#set document(title: "10.5 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")) == 10.5#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 thousand ] ] #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 The #strong[Quotient Property of Exponents], introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Quotient Property of 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 , n")[$m , #h(0.2em) n$] 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")[$frac(a^(m), a^(n)) = a^(m − n) , #h(1.5em) m > n #h(1.5em) "and" #h(1.5em) frac(a^(m), a^(n)) = frac(1, a^(n − m)) , #h(1.5em) n > m$] ] 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")[$frac(x^(2), x^(5))$]#math.equation(block: true, alt: "x to the power 2 minus 5")[$x^(2 − 5)$]#math.equation(block: true, alt: "x to the power −3")[$x^(−3)$]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: #math.equation(block: false, alt: "the fraction x squared over x to the power 5 .")[$frac(x^(2), x^(5)) .$] #figure(figph[A fraction is shown. The numerator is x times x, the denominator is x times x times x times x times x. Two x's are crossed out in red on the top and on the bottom. Below that, the fraction 1 over x cubed is shown.], alt: "A fraction is shown. The numerator is x times x, the denominator is x times x times x times x times x. Two x's are crossed out in red on the top and on the bottom. Below that, the fraction 1 over x cubed is shown.", 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 #strong[negative exponent]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Negative Exponent] 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: "a to the power minus n equals the fraction 1 over a to the power n .")[$a^(− n) = frac(1, a^(n)) .$] ] The negative exponent tells us to re-write the expression by taking the #strong[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 negative exponent and other properties of exponents to write an expression with only positive exponents. #examplebox("Example 1")[][ Simplify: + ⓐ #math.equation(block: false, alt: "4 to the power −2")[$#h(0.2em) 4^(−2)$] + ⓑ #math.equation(block: false, alt: "10 to the power −3")[$#h(0.2em) 10^(−3)$] #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 minus 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")[$frac(1, 16)$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [], [#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 minus 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")[$#h(0.2em) 2^(−3)$] + ⓑ #math.equation(block: false, alt: "10 to the power −2")[$#h(0.2em) 10^(−2)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 8")[$#h(0.2em) frac(1, 8)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 100")[$#h(0.2em) frac(1, 100)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: + ⓐ #math.equation(block: false, alt: "3 to the power −2")[$#h(0.2em) 3^(−2)$] + ⓑ #math.equation(block: false, alt: "10 to the power −4")[$#h(0.2em) 10^(−4)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 9")[$#h(0.2em) frac(1, 9)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 10,000")[$#h(0.2em) frac(1, 10","000)$] ] ] When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent. #examplebox("Example 2")[][ Simplify: + ⓐ #math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power −2")[$#h(0.2em) attach(( −3 ), t: −2)$] + ⓑ #math.equation(block: false, alt: "−3 to the power −2")[$#h(0.2em) −3^(−2)$] #solutionbox[ The negative in the exponent does not affect the sign of the base. #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [The exponent applies to the base, #math.equation(block: false, alt: "minus 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")[$frac(1, 9)$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [The expression #math.equation(block: false, alt: "minus 3 to the power −2")[$− 3^(−2)$] means "find the opposite of #math.equation(block: false, alt: "3 to the power −2")[$3^(−2)$]". #linebreak() The exponent applies only to the base, 3.], [#math.equation(block: false, alt: "minus 3 to the power −2")[$− 3^(−2)$]], [Rewrite as a product with −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")[$− frac(1, 9)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: + ⓐ #math.equation(block: false, alt: "open parenthesis −5 close parenthesis to the power −2")[$#h(0.2em) attach(( −5 ), t: −2)$] + ⓑ #math.equation(block: false, alt: "minus 5 to the power −2")[$#h(0.2em) − 5^(−2)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 25")[$#h(0.2em) frac(1, 25)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 25")[$#h(0.2em) − frac(1, 25)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: + ⓐ #math.equation(block: false, alt: "open parenthesis −2 close parenthesis to the power −2")[$#h(0.2em) attach(( −2 ), t: −2)$] + ⓑ #math.equation(block: false, alt: "−2 to the power −2")[$#h(0.2em) −2^(−2)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 4")[$#h(0.2em) frac(1, 4)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 4")[$#h(0.2em) − frac(1, 4)$] ] ] We must be careful to follow the #strong[order of operations]. In the next example, parts ⓐ and ⓑ look similar, but we get different results. #examplebox("Example 3")[][ Simplify: + ⓐ #math.equation(block: false, alt: "4 times 2 to the power −1")[$#h(0.2em) 4 · 2^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 4 times 2 close parenthesis to the power −1")[$#h(0.2em) attach(( 4 · 2 ), t: −1)$] #solutionbox[ Remember to always follow the order of operations. #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [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 minus 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")[$2$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], [#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 minus 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")[$#h(0.2em) 6 · 3^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 6 times 3 close parenthesis to the power −1")[$#h(0.2em) attach(( 6 · 3 ), t: −1)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "2")[$#h(0.2em) #h(0.2em) 2$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 18")[$#h(0.2em) frac(1, 18)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: + ⓐ #math.equation(block: false, alt: "8 times 2 to the power −2")[$#h(0.2em) 8 · 2^(−2)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 8 times 2 close parenthesis to the power −2")[$#h(0.2em) attach(( 8 · 2 ), t: −2)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "2")[$#h(0.2em) 2$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 256")[$#h(0.2em) frac(1, 256)$] ] ] When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. #examplebox("Example 4")[][ Simplify: #math.equation(block: false, alt: "x to the power −6 .")