + 7 )$] #math.equation(block: true, alt: "open parenthesis y plus the fraction 2 over 5 close parenthesis open parenthesis y minus the fraction 2 over 5 close parenthesis")[$( y + frac(2, 5) ) ( y − frac(2, 5) )$] #solutionbox[ #math.equation(block: true, alt: "y squared minus the fraction 4 over 25")[$y^(2) − frac(4, 25)$] ] #math.equation(block: true, alt: "open parenthesis 12 c plus 13 close parenthesis open parenthesis 12 c minus 13 close parenthesis")[$( 12 c + 13 ) ( 12 c − 13 )$] #math.equation(block: true, alt: "open parenthesis 6 minus r close parenthesis open parenthesis 6 plus r close parenthesis")[$( 6 − r ) ( 6 + r )$] #solutionbox[ #math.equation(block: true, alt: "36 minus r squared")[$36 − r^(2)$] ] #math.equation(block: true, alt: "open parenthesis u plus the fraction 3 over 4 v close parenthesis open parenthesis u minus the fraction 3 over 4 v close parenthesis")[$( u + frac(3, 4) v ) ( u − frac(3, 4) v )$] #math.equation(block: true, alt: "open parenthesis 5 p to the power 4 minus 4 q cubed close parenthesis open parenthesis 5 p to the power 4 plus 4 q cubed close parenthesis")[$( 5 p^(4) − 4 q^(3) ) ( 5 p^(4) + 4 q^(3) )$] #solutionbox[ #math.equation(block: true, alt: "25 p to the power 8 minus 16 q to the power 6")[$25 p^(8) − 16 q^(6)$] ] #strong[Recognize and Use the Appropriate Special Product Pattern] In the following exercises, find each product. #math.equation(block: true, alt: "open parenthesis 3 m plus 10 close parenthesis squared")[$attach(( 3 m + 10 ), t: 2)$] #math.equation(block: true, alt: "open parenthesis 6 a plus 11 close parenthesis open parenthesis 6 a minus 11 close parenthesis")[$( 6 a + 11 ) ( 6 a − 11 )$] #solutionbox[ #math.equation(block: true, alt: "36 a squared minus 121")[$36 a^(2) − 121$] ] #math.equation(block: true, alt: "open parenthesis 5 x plus y close parenthesis open parenthesis x minus 5 y close parenthesis")[$( 5 x + y ) ( x − 5 y )$] #math.equation(block: true, alt: "open parenthesis c to the power 4 plus 9 d close parenthesis squared")[$attach(( c^(4) + 9 d ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "c to the power 8 plus 18 c to the power 4 d plus 81 d squared")[$c^(8) + 18 c^(4) d + 81 d^(2)$] ] #math.equation(block: true, alt: "open parenthesis p to the power 5 plus q to the power 5 close parenthesis open parenthesis p to the power 5 minus q to the power 5 close parenthesis")[$( p^(5) + q^(5) ) ( p^(5) − q^(5) )$] #math.equation(block: true, alt: "open parenthesis a squared plus 4 b close parenthesis open parenthesis 4 a minus b squared close parenthesis")[$( a^(2) + 4 b ) ( 4 a − b^(2) )$] #solutionbox[ #math.equation(block: true, alt: "4 a cubed minus a squared b squared plus 16 a b minus 4 b cubed")[$4 a^(3) − a^(2) b^(2) + 16 a b − 4 b^(3)$] ] ==== Divide Monomials #strong[Simplify Expressions Using the Quotient Property for Exponents] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction u to the power 24 over u to the power 6")[$frac(u^(24), u^(6))$] #math.equation(block: true, alt: "the fraction 10 to the power 25 over 10 to the power 5")[$frac(10^(25), 10^(5))$] #solutionbox[ #math.equation(block: true, alt: "10 to the power 20")[$10^(20)$] ] #math.equation(block: true, alt: "the fraction 3 to the power 4 over 3 to the power 6")[$frac(3^(4), 3^(6))$] #math.equation(block: true, alt: "the fraction v to the power 12 over v to the power 48")[$frac(v^(12), v^(48))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over v to the power 36")[$frac(1, v^(36))$] ] #math.equation(block: true, alt: "the fraction x over x to the power 5")[$frac(x, x^(5))$] #math.equation(block: true, alt: "the fraction 5 over 5 to the power 8")[$frac(5, 5^(8))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 5 to the power 7")[$frac(1, 5^(7))$] ] #strong[Simplify Expressions with Zero Exponents] In the following exercises, simplify. #math.equation(block: true, alt: "75 to the power 0")[$75^(0)$] #math.equation(block: true, alt: "x to the power 0")[$x^(0)$] #solutionbox[ 1 ] #math.equation(block: true, alt: "− 12 to the power 0")[$"−" 12^(0)$] #math.equation(block: true, alt: "open parenthesis − 12 to the power 0 close parenthesis")[$( "−" 12^(0) )$]#math.equation(block: true, alt: "open parenthesis −12 close parenthesis to the power 0")[$attach(( −12 ), t: 0)$] #solutionbox[ −1 ] #math.equation(block: true, alt: "25 x to the power 0")[$25 x^(0)$] #math.equation(block: true, alt: "open parenthesis 25 x close parenthesis to the power 0")[$attach(( 25 x ), t: 0)$] #solutionbox[ 1 ] #math.equation(block: true, alt: "19 n to the power 0 minus 25 m to the power 0")[$19 n^(0) − 25 m^(0)$] #math.equation(block: true, alt: "open parenthesis 19 n close parenthesis to the power 0 minus open parenthesis 25 m close parenthesis to the power 0")[$attach(( 19 n ), t: 0) − attach(( 25 m ), t: 0)$] #solutionbox[ 0 ] #strong[Simplify Expressions Using the Quotient to a Power Property] In the following exercises, simplify. #math.equation(block: true, alt: "open parenthesis the fraction 2 over 5 close parenthesis cubed")[$attach(( frac(2, 5) ), t: 3)$] #math.equation(block: true, alt: "open parenthesis the fraction m over 3 close parenthesis to the power 4")[$attach(( frac(m, 3) ), t: 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction m to the power 4 over 81")[$frac(m^(4), 81)$] ] #math.equation(block: true, alt: "open parenthesis the fraction r over s close parenthesis to the power 8")[$attach(( frac(r, s) ), t: 8)$] #math.equation(block: true, alt: "open parenthesis the fraction x over 2 y close parenthesis to the power 6")[$attach(( frac(x, 2 y) ), t: 6)$] #solutionbox[ #math.equation(block: true, alt: "the fraction x to the power 6 over 64 y to the power 6")[$frac(x^(6), 64 y^(6))$] ] #strong[Simplify Expressions by Applying Several Properties] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction open parenthesis x cubed close parenthesis to the power 5 over x to the power 9")[$frac(attach(( x^(3) ), t: 5), x^(9))$] #math.equation(block: true, alt: "the fraction n to the power 10 over open parenthesis n to the power 5 close parenthesis squared")[$frac(n^(10), attach(( n^(5) ), t: 2))$] #solutionbox[ 1 ] #math.equation(block: true, alt: "open parenthesis the fraction q to the power 6 over q to the power 8 close parenthesis cubed")[$attach(( frac(q^(6), q^(8)) ), t: 3)$] #math.equation(block: true, alt: "open parenthesis the fraction r to the power 8 over r cubed close parenthesis to the power 4")[$attach(( frac(r^(8), r^(3)) ), t: 4)$] #solutionbox[ #math.equation(block: true, alt: "r to the power 20")[$r^(20)$] ] #math.equation(block: true, alt: "open parenthesis the fraction c squared over d to the power 5 close