[$x^(−6) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "x to the power −6")[$x^(−6)$]]), [Use the definition of a negative exponent, #math.equation(block: false, alt: "a to the power minus 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 x to the power 6")[$frac(1, x^(6))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "y to the power −7 .")[$y^(−7) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over y to the power 7")[$frac(1, y^(7))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "z to the power −8 .")[$z^(−8) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over z to the power 8")[$frac(1, z^(8))$] ] ] When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the #strong[order of operations], expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example. #examplebox("Example 5")[][ Simplify: + ⓐ #math.equation(block: false, alt: "5 y to the power −1")[$#h(0.2em) 5 y^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 5 y close parenthesis to the power −1")[$#h(0.2em) attach(( 5 y ), t: −1)$] + ⓒ #math.equation(block: false, alt: "open parenthesis −5 y close parenthesis to the power −1")[$#h(0.2em) attach(( −5 y ), t: −1)$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [Notice the exponent applies to just the base #math.equation(block: false, alt: "y")[$y$].], [#math.equation(block: false, alt: "5 y to the power −1")[$5 y^(−1)$]], [Take the reciprocal of #math.equation(block: false, alt: "y")[$y$] 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")[$frac(5, y)$]], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [Here the parentheses make the exponent apply to the base #math.equation(block: false, alt: "5 y")[$5 y$].], [#math.equation(block: false, alt: "open parenthesis 5 y close parenthesis to the power −1")[$attach(( 5 y ), t: −1)$]], [Take the reciprocal of #math.equation(block: false, alt: "5 y")[$5 y$] 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")[$frac(1, 5 y)$]], )) #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 is #math.equation(block: false, alt: "minus 5 y")[$− 5 y$]. Take the reciprocal of #math.equation(block: false, alt: "minus 5 y")[$− 5 y$] 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 minus 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")[$#h(0.2em) 8 p^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 8 p close parenthesis to the power −1")[$#h(0.2em) attach(( 8 p ), t: −1)$] + ⓒ #math.equation(block: false, alt: "open parenthesis −8 p close parenthesis to the power −1")[$#h(0.2em) attach(( −8 p ), t: −1)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 8 over p")[$#h(0.2em) frac(8, p)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 8 p")[$#h(0.2em) frac(1, 8 p)$] + ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 8 p")[$#h(0.2em) − frac(1, 8 p)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: + ⓐ #math.equation(block: false, alt: "11 q to the power −1")[$#h(0.2em) 11 q^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 11 q close parenthesis to the power −1")[$#h(0.2em) attach(( 11 q ), t: −1)$] + ⓒ #math.equation(block: false, alt: "open parenthesis −11 q close parenthesis to the power −1")[$#h(0.2em) attach(( −11 q ), t: −1)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 11 over q")[$#h(0.2em) frac(11, q)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 11 q")[$#h(0.2em) frac(1, 11 q)$] + ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 11 q")[$#h(0.2em) − frac(1, 11 q)$] ] ] Now that we have defined negative exponents, the #strong[Quotient Property of Exponents] 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 ,")[$frac(a^(m), a^(n)) = a^(m − n) ,$] where #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$] and #emph[m] and #emph[n] are integers. When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, #math.equation(block: false, alt: "a to the power minus n equals the fraction 1 over a to the power n .")[$a^(− n) = frac(1, a^(n)) .$] === Simplify Expressions with Integer Exponents All the exponent properties we developed earlier in this 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 , b")[$a , b$] are real numbers and #math.equation(block: false, alt: "m , n")[$m , n$] 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 Property, 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; Definition of Negative Exponent, a to the power minus n equals the fraction 1 over a to the pow")[$"Product Property" & & & a^(m) · a^(n) = a^(m + n) \ "Power Property" & & & attach(( a^(m) ), t: n) = a^(m · n) \ "Product to a Power Property" & & & attach(( a b ), t: m) = a^(m) b^(m) \ "Quotient Property" & & & frac(a^(m), a^(n)) = a^(m − n) , #h(0.2em) a ≠ 0 \ "Zero Exponent Property" & & & a^(0) = 1 , #h(0.2em) a ≠ 0 \ "Quotient to a Power Property" & & & attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m)) , #h(0.2em) b ≠ 0 \ "Definition of Negative Exponent" & & & a^(− n) = frac(1, a^(n))$] ] #examplebox("Example 6")[][ Simplify: + ⓐ #math.equation(block: false, alt: "x to the power −4 times x to the power 6")[$#h(0.2em) x^(−4) · x^(6)$] + ⓑ #math.equation(block: false, alt: "y to the power −6 times y to the power 4")[$#h(0.2em) y^(−6) · y^(4)$] + ⓒ #math.equation(block: false, alt: "z to the power −5 times z to the power −3")[$#h(0.2em) z^(−5) · z^(−3)$] #solutionbox[ #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")[$x^(2)$]], )) #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")[$y^(−6) · y^(4)$]], [The bases are the same, 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 minus 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")[$frac(1, y^(2))$]], )) #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")[$z^(−5) · z^(−3)$]], [The bases are the same, so add the exponents.], [#math.equation(block: false, alt: "z to the power −5−3")[$z^(−5−3)$]], [Simplify.], [#math.equation(block: false, alt: "z to the power −8")[$z^(−8)$]], [Use the definition of a negative exponent, #math.equation(block: false, alt: "a to the power minus 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 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")[$#h(0.2em) x^(−3) · x^(7)$] + ⓑ #math.equation(block: false, alt: "y to the power −7 times y squared")[$#h(0.2em) y^(−7) · y^(2)$] + ⓒ #math.equation(block: false, alt: "z to the power −4 times z to the power −5")[$#h(0.2em) z^(−4) · z^(−5)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "x to the power 4")[$#h(0.2em) x^(4)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over y to the power 5")[$#h(0.2em) frac(1, y^(5))$] + ⓒ #math.equation(block: false, alt: "the fraction 1 over z to the power 9")[$#h(0.2em) frac(1, z^(9))$] ] ] #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")[$#h(0.2em) a^(−1) · a^(6)$] + ⓑ #math.equation(block: false, alt: "b to the power −8 times b to the power 4")[$#h(0.2em) b^(−8) · b^(4)$] + ⓒ #math.equation(block: false, alt: "c to the power −8 times c to the power −7")[$#h(0.2em) c^(−8) · c^(−7)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "a to the power 5")[$#h(0.2em) a^(5)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over b to the power 4")[$#h(0.2em) frac(1, b^(4))$] + ⓒ #math.equation(block: false, alt: "the fraction 1 over c to the power 15")[$#h(0.2em) frac(1, c^(15))$] ] ] In the next two examples, we’ll start by using the #strong[Commutative Property] to group the same variables together. This makes it easier to identify the like bases before using the #strong[Product Property of Exponents]. #examplebox("Example 7")[][ 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")[$frac(1, r^(2) s^(8))$] ] ] If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Use Multiplication Properties of Exponents. #examplebox("Example 8")[][ Simplify: #math.equation(block: false, alt: "open parenthesis 2 x to the power −6 y to the power 8 close parenthesis open parenthesis −5 x to the power 5 y to the power −3 close parenthesis .")[$( 2 x^(−6) y^(8) ) ( −5 x^(5) y^(−3) ) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis 2 x to the power −6 y to the power 8 close parenthesis open parenthesis −5 x to the power 5 y to the power −3 close parenthesis")[$( 2 x^(−6) y^(8) ) ( −5 x^(5) y^(−3) )$]]), [Rewrite with the like bases together.], [#math.equation(block: false, alt: "2 open parenthesis −5 close parenthesis times open parenthesis x to the power −6 x to the power 5 close parenthesis times open parenthesis y to the power 8 y to the power −3 close parenthesis")[$2 ( −5 ) · ( x^(−6) x^(5) ) · ( y^(8) y^(−3) )$]], [Simplify.], [#math.equation(block: false, alt: "−10 times x to the power −1 times y to the power 5")[$−10 · x^(−1) · y^(5)$]], [Use the definition of a negative exponent, #math.equation(block: false, alt: "a to the power minus n equals the fraction 1 over a to the power n .")[$a^(− n) = frac(1, a^(n)) .$]], [#math.equation(block: false, alt: "−10 times the fraction 1 over x to the power 1 times y to the power 5")[$−10 · frac(1, x^(1)) · y^(5)$]], [Simplify.], [#math.equation(block: false, alt: "the fraction −10 y to the power 5 over x")[$frac(−10 y^(5), x)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis 3 u to the power −5 v to the power 7 close parenthesis open parenthesis −4 u to the power 4 v to the power −2 close parenthesis .")[$( 3 u^(−5) v^(7) ) ( −4 u^(4) v^(−2) ) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 12 v to the power 5 over u")[$− frac(12 v^(5), u)$] ] ] #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 #strong[Power Property] and the #strong[Product to a Power Property]. #examplebox("Example 9")[][ Simplify: #math.equation(block: false, alt: "open parenthesis k cubed close parenthesis to the power −2 .")[$attach(( k^(3) ), t: −2) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis k cubed close parenthesis to the power −2")[$attach(( k^(3) ), t: −2)$]]), [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: "k to the power 3 open parenthesis −2 close parenthesis")[$k^(3 ( −2 ))$]], [Simplify.], [#math.equation(block: false, alt: "k to the power −6")[$k^(−6)$]], [Rewrite with a positive exponent.], [#math.equation(block: false, alt: "the fraction 1 over k to the power 6")[$frac(1, k^(6))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis x to the power 4 close parenthesis to the power −1 .")[$attach(( x^(4) ), t: −1) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over x to the power 4")[$frac(1, x^(4))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis y squared close parenthesis to the power −2 .")[$attach(( y^(2) ), t: −2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over y to the power 4")[$frac(1, y^(4))$] ] ] #examplebox("Example 10")[][ 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 #math.equation(block: false, alt: "5 squared")[$5^(2)$] and multiply the exponents of #math.equation(block: false, alt: "x")[$x$] using the #linebreak() 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: "25 x to the power −6")[$25 x^(−6)$]], [Rewrite #math.equation(block: false, alt: "x to the power −6")[$x^(−6)$] by using the definition of a negative #linebreak() exponent, #math.equation(block: false, alt: "a to the power minus 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 #strong[Quotient Property]. #examplebox("Example 11")[][ 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([], [#figure(figph[The exponent expression r^5 over r^-4.], alt: "The exponent expression r^5 over r^-4.", caption: none)]), [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")[$frac(a^(m), a^(n)) = a^(m − n)$].], [#figure(figph[A mathematical expression reads 'r to the power of 5 minus negative 4,' with '-4' highlighted in red.], alt: "A mathematical expression reads 'r to the power of 5 minus negative 4,' with '-4' highlighted in red.", caption: none)], [#figure(figph[The text reads 'Be careful to subtract 5 - (-4).', highlighting the subtraction of a negative number.], alt: "The text reads 'Be careful to subtract 5 - (-4).', highlighting the subtraction of a negative number.", caption: none)], [], [Simplify.], [#figure(figph[The expression r^9 is displayed on a white background.], alt: "The expression r^9 is displayed on a white background.", caption: none)], )) ] ] #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[ #emph[x]#super[11] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction y to the power 7 over y to the power −6 .")[$frac(y^(7), y^(−6)) .$] #solutionbox[ #emph[y]#super[13] ] ] === 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 #math.equation(block: false, alt: "10 .")[$10 .$] 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 #math.equation(block: false, alt: "4000")[$4000$] and #math.equation(block: false, alt: "0.004 .")[$0.004 .$] We know that #math.equation(block: false, alt: "4000")[$4000$] means #math.equation(block: false, alt: "4 times 1000")[$4 #h(0.2em) × #h(0.2em) 1000$] and #math.equation(block: false, alt: "0.004")[$0.004$] means #math.equation(block: false, alt: "4 times the fraction 1 over 1000 .")[$4 #h(0.2em) × #h(0.2em) frac(1, 1000) .