parenthesis to the power 9")[$attach(( frac(c^(2), d^(5)) ), t: 9)$] #math.equation(block: true, alt: "open parenthesis the fraction 3 x to the power 4 over 2 y squared close parenthesis to the power 5")[$attach(( frac(3 x^(4), 2 y^(2)) ), t: 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 243 x to the power 20 over 32 y to the power 10")[$frac(243 x^(20), 32 y^(10))$] ] #math.equation(block: true, alt: "open parenthesis the fraction v cubed v to the power 9 over v to the power 6 close parenthesis to the power 4")[$attach(( frac(v^(3) v^(9), v^(6)) ), t: 4)$] #math.equation(block: true, alt: "the fraction open parenthesis 3 n squared close parenthesis to the power 4 open parenthesis −5 n to the power 4 close parenthesis cubed over open parenthesis −2 n to the power 5 close parenthesis squared")[$frac(attach(( 3 n^(2) ), t: 4) attach(( −5 n^(4) ), t: 3), attach(( −2 n^(5) ), t: 2))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 10,125 n to the power 10 over 4")[$− frac(10","125 n^(10), 4)$] ] #strong[Divide Monomials] In the following exercises, divide the monomials. #math.equation(block: true, alt: "−65 y to the power 14 divided by 5 y squared")[$−65 y^(14) ÷ 5 y^(2)$] #math.equation(block: true, alt: "the fraction 64 a to the power 5 b to the power 9 over −16 a to the power 10 b cubed")[$frac(64 a^(5) b^(9), −16 a^(10) b^(3))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 4 b to the power 6 over a to the power 5")[$− frac(4 b^(6), a^(5))$] ] #math.equation(block: true, alt: "the fraction 144 x to the power 15 y to the power 8 z cubed over 18 x to the power 10 y squared z to the power 12")[$frac(144 x^(15) y^(8) z^(3), 18 x^(10) y^(2) z^(12))$] #math.equation(block: true, alt: "the fraction open parenthesis 8 p to the power 6 q squared close parenthesis open parenthesis 9 p cubed q to the power 5 close parenthesis over 16 p to the power 8 q to the power 7")[$frac(( 8 p^(6) q^(2) ) ( 9 p^(3) q^(5) ), 16 p^(8) q^(7))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 9 p over 2")[$frac(9 p, 2)$] ] ==== Divide Polynomials #strong[Divide a Polynomial by a Monomial] In the following exercises, divide each polynomial by the monomial. #math.equation(block: true, alt: "the fraction 42 z squared minus 18 z over 6")[$frac(42 z^(2) − 18 z, 6)$] #math.equation(block: true, alt: "open parenthesis 35 x squared minus 75 x close parenthesis divided by 5 x")[$( 35 x^(2) − 75 x ) ÷ 5 x$] #solutionbox[ #math.equation(block: true, alt: "7 x minus 15")[$7 x − 15$] ] #math.equation(block: true, alt: "the fraction 81 n to the power 4 plus 105 n squared over −3")[$frac(81 n^(4) + 105 n^(2), −3)$] #math.equation(block: true, alt: "the fraction 550 p to the power 6 minus 300 p to the power 4 over 10 p cubed")[$frac(550 p^(6) − 300 p^(4), 10 p^(3))$] #solutionbox[ #math.equation(block: true, alt: "55 p cubed minus 30 p")[$55 p^(3) − 30 p$] ] #math.equation(block: true, alt: "open parenthesis 63 x y cubed plus 56 x squared y to the power 4 close parenthesis divided by open parenthesis 7 x y close parenthesis")[$( 63 x y^(3) + 56 x^(2) y^(4) ) ÷ ( 7 x y )$] #math.equation(block: true, alt: "the fraction 96 a to the power 5 b squared minus 48 a to the power 4 b cubed minus 56 a squared b to the power 4 over 8 a b squared")[$frac(96 a^(5) b^(2) − 48 a^(4) b^(3) − 56 a^(2) b^(4), 8 a b^(2))$] #solutionbox[ #math.equation(block: true, alt: "12 a to the power 4 minus 6 a cubed b minus 7 a b squared")[$12 a^(4) − 6 a^(3) b − 7 a b^(2)$] ] #math.equation(block: true, alt: "the fraction 57 m squared minus 12 m plus 1 over −3 m")[$frac(57 m^(2) − 12 m + 1, −3 m)$] #math.equation(block: true, alt: "the fraction 105 y to the power 5 plus 50 y cubed minus 5 y over 5 y cubed")[$frac(105 y^(5) + 50 y^(3) − 5 y, 5 y^(3))$] #solutionbox[ #math.equation(block: true, alt: "21 y squared plus 10 minus the fraction 1 over y squared")[$21 y^(2) + 10 − frac(1, y^(2))$] ] #strong[Divide a Polynomial by a Binomial] In the following exercises, divide each polynomial by the binomial. #math.equation(block: true, alt: "open parenthesis k squared minus 2 k minus 99 close parenthesis divided by open parenthesis k plus 9 close parenthesis")[$( k^(2) − 2 k − 99 ) ÷ ( k + 9 )$] #math.equation(block: true, alt: "open parenthesis v squared minus 16 v plus 64 close parenthesis divided by open parenthesis v minus 8 close parenthesis")[$( v^(2) − 16 v + 64 ) ÷ ( v − 8 )$] #solutionbox[ #math.equation(block: true, alt: "v minus 8")[$v − 8$] ] #math.equation(block: true, alt: "open parenthesis 3 x squared minus 8 x minus 35 close parenthesis divided by open parenthesis x minus 5 close parenthesis")[$( 3 x^(2) − 8 x − 35 ) ÷ ( x − 5 )$] #math.equation(block: true, alt: "open parenthesis n squared minus 3 n minus 14 close parenthesis divided by open parenthesis n plus 3 close parenthesis")[$( n^(2) − 3 n − 14 ) ÷ ( n + 3 )$] #solutionbox[ #math.equation(block: true, alt: "n minus 6 plus the fraction 4 over n plus 3")[$n − 6 + frac(4, n + 3)$] ] #math.equation(block: true, alt: "open parenthesis 4 m cubed plus m minus 5 close parenthesis divided by open parenthesis m minus 1 close parenthesis")[$( 4 m^(3) + m − 5 ) ÷ ( m − 1 )$] #math.equation(block: true, alt: "open parenthesis u cubed minus 8 close parenthesis divided by open parenthesis u minus 2 close parenthesis")[$( u^(3) − 8 ) ÷ ( u − 2 )$] #solutionbox[ #math.equation(block: true, alt: "u squared plus 2 u plus 4")[$u^(2) + 2 u + 4$] ] ==== Integer Exponents and Scientific Notation #strong[Use the Definition of a Negative Exponent] In the following exercises, simplify. #math.equation(block: true, alt: "9 to the power −2")[$9^(−2)$] #math.equation(block: true, alt: "open parenthesis −5 close parenthesis to the power −3")[$attach(( −5 ), t: −3)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 125")[$− frac(1, 125)$] ] #math.equation(block: true, alt: "3 times 4 to the power −3")[$3 · 4^(−3)$] #math.equation(block: true, alt: "open parenthesis 6 u close parenthesis to the power −3")[$attach(( 6 u ), t: −3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 216 u cubed")[$frac(1, 216 u^(3))$] ] #math.equation(block: true, alt: "open parenthesis the fraction 2 over 5 close parenthesis to the power −1")[$attach(( frac(2, 5) ), t: −1)$] #math.equation(block: true, alt: "open parenthesis the fraction 3 over 4 close parenthesis to the power −2")[$attach(( frac(3, 4) ), t: −2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 16 over 9")[$frac(16, 9)$] ] #strong[Simplify Expressions with Integer Exponents] In the following exercises, simplify. #math.equation(block: true, alt: "p to the power −2 times p to the power 8")[$p^(−2) · p^(8)$] #math.equation(block: true, alt: "q to the power −6 times q to the power −5")[$q^(−6) · q^(−5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over q to the power 11")[$frac(1, q^(11))$] ] #math.equation(block: true, alt: "open parenthesis c to the power −2 d close parenthesis open parenthesis c to the power −3 d to the power −2 close parenthesis")[$( c^(−2) d ) ( c^(−3) d^(−2) )$] #math.equation(block: true, alt: "open parenthesis y to the power 8 close parenthesis to the power −1")[$attach(( y^(8) ), t: −1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over y to the power 8")[$frac(1, y^(8))$] ] #math.equation(block: true, alt: "open parenthesis q to the power −4 close parenthesis to the power −3")[$attach(( q^(−4) ), t: −3)$] #math.equation(block: true, alt: "the fraction a to the power 8 over a to the power 12")[$frac(a^(8), a^(12))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over a to the power 4")[$frac(1, a^(4))$] ] #math.equation(block: true, alt: "the fraction n to the power 5 over n to the power −4")[$frac(n^(5), n^(−4))$] #math.equation(block: true, alt: "the fraction r to the power −2 over r to the power −3")[$frac(r^(−2), r^(−3))$] #solutionbox[ #math.equation(block: true, alt: "r")[$r$] ] #strong[Convert from Decimal Notation to Scientific Notation] In the following exercises, write each number in scientific notation. 8,500,000 0.00429 #solutionbox[ #math.equation(block: true, alt: "4.29 times 10 to the power −3")[$4.29 #h(0.2em) × #h(0.2em) 10^(−3)$] ] The thickness of a dime is about 0.053 inches. In 2015, the population of the world was about 7,200,000,000 people. #solutionbox[ #math.equation(block: true, alt: "7.2 times 10 to the power 9")[$7.2 #h(0.2em) × #h(0.2em) 10^(9)$] ] #strong[Convert Scientific Notation to Decimal Form] In the following exercises, convert each number to decimal form. #math.equation(block: true, alt: "3.8 times 10 to the power 5")[$3.8 #h(0.2em) × #h(0.2em) 10^(5)$] #math.equation(block: true, alt: "1.5 times 10 to the power 10")[$1.5 #h(0.2em) × #h(0.2em) 10^(10)$] #solutionbox[ #math.equation(block: true, alt: "15,000,000,000")[$15,000,000,000$] ] #math.equation(block: true, alt: "9.1 times 10 to the power −7")[$9.1 #h(0.2em) × #h(0.2em) 10^(−7)$] #math.equation(block: true, alt: "5.5 times 10 to the power −1")[$5.5 #h(0.2em) × #h(0.2em) 10^(−1)$] #solutionbox[ #math.equation(block: true, alt: "0.55")[$0.55$] ] #strong[Multiply and Divide Using Scientific Notation] In the following exercises, multiply and write your answer in decimal form. #math.equation(block: true, alt: "open parenthesis 2 times 10 to the power 5 close parenthesis open parenthesis 4 times 10 to the power −3 close parenthesis")[$( 2 #h(0.2em) × #h(0.2em) 10^(5) ) ( 4 #h(0.2em) × #h(0.2em) 10^(−3) )$] #math.equation(block: true, alt: "open parenthesis 3.5 times 10 to the power −2 close parenthesis open parenthesis 6.2 times 10 to the power −1 close parenthesis")[$( 3.5 #h(0.2em) × #h(0.2em) 10^(−2) ) ( 6.2 #h(0.2em) × #h(0.2em) 10^(−1) )$] #solutionbox[ #math.equation(block: true, alt: "0.0217")[$0.0217$] ] In the following exercises, divide and write your answer in decimal form. #math.equation(block: true, alt: "the fraction 8 times 10 to the power 5 over 4 times 10 to the power −1")[$frac(8 #h(0.2em) × #h(0.2em) 10^(5), 4 #h(0.2em) × #h(0.2em) 10^(−1))$] #math.equation(block: true, alt: "the fraction 