$] If we write the #math.equation(block: false, alt: "1000")[$1000$] as a power of ten in exponential form, we can rewrite these numbers in this way: #math.equation(block: true, alt: "4000, , 0.004; 4 times 1000, , 4 times the fraction 1 over 1000; 4 times 10 cubed, , 4 times the fraction 1 over 10 cubed; , 4 times 10 to the power −3")[$4000 & #h(2em) & & 0.004 \ 4 #h(0.2em) × #h(0.2em) 1000 & #h(2em) & & 4 #h(0.2em) × #h(0.2em) frac(1, 1000) \ 4 #h(0.2em) × #h(0.2em) 10^(3) & #h(2em) & & 4 #h(0.2em) × #h(0.2em) frac(1, 10^(3)) \ & #h(2em) & & 4 #h(0.2em) × #h(0.2em) 10^(−3)$]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 #math.equation(block: false, alt: "10 ,")[$10 ,$] and the second factor is a power of #math.equation(block: false, alt: "10")[$10$] 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")[$a #h(0.2em) × #h(0.2em) 10^(n)$]where #math.equation(block: false, alt: "a greater than or equal to 1")[$a ≥ 1$] and #math.equation(block: false, alt: "a less than 10")[$a < 10$] and #math.equation(block: false, alt: "n")[$n$] 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. Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier. If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation. #figure(figph[On the left, we see 4000 equals 4 times 10 cubed. Beneath that is the same thing, but there is an arrow from after the last 0 in 4000 to between the 4 and the first 0. Beneath, it says, “Moved the decimal point 3 places to the left.” On the right, we see 0.004 equals 4 times 10 to the negative 3. Beneath that is the same thing, but there is an arrow from the decimal point to after the 4. Beneath, it says, “Moved the decimal point 3 places to the right.”], alt: "On the left, we see 4000 equals 4 times 10 cubed. Beneath that is the same thing, but there is an arrow from after the last 0 in 4000 to between the 4 and the first 0. Beneath, it says, “Moved the decimal point 3 places to the left.” On the right, we see 0.004 equals 4 times 10 to the negative 3. Beneath that is the same thing, but there is an arrow from the decimal point to after the 4. Beneath, it says, “Moved the decimal point 3 places to the right.”", caption: none) In both cases, the decimal was moved #math.equation(block: false, alt: "3")[$3$] places to get the first factor, #math.equation(block: false, alt: "4 ,")[$4 ,$] by itself. - The power of #math.equation(block: false, alt: "10")[$10$] is positive when the number is larger than #math.equation(block: false, alt: "1 : 4000 equals 4 times 10 cubed .")[$1 ":" #h(0.2em) 4000 = 4 #h(0.2em) × #h(0.2em) 10^(3) .$] - The power of #math.equation(block: false, alt: "10")[$10$] is negative when the number is between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "1 : 0.004 equals 4 times 10 to the power minus 3 .")[$1 ":" #h(0.2em) 0.004 = 4 #h(0.2em) × #h(0.2em) 10^(− 3) .$] #examplebox("Example 12")[][ Write #math.equation(block: false, alt: "37,000")[$37,000$] in scientific notation. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([#emph[Step 1]: Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.], [#figure(figph[The number 37000. is displayed in black text. Underneath the last three zeros of the number, there are three small, light blue wavy arrows pointing downwards, indicating a numerical operation or a specific feature of the zeros.], alt: "The number 37000. is displayed in black text. Underneath the last three zeros of the number, there are three small, light blue wavy arrows pointing downwards, indicating a numerical operation or a specific feature of the zeros.", caption: none)]), [#emph[Step 2]: Count the number of decimal places, #math.equation(block: false, alt: "n")[$n$], that the decimal point was moved.], [3.70000 #linebreak() 4 places], [#emph[Step 3]: Write the number as a product with a power of 10.], [#math.equation(block: false, alt: "3.7 times 10 to the power 4")[$3.7 × 10^(4)$]], [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")[$10^("−n")$]], [], [#emph[Step 4]: Check.], [], [#math.equation(block: false, alt: "10 to the power 4")[$10^(4)$] is 10,000 and 10,000 times 3.7 will be 37,000.], [], [], [#math.equation(block: false, alt: "37,000 equals 3.7 times 10 to the power 4")[$37,000 = 3.7 × 10^(4)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in scientific notation: #math.equation(block: false, alt: "96,000 .")[$96,000 .$] #solutionbox[ 9.6 × 10#super[4] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in scientific notation: #math.equation(block: false, alt: "48,300 .")[$48,300 .$] #solutionbox[ 4.83 × 10#super[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 #math.equation(block: false, alt: "1")[$1$] but less than #math.equation(block: false, alt: "10 .")[$10 .$] + 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 #math.equation(block: false, alt: "10 .")[$10 .$] - If the original number is: - greater than #math.equation(block: false, alt: "1 ,")[$1 ,$] the power of #math.equation(block: false, alt: "10")[$10$] will be #math.equation(block: false, alt: "10 to the power n .")[$10^(n) .$] - between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "1 ,")[$1 ,$] the power of #math.equation(block: false, alt: "10")[$10$] will be #math.equation(block: false, alt: "10 to the power minus n .")[$10^(− n) .$] + Check. ] #examplebox("Example 13")[][ Write in scientific notation: #math.equation(block: false, alt: "0.0052 .")[$0.0052 .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [0.0052]), [Move the decimal point to get 5.2, a number between 1 and 10.], [#figure(figph[The number 0.0052 is shown with three blue wavy arrows underneath, indicating a shift of the decimal point three places to the right.], alt: "The number 0.0052 is shown with three blue wavy arrows underneath, indicating a shift of the decimal point three places to the right.", caption: none)], [Count the number of decimal places the point was moved.], [3 places], [Write as a product with a power of 10.], [#math.equation(block: false, alt: "5.2 times 10 to the power −3")[$5.2 × 10^(−3)$]], [Check your answer: #linebreak() #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; 0.0052")[$5.2 #h(0.2em) × #h(0.2em) 10^(−3) \ 5.2 #h(0.2em) × #h(0.2em) frac(1, 10^(3)) #h(0.35em) \ \ \ #h(0.15em) 5.2 #h(0.2em) × #h(0.2em) frac(1, 1000) \ #h(0.25em) 5.2 #h(0.2em) × #h(0.2em) 0.001 \ 0.0052$]], [], [], [#math.equation(block: false, alt: "0.0052 equals 5.2 times 10 to the power −3")[$0.0052 = 5.2 #h(0.2em) × #h(0.2em) 10^(−3)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in scientific notation: #math.equation(block: false, alt: "0.0078 .")[$0.0078 .$] #solutionbox[ 7.8 × 10#super[−3] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Write in scientific notation: #math.equation(block: false, alt: "0.0129 .")