9 times 10 to the power −5 over 3 times 10 squared")[$frac(9 #h(0.2em) × #h(0.2em) 10^(−5), 3 #h(0.2em) × #h(0.2em) 10^(2))$] #solutionbox[ #math.equation(block: true, alt: "0.0000003")[$0.0000003$] ] === Chapter Practice Test For the polynomial #math.equation(block: false, alt: "10 x to the power 4 plus 9 y squared minus 1")[$10 x^(4) + 9 y^(2) − 1$] #linebreak() ⓐ Is it a monomial, binomial, or trinomial? #linebreak() ⓑ What is its degree? In the following exercises, simplify each expression. #math.equation(block: true, alt: "open parenthesis 12 a squared minus 7 a plus 4 close parenthesis plus open parenthesis 3 a squared plus 8 a minus 10 close parenthesis")[$( 12 a^(2) − 7 a + 4 ) + ( 3 a^(2) + 8 a − 10 )$] #solutionbox[ #math.equation(block: true, alt: "15 a squared plus a minus 6")[$15 a^(2) + a − 6$] ] #math.equation(block: true, alt: "open parenthesis 9 p squared minus 5 p plus 1 close parenthesis minus open parenthesis 2 p squared minus 6 close parenthesis")[$( 9 p^(2) − 5 p + 1 ) − ( 2 p^(2) − 6 )$] #math.equation(block: true, alt: "open parenthesis minus the fraction 2 over 5 close parenthesis cubed")[$attach(( − frac(2, 5) ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 8 over 125")[$− frac(8, 125)$] ] #math.equation(block: true, alt: "u times u to the power 4")[$u · u^(4)$] #math.equation(block: true, alt: "open parenthesis 4 a cubed b to the power 5 close parenthesis squared")[$attach(( 4 a^(3) b^(5) ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "16 a to the power 6 b to the power 10")[$16 a^(6) b^(10)$] ] #math.equation(block: true, alt: "open parenthesis −9 r to the power 4 s to the power 5 close parenthesis open parenthesis 4 r s to the power 7 close parenthesis")[$( −9 r^(4) s^(5) ) ( 4 r s^(7) )$] #math.equation(block: true, alt: "3 k open parenthesis k squared minus 7 k plus 13 close parenthesis")[$3 k ( k^(2) − 7 k + 13 )$] #solutionbox[ #math.equation(block: true, alt: "3 k cubed minus 21 k squared plus 39 k")[$3 k^(3) − 21 k^(2) + 39 k$] ] #math.equation(block: true, alt: "open parenthesis m plus 6 close parenthesis open parenthesis m plus 12 close parenthesis")[$( m + 6 ) ( m + 12 )$] #math.equation(block: true, alt: "open parenthesis v minus 9 close parenthesis open parenthesis 9 v minus 5 close parenthesis")[$( v − 9 ) ( 9 v − 5 )$] #solutionbox[ #math.equation(block: true, alt: "9 v squared minus 86 v plus 45")[$9 v^(2) − 86 v + 45$] ] #math.equation(block: true, alt: "open parenthesis 4 c minus 11 close parenthesis open parenthesis 3 c minus 8 close parenthesis")[$( 4 c − 11 ) ( 3 c − 8 )$] #math.equation(block: true, alt: "open parenthesis n minus 6 close parenthesis open parenthesis n squared minus 5 n plus 4 close parenthesis")[$( n − 6 ) ( n^(2) − 5 n + 4 )$] #solutionbox[ #math.equation(block: true, alt: "n cubed minus 11 n squared plus 34 n minus 24")[$n^(3) − 11 n^(2) + 34 n − 24$] ] #math.equation(block: true, alt: "open parenthesis 2 x minus 15 y close parenthesis open parenthesis 5 x plus 7 y close parenthesis")[$( 2 x − 15 y ) ( 5 x + 7 y )$] #math.equation(block: true, alt: "open parenthesis 7 p minus 5 close parenthesis open parenthesis 7 p plus 5 close parenthesis")[$( 7 p − 5 ) ( 7 p + 5 )$] #solutionbox[ #math.equation(block: true, alt: "49 p squared minus 25")[$49 p^(2) − 25$] ] #math.equation(block: true, alt: "open parenthesis 9 v minus 2 close parenthesis squared")[$attach(( 9 v − 2 ), t: 2)$] #math.equation(block: true, alt: "the fraction 3 to the power 8 over 3 to the power 10")[$frac(3^(8), 3^(10))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 9")[$frac(1, 9)$] ] #math.equation(block: true, alt: "open parenthesis the fraction m to the power 4 times m over m cubed close parenthesis to the power 6")[$attach(( frac(m^(4) · m, m^(3)) ), t: 6)$] #math.equation(block: true, alt: "open parenthesis 87 x to the power 15 y cubed z to the power 22 close parenthesis to the power 0")[$attach(( 87 x^(15) y^(3) z^(22) ), t: 0)$] #solutionbox[ #math.equation(block: true, alt: "1")[$1$] ] #math.equation(block: true, alt: "the fraction 80 c to the power 8 d squared over 16 c d to the power 10")[$frac(80 c^(8) d^(2), 16 c d^(10))$] #math.equation(block: true, alt: "the fraction 12 x squared plus 42 x minus 6 over 2 x")[$frac(12 x^(2) + 42 x − 6, 2 x)$] #solutionbox[ #math.equation(block: true, alt: "6 x plus 21 minus the fraction 3 over x")[$6 x + 21 − frac(3, x)$] ] #math.equation(block: true, alt: "open parenthesis 70 x y to the power 4 plus 95 x cubed y close parenthesis divided by 5 x y")[$( 70 x y^(4) + 95 x^(3) y ) ÷ 5 x y$] #math.equation(block: true, alt: "the fraction 64 x cubed minus 1 over 4 x minus 1")[$frac(64 x^(3) − 1, 4 x − 1)$] #solutionbox[ #math.equation(block: true, alt: "16 x squared plus 4 x plus 1")[$16 x^(2) + 4 x + 1$] ] #math.equation(block: true, alt: "open parenthesis y squared minus 5 y minus 18 close parenthesis divided by open parenthesis y plus 3 close parenthesis")[$( y^(2) − 5 y − 18 ) ÷ ( y + 3 )$] #math.equation(block: true, alt: "5 to the power −2")[$5^(−2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 25")[$frac(1, 25)$] ] #math.equation(block: true, alt: "open parenthesis 4 m close parenthesis to the power −3")[$attach(( 4 m ), t: −3)$] #math.equation(block: true, alt: "q to the power −4 times q to the power −5")[$q^(−4) · q^(−5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over q to the power 9")[$frac(1, q^(9))$] ] #math.equation(block: true, alt: "the fraction n to the power −2 over n to the power −10")[$frac(n^(−2), n^(−10))$] Convert 83,000,000 to scientific notation. #solutionbox[ #math.equation(block: true, alt: "8.3 times 10 to the power 7")[$8.3 #h(0.2em) × #h(0.2em) 10^(7)$] ] Convert #math.equation(block: false, alt: "6.91 times 10 to the power −5")[$6.91 #h(0.2em) × #h(0.2em) 10^(−5)$] to decimal form. In the following exercises, simplify, and write your answer in decimal form. #math.equation(block: true, alt: "open parenthesis 3.4 times 10 to the power 9 close parenthesis open parenthesis 2.2 times 10 to the power −5 close parenthesis")[$( 3.4 #h(0.2em) × #h(0.2em) 10^(9) ) ( 2.2 #h(0.2em) × #h(0.2em) 10^(−5) )$] #solutionbox[ 74,800 ] #math.equation(block: true, alt: "the fraction 8.4 times 10 to the power −3 over 4 times 10 cubed")[$frac(8.4 #h(0.2em) × #h(0.2em) 10^(−3), 4 #h(0.2em) × #h(0.2em) 10^(3))$] A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial #math.equation(block: false, alt: "−16 t squared plus 1000")[$−16 t^(2) + 1000$] gives the height of the package #math.equation(block: false, alt: "t")[$t$] seconds a after it was dropped. Find the height when #math.equation(block: false, alt: "t equals 6")[$t = 6$] seconds. #solutionbox[ 424 feet ]