[$0.0129 .$] #solutionbox[ 1.29 × 10#super[−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")[$9.12 #h(0.2em) × #h(0.2em) 10^(4) & #h(2em) & & 9.12 #h(0.2em) × #h(0.2em) 10^(−4) \ 9.12 #h(0.2em) × #h(0.2em) 10,000 & #h(2em) & & 9.12 #h(0.2em) × #h(0.2em) 0.0001 \ 91,200 & #h(2em) & & 0.000912$]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. #figure(figph[On the left, we see 9.12 times 10 to the 4th equals 91,200. Beneath that is 9.12 followed by 2 spaces, with an arrow from the decimal to after the second space, times 10 to the 4th equals 91,200. On the right, we see 9.12 times 10 to the negative 4 equals 0.000912. Beneath that is three spaces followed by 9.12 with an arrow from the decimal to after the first space, times 10 to the negative 4 equals 0.000912.], alt: "On the left, we see 9.12 times 10 to the 4th equals 91,200. Beneath that is 9.12 followed by 2 spaces, with an arrow from the decimal to after the second space, times 10 to the 4th equals 91,200. On the right, we see 9.12 times 10 to the negative 4 equals 0.000912. Beneath that is three spaces followed by 9.12 with an arrow from the decimal to after the first space, times 10 to the negative 4 equals 0.000912.", 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 14")[][ 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(table( columns: 2, align: left, inset: 6pt, table.header([#emph[Step 1]: Determine the exponent, #math.equation(block: false, alt: "n")[$n$], on the factor 10.], [#math.equation(block: false, alt: "6.2 times 10 cubed")[$6.2 × 10^(3)$]]), [#emph[Step 2]: Move the decimal point #math.equation(block: false, alt: "n")[$n$] places, adding zeros if needed.], [#figure(figph[The number 6.200 is shown with three blue curved arrows pointing left under the digits '200', visually representing a three-place decimal shift or counting of digits.], alt: "The number 6.200 is shown with three blue curved arrows pointing left under the digits '200', visually representing a three-place decimal shift or counting of digits.", caption: none)], [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: "| n |")[$| n |$] places to the left.], [6,200], [#emph[Step 3]: Check to see if your answer makes sense.], [], [#math.equation(block: false, alt: "10 cubed")[$10^(3)$] is 1000 and 1000 times 6.2 will be 6,200.], [#math.equation(block: false, alt: "6.2 times 10 cubed equals 6,200")[$6.2 × 10^(3) = 6,200$]], )) ] ] #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 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Convert scientific notation to decimal form.] + Determine the exponent, #math.equation(block: false, alt: "n ,")[$n ,$] on the factor #math.equation(block: false, alt: "10 .")[$10 .$] + 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: "| n |")[$| n |$] places to the left. + Check. ] #examplebox("Example 15")[][ 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([], [#math.equation(block: false, alt: "8.9 times 10 to the power −2")[$8.9 × 10^(−2)$]]), [Determine the exponent #math.equation(block: false, alt: "n")[$n$], on the factor 10.], [The exponent is −2.], [Move the decimal point 2 places to the left.], [#figure(figph[The number -8.9 is displayed in black text on a white background, with light blue wavy lines underneath the -8 portion of the number, suggesting a focus or grouping on the integer part.], alt: "The number -8.9 is displayed in black text on a white background, with light blue wavy lines underneath the -8 portion of the number, suggesting a focus or grouping on the integer part.", caption: none)], [Add zeros as needed for placeholders.], [0.089], [], [#math.equation(block: false, alt: "8.9 times 10 to the power −2 equals 0.089")[$8.9 × 10^(−2) = 0.089$]], [The Check is left to you.], [], )) ] ] #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 We use the Properties of Exponents to multiply and divide numbers in scientific notation. #examplebox("Example 16")[][ 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[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#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")[$( 4 #h(0.2em) × #h(0.2em) 10^(5) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−7) )$]]), [Use the Commutative Property to rearrange the factors.], [#math.equation(block: false, alt: "4 times 2 times 10 to the power 5 times 10 to the power −7")[$4 · 2 · 10^(5) · 10^(−7)$]], [Multiply 4 by 2 and use the Product Property to multiply #math.equation(block: false, alt: "10 to the power 5")[$10^(5)$] by #math.equation(block: false, alt: "10 to the power −7")[$10^(−7)$].], [#math.equation(block: false, alt: "8 times 10 to the power −2")[$8 #h(0.2em) × #h(0.2em) 10^(−2)$]], [Change to decimal form by moving the decimal two places left.], [#math.equation(block: false, alt: "0.08")[$0.08$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Multiply. Write answers in decimal form: #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 .")[$( 3 #h(0.2em) × #h(0.2em) 10^(6) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−8) ) .$] #solutionbox[ 0.06 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Multiply. Write answers in decimal form: #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 .")[$( 3 #h(0.2em) × #h(0.2em) 10^(−2) ) ( 3 #h(0.2em) × #h(0.2em) 10^(−1) ) .$] #solutionbox[ 0.009 ] ] #examplebox("Example 17")[][ 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.], [#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 9 by 3 and use the Quotient Property to divide #math.equation(block: false, alt: "10 cubed")[$10^(3)$] by #math.equation(block: false, alt: "10 to the power −2")[$10^(−2)$].], [#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.], [#math.equation(block: false, alt: "300,000")[$300,000$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Divide. Write answers in decimal form: #math.equation(block: false, alt: "the fraction 8 times 10 to the power 4 over 2 times 10 to the power −1 .")[$frac(8 #h(0.2em) × #h(0.2em) 10^(4), 2 #h(0.2em) × #h(0.2em) 10^(−1)) .$] #solutionbox[ 400,000 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Divide. Write answers in decimal form: #math.equation(block: false, alt: "the fraction 8 times 10 squared over 4 times 10 to the power −2 .")[$frac(8 #h(0.2em) × #h(0.2em) 10^(2), 4 #h(0.2em) × #h(0.2em) 10^(−2)) .$] #solutionbox[ 20,000 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://www.openstax.org/l/24negexponents")[Negative Exponents] - #link("https://www.openstax.org/l/24simpexprnegex")[Examples of Simplifying Expressions with Negative Exponents] - #link("https://www.openstax.org/l/24scnotation")[Scientific Notation] ] === Key Concepts - #emph[Summary of Exponent Properties] - If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers and #math.equation(block: false, alt: "m , n")[$m , n$] 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 Property, 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; Definition of Negative Exponent, a to the power minus n equals the fraction 1 over a to the pow")[$"Product Property" & & & a^(m) · a^(n) = a^(m + n) \ "Power Property" & & & attach(( a^(m) ), t: n) = a^(m · n) \ "Product to a Power Property" & & & attach(( a b ), t: m) = a^(m) b^(m) \ "Quotient Property" & & & frac(a^(m), a^(n)) = a^(m − n) , #h(0.2em) a ≠ 0 \ "Zero Exponent Property" & & & a^(0) = 1 , #h(0.2em) a ≠ 0 \ "Quotient to a Power Property" & & & attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m)) , #h(0.2em) b ≠ 0 \ "Definition of Negative Exponent" & & & a^(− n) = frac(1, a^(n))$] - #emph[Convert from Decimal Notation to Scientific Notation:] 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")[$10^(n)$]. - If the original number is between 0 and 1, the power of 10 will be #math.equation(block: false, alt: "10 to the power - n")[$10^(- n)$]. + Check. - #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: "| n |")[$| n |$] places to the left. + Check. ==== Practice Makes Perfect #strong[Use the Definition of a Negative Exponent] In the following exercises, simplify. #math.equation(block: true, alt: "5 to the power −3")[$5^(−3)$] #math.equation(block: true, alt: "8 to the power −2")[$8^(−2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 64")[$frac(1, 64)$] ] #math.equation(block: true, alt: "3 to the power −4")[$3^(−4)$] #math.equation(block: true, alt: "2 to the power −5")[$2^(−5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 32")[$frac(1, 32)$] ] #math.equation(block: true, alt: "7 to the power −1")[$7^(−1)$] #math.equation(block: true, alt: "10 to the power −1")[$10^(−1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 10")[$frac(1, 10)$] ] #math.equation(block: true, alt: "2 to the power −3 plus 2 to the power −2")[$2^(−3) + 2^(−2)$] #math.equation(block: true, alt: "3 to the power −2 plus 3 to the power −1")[$3^(−2) + 3^(−1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 9")[$frac(4, 9)$] ] #math.equation(block: true, alt: "3 to the power −1 plus 4 to the power −1")[$3^(−1) + 4^(−1)$] #math.equation(block: true, alt: "10 to the power −1 plus 2 to the power −1")[$10^(−1) + 2^(−1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 5")[$frac(3, 5)$] ] #math.equation(block: true, alt: "10 to the power 0 minus 10 to the power −1 plus 10 to the power −2")[$10^(0) − 10^(−1) + 10^(−2)$] #math.equation(block: true, alt: "2 to the power 0 minus 2 to the power −1 plus 2 to the power −2")[$2^(0) − 2^(−1) + 2^(−2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 4")[$frac(3, 4)$] ] + ⓐ #math.equation(block: false, alt: "open parenthesis −6 close parenthesis to the power −2")[$#h(0.2em) attach(( −6 ), t: −2)$] + ⓑ #math.equation(block: false, alt: "minus 6 to the power −2")[$#h(0.2em) − 6^(−2)$] + ⓐ #math.equation(block: false, alt: "open parenthesis −8 close parenthesis to the power −2")[$#h(0.2em) attach(( −8 ), t: −2)$] + ⓑ #math.equation(block: false, alt: "minus 8 to the power −2")[$#h(0.2em) − 8^(−2)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 64")[$#h(0.2em) frac(1, 64)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 64")[$#h(0.2em) − frac(1, 64)$] ] + ⓐ #math.equation(block: false, alt: "open parenthesis −10 close parenthesis to the power −4")[$#h(0.2em) attach(( −10 ), t: −4)$] + ⓑ #math.equation(block: false, alt: "minus 10 to the power −4")[$#h(0.2em) − 10^(−4)$] + ⓐ #math.equation(block: false, alt: "open parenthesis −4 close parenthesis to the power −6")[$#h(0.2em) attach(( −4 ), t: −6)$] + ⓑ #math.equation(block: false, alt: "minus 4 to the power −6")[$#h(0.2em) − 4^(−6)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 1 over 4096")[$#h(0.2em) frac(1, 4096)$] + ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 4096")[$#h(0.2em) − frac(1, 4096)$] ] + ⓐ #math.equation(block: false, alt: "5 times 2 to the power −1")[$#h(0.2em) 5 · 2^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 5 times 2 close parenthesis to the power −1")[$#h(0.2em) attach(( 5 · 2 ), t: −1)$] + ⓐ #math.equation(block: false, alt: "10 times 3 to the power −1")[$#h(0.2em) 10 · 3^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 10 times 3 close parenthesis to the power −1")[$#h(0.2em) attach(( 10 · 3 ), t: −1)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 10 over 3")[$#h(0.2em) frac(10, 3)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 30")[$#h(0.2em) frac(1, 30)$] ] + ⓐ #math.equation(block: false, alt: "4 times 10 to the power −3")[$#h(0.2em) 4 · 10^(−3)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 4 times 10 close parenthesis to the power −3")[$#h(0.2em) attach(( 4 · 10 ), t: −3)$] + ⓐ #math.equation(block: false, alt: "3 times 5 to the power −2")[$#h(0.2em) 3 · 5^(−2)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 3 times 5 close parenthesis to the power −2")[$#h(0.2em) attach(( 3 · 5 ), t: −2)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 3 over 25")[$#h(0.2em) frac(3, 25)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 225")[$#h(0.2em) frac(1, 225)$] ] #math.equation(block: true, alt: "n to the power −4")[$n^(−4)$] #math.equation(block: true, alt: "p to the power −3")[$p^(−3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over p cubed")[$frac(1, p^(3))$] ] #math.equation(block: true, alt: "c to the power −10")[$c^(−10)$] #math.equation(block: true, alt: "m to the power −5")[$m^(−5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over m to the power 5")[$frac(1, m^(5))$] ] + ⓐ #math.equation(block: false, alt: "4 x to the power −1")[$#h(0.2em) 4 x^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 4 x close parenthesis to the power −1")[$#h(0.2em) attach(( 4 x ), t: −1)$] + ⓒ #math.equation(block: false, alt: "open parenthesis −4 x close parenthesis to the power −1")[$#h(0.2em) attach(( −4 x ), t: −1)$] + ⓐ #math.equation(block: false, alt: "3 q to the power −1")[$#h(0.2em) 3 q^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 3 q close parenthesis to the power −1")[$#h(0.2em) attach(( 3 q ), t: −1)$] + ⓒ #math.equation(block: false, alt: "open parenthesis −3 q close parenthesis to the power −1")[$#h(0.2em) attach(( −3 q ), t: −1)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 3 over q")[$#h(0.2em) frac(3, q)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 3 q")[$#h(0.2em) frac(1, 3 q)$] + ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 3 q")[$#h(0.2em) − frac(1, 3 q)$] ] + ⓐ #math.equation(block: false, alt: "6 m to the power −1")[$#h(0.2em) 6 m^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 6 m close parenthesis to the power −1")[$#h(0.2em) attach(( 6 m ), t: −1)$] + ⓒ #math.equation(block: false, alt: "open parenthesis −6 m close parenthesis to the power −1")[$#h(0.2em) attach(( −6 m ), t: −1)$] + ⓐ #math.equation(block: false, alt: "10 k to the power −1")[$#h(0.2em) 10 k^(−1)$] + ⓑ #math.equation(block: false, alt: "open parenthesis 10 k close parenthesis to the power −1")[$#h(0.2em) attach(( 10 k ), t: −1)$] + ⓒ #math.equation(block: false, alt: "open parenthesis −10 k close parenthesis to the power −1")[$#h(0.2em) attach(( −10 k ), t: −1)$] #solutionbox[ + ⓐ #math.equation(block: false, alt: "the fraction 10 over k")[$#h(0.2em) frac(10, k)$] + ⓑ #math.equation(block: false, alt: "the fraction 1 over 10 k")[$#h(0.2em) frac(1, 10 k)$] + ⓒ #math.equation(block: false, alt: "minus the fraction 1 over 10 k")[$#h(0.2em) − frac(1, 10 k)$] ] #strong[Simplify Expressions with Integer Exponents] In the following exercises, simplify#emph[.] #math.equation(block: true, alt: "p to the power −4 times p to the power 8")[$p^(−4) · p^(8)$] #math.equation(block: true, alt: "r to the power −2 times r to the power 5")[$r^(−2) · r^(5)$] #solutionbox[ #emph[r]#super[3] ] #math.equation(block: true, alt: "n to the power −10 times n squared")[$n^(−10) · n^(2)$] #math.equation(block: true, alt: "q to the power −8 times q cubed")[$q^(−8) · q^(3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over q to the power 5")[$frac(1, q^(5))$] ] #math.equation(block: true, alt: "k to the power −3 times k to the power −2")[$k^(−3) · k^(−2)$] #math.equation(block: true, alt: "z to the power −6 times z to the power −2")[$z^(−6) · z^(−2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over z to the power 8")[$frac(1, z^(8))$] ] #math.equation(block: true, alt: "a times a to the power −4")[$a · a^(−4)$] #math.equation(block: true, alt: "m times m to the power −2")[$m · m^(−2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over m")[$frac(1, m)$] ] #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: "a cubed b to the power −3")[$a^(3) b^(−3)$] #math.equation(block: true, alt: "u squared v to the power −2")[$u^(2) v^(−2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction u squared over v squared")[$frac(u^(2), v^(2))$] ] #math.equation(block: true, alt: "open parenthesis x to the power 5 y to the power −1 close parenthesis open parenthesis x to the power −10 y to the power −3 close parenthesis")[$( x^(5) y^(−1) ) ( x^(−10) y^(−3) )$] #math.equation(block: true, alt: "open parenthesis a cubed b to the power −3 close parenthesis open parenthesis a to the power −5 b to the power −1 close parenthesis")[$( a^(3) b^(−3) ) ( a^(−5) b^(−1) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over a squared b to the power 4")[$frac(1, a^(2) b^(4))$] ] #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 −4 close parenthesis")[$( u v^(−2) ) ( u^(−5) v^(−4) )$] #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 −2 r to the power −3 s to the power 9 close parenthesis open parenthesis 6 r to the power 4 s to the power −5 close parenthesis")[$( −2 r^(−3) s^(9) ) ( 6 r^(4) s^(−5) )$] #math.equation(block: true, alt: "open parenthesis −3 p to the power −5 q to the power 8 close parenthesis open parenthesis 7 p squared q to the power −3 close parenthesis")[$( −3 p^(−5) q^(8) ) ( 7 p^(2) q^(−3) )$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 21 q to the power 5 over p cubed")[$− frac(21 q^(5), p^(3))$] ] #math.equation(block: true, alt: "open parenthesis −6 m to the power −8 n to the power −5 close parenthesis open parenthesis −9 m to the power 4 n squared close parenthesis")[$( −6 m^(−8) n^(−5) ) ( −9 m^(4) n^(2) )$] #math.equation(block: true, alt: "open parenthesis −8 a to the power −5 b to the power −4 close parenthesis open parenthesis −4 a squared b cubed close parenthesis")[$( −8 a^(−5) b^(−4) ) ( −4 a^(2) b^(3) )$] #solutionbox[ #math.equation(block: true, alt: "the fraction 32 over a cubed b")[$frac(32, a^(3) b)$] ] #math.equation(block: true, alt: "open parenthesis a cubed close parenthesis to the power −3")[$attach(( a^(3) ), t: −3)$] #math.equation(block: true, alt: "open parenthesis q to the power 10 close parenthesis to the power −10")[$attach(( q^(10) ), t: −10)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over q to the power 100")[$frac(1, q^(100))$] ] #math.equation(block: true, alt: "open parenthesis n squared close parenthesis to the power −1")[$attach(( n^(2) ), t: −1)$] #math.equation(block: true, alt: "open parenthesis x to the power 4 close parenthesis to the power −1")[$attach(( x^(4) ), t: −1)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over x to the power 4")[$frac(1, x^(4))$] ] #math.equation(block: true, alt: "open parenthesis y to the power −5 close parenthesis to the power 4")[$attach(( y^(−5) ), t: 4)$] #math.equation(block: true, alt: "open parenthesis p to the power −3 close parenthesis squared")[$attach(( p^(−3) ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over p to the power 6")[$frac(1, p^(6))$] ] #math.equation(block: true, alt: "open parenthesis q to the power −5 close parenthesis to the power −2")[$attach(( q^(−5) ), t: −2)$] #math.equation(block: true, alt: "open parenthesis m to the power −2 close parenthesis to the power −3")[$attach(( m^(−2) ), t: −3)$] #solutionbox[ #emph[m]#super[6] ] #math.equation(block: true, alt: "open parenthesis 4 y to the power −3 close parenthesis squared")[$attach(( 4 y^(−3) ), t: 2)$] #math.equation(block: true, alt: "open parenthesis 3 q to the power −5 close parenthesis squared")[$attach(( 3 q^(−5) ), t: 2)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 9 over q to the power 10")[$frac(9, q^(10))$] ] #math.equation(block: true, alt: "open parenthesis 10 p to the power −2 close parenthesis to the power −5")[$attach(( 10 p^(−2) ), t: −5)$] #math.equation(block: true, alt: "open parenthesis 2 n to the power −3 close parenthesis to the power −6")[$attach(( 2 n^(−3) ), t: −6)$] #solutionbox[ #math.equation(block: true, alt: "the fraction n to the power 18 over 64")[$frac(n^(18), 64)$] ] #math.equation(block: true, alt: "the fraction u to the power 9 over u to the power −2")[$frac(u^(9), u^(−2))$] #math.equation(block: true, alt: "the fraction b to the power 5 over b to the power −3")[$frac(b^(5), b^(−3))$] #solutionbox[ #emph[b]#super[8] ] #math.equation(block: true, alt: "the fraction x to the power −6 over x to the power 4")[$frac(x^(−6), x^(4))$] #math.equation(block: true, alt: "the fraction m to the power 5 over m to the power −2")[$frac(m^(5), m^(−2))$] #solutionbox[ #emph[m]#super[7] ] #math.equation(block: true, alt: "the fraction q cubed over q to the power 12")[$frac(q^(3), q^(12))$] #math.equation(block: true, alt: "the fraction r to the power 6 over r to the power 9")[$frac(r^(6), r^(9))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over r cubed")[$frac(1, r^(3))$] ] #math.equation(block: true, alt: "the fraction n to the power −4 over n to the power −10")[$frac(n^(−4), n^(−10))$] #math.equation(block: true, alt: "the fraction p to the power −3 over p to the power −6")[$frac(p^(−3), p^(−6))$] #solutionbox[ #emph[p]#super[3] ] #strong[Convert from Decimal Notation to Scientific Notation] In the following exercises, write each number in scientific notation. 45,000 280,000 #solutionbox[ 2.8 × 10#super[5] ] 8,750,000 1,290,000 #solutionbox[ 1.29 × 10#super[6] ] 0.036 0.041 #solutionbox[ 4.1 × 10#super[−2] ] 0.00000924 0.0000103 #solutionbox[ 1.03 × 10#super[−5] ] The population of the United States on July 4, 2010 was almost #math.equation(block: false, alt: "310,000,000 .")[$310,000,000 .$] The population of the world on July 4, 2010 was more than #math.equation(block: false, alt: "6,850,000,000 .")[$6,850,000,000 .$] #solutionbox[ 6.85 × 10#super[9] ] The average width of a human hair is #math.equation(block: false, alt: "0.0018")[$0.0018$] centimeters. The probability of winning the #math.equation(block: false, alt: "2010")[$2010$] Megamillions lottery is about #math.equation(block: false, alt: "0.0000000057 .")[$0.0000000057 .$] #solutionbox[ 5.7 × 10#super[−9] ] #strong[Convert Scientific Notation to Decimal Form] In the following exercises, convert each number to decimal form. #math.equation(block: true, alt: "4.1 times 10 squared")[$4.1 #h(0.2em) × #h(0.2em) 10^(2)$] #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: "5.5 times 10 to the power 8")[$5.5 #h(0.2em) × #h(0.2em) 10^(8)$] #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: "3.5 times 10 to the power −2")[$3.5 #h(0.2em) × #h(0.2em) 10^(−2)$] #math.equation(block: true, alt: "2.8 times 10 to the power −2")[$2.8 #h(0.2em) × #h(0.2em) 10^(−2)$] #solutionbox[ 0.028 ] #math.equation(block: true, alt: "1.93 times 10 to the power −5")[$1.93 #h(0.2em) × #h(0.2em) 10^(−5)$] #math.equation(block: true, alt: "6.15 times 10 to the power −8")[$6.15 #h(0.2em) × #h(0.2em) 10^(−8)$] #solutionbox[ 0.0000000615 ] 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 .")[$2 #h(0.2em) × #h(0.2em) 10^(4) .$] At the start of 2012, the US federal budget had a deficit of more than #math.equation(block: false, alt: "$1.5 times 10 to the power 13 .")[$"$1.5" #h(0.2em) × #h(0.2em) 10^(13) .$] #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 .")[$3.9 #h(0.2em) × #h(0.2em) 10^(−4) .$] 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. #solutionbox[ 0.00001 ] #strong[Multiply and Divide Using Scientific Notation] In the following exercises, multiply or divide 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 2 times 10 to the power −9 close parenthesis")[$( 2 #h(0.2em) × #h(0.2em) 10^(5) ) ( 2 #h(0.2em) × #h(0.2em) 10^(−9) )$] #math.equation(block: true, alt: "open parenthesis 3 times 10 squared close parenthesis open parenthesis 1 times 10 to the power −5 close parenthesis")[$( 3 #h(0.2em) × #h(0.2em) 10^(2) ) ( 1 #h(0.2em) × #h(0.2em) 10^(−5) )$] #solutionbox[ 0.003 ] #math.equation(block: true, alt: "open parenthesis 1.6 times 10 to the power −2 close parenthesis open parenthesis 5.2 times 10 to the power −6 close parenthesis")[$( 1.6 #h(0.2em) × #h(0.2em) 10^(−2) ) ( 5.2 #h(0.2em) × #h(0.2em) 10^(−6) )$] #math.equation(block: true, alt: "open parenthesis 2.1 times 10 to the power −4 close parenthesis open parenthesis 3.5 times 10 to the power −2 close parenthesis")[$( 2.1 #h(0.2em) × #h(0.2em) 10^(−4) ) ( 3.5 #h(0.2em) × #h(0.2em) 10^(−2) )$] #solutionbox[ 0.00000735 ] #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 ] #math.equation(block: true, alt: "the fraction 7 times 10 to the power −2 over 1 times 10 to the power −8")[$frac(7 #h(0.2em) × #h(0.2em) 10^(−2), 1 #h(0.2em) × #h(0.2em) 10^(−8))$] #math.equation(block: true, alt: "the fraction 5 times 10 to the power −3 over 1 times 10 to the power −10")[$frac(5 #h(0.2em) × #h(0.2em) 10^(−3), 1 #h(0.2em) × #h(0.2em) 10^(−10))$] #solutionbox[ 50,000,000 ] ==== Everyday Math #strong[Calories] In May 2010 the Food and Beverage Manufacturers pledged to reduce their products by #math.equation(block: false, alt: "1.5")[$1.5$] trillion calories by the end of 2015. + ⓐ Write #math.equation(block: false, alt: "1.5")[$1.5$] trillion in decimal notation. + ⓑ Write #math.equation(block: false, alt: "1.5")[$1.5$] trillion in scientific notation. #strong[Length of a year] The difference between the calendar year and the astronomical year is #math.equation(block: false, alt: "0.000125")[$0.000125$] day. + ⓐ Write this number in scientific notation. + ⓑ How many years does it take for the difference to become 1 day? #solutionbox[ + ⓐ 1.25 × 10#super[−4] + ⓐ 8,000 ] #strong[Calculator display] Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the probability of getting a particular 5-card hand from a deck of cards, Mario divided #math.equation(block: false, alt: "1")[$1$] by #math.equation(block: false, alt: "2,598,960")[$2,598,960$] and saw the answer #math.equation(block: false, alt: "3.848 times 10 to the power −7 .")[$3.848 #h(0.2em) × #h(0.2em) 10^(−7) .$] Write the number in decimal notation. #strong[Calculator display] Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with #math.equation(block: false, alt: "6")[$6$] of her #math.equation(block: false, alt: "50")[$50$] favorite photographs, she multiplied #math.equation(block: false, alt: "50 times 49 times 48 times 47 times 46 times 45 .")[$50 · 49 · 48 · 47 · 46 · 45 .$] Her calculator gave the answer #math.equation(block: false, alt: "1.1441304 times 10 to the power 10 .")[$1.1441304 #h(0.2em) × #h(0.2em) 10^(10) .$] Write the number in decimal notation. #solutionbox[ 11,441,304,000 ] ==== 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[A self-assessment checklist for math skills, including exponents and scientific notation, with options to mark 'Confidently,' 'With some help,' or 'No-I don't get it!'], alt: "A self-assessment checklist for math skills, including exponents and scientific notation, with options to mark 'Confidently,' 'With some help,' or 'No-I don't get it!'", caption: none) ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?