#set document(title: "6.5 Divide Monomials", 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")) == 6.5#h(0.6em)Divide Monomials #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Simplify: #math.equation(block: false, alt: "the fraction 8 over 24 .")[$frac(8, 24) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 3")[$frac(1, 3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis 2 m cubed close parenthesis to the power 5 .")[$attach(( 2 m^(3) ), t: 5) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "32 m to the power 15")[$32 m^(15)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 12 x over 12 y .")[$frac(12 x, 12 y) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "the fraction x over y")[$frac(x, y)$] ] ] === Simplify Expressions Using the Quotient Property for Exponents Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Summary of Exponent Properties for Multiplication] 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 whole numbers, then #figure(table( columns: 2, align: left, inset: 6pt, table.header([#emph[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")[$a^(m) · a^(n) = a^(m + n)$]]), [#emph[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")[$attach(( a^(m) ), t: n) = a^(m · n)$]], [#emph[Product to a Power]], [#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")[$attach(( a b ), t: m) = a^(m) b^(m)$]], )) ] Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Equivalent Fractions Property] If #math.equation(block: false, alt: "a , b , and c")[$a , b , #h(0.2em) "and" #h(0.2em) c$] are whole numbers where #math.equation(block: false, alt: "b not equal to 0 , c not equal to 0")[$b ≠ 0 , c ≠ 0$], #math.equation(block: true, alt: "then the fraction a over b equals the fraction a times c over b times c and the fraction a times c over b times c equals the fraction a over b")[$"then" #h(1em) frac(a, b) = frac(a · c, b · c) #h(1em) "and" #h(1em) frac(a · c, b · c) = frac(a, b)$] ] As before, we’ll try to discover a property by looking at some examples. #figure(table( columns: 4, align: left, inset: 6pt, table.header([Consider], [#math.equation(block: false, alt: "the fraction x to the power 5 over x squared")[$frac(x^(5), x^(2))$]], [and], [#math.equation(block: false, alt: "the fraction x squared over x cubed")[$frac(x^(2), x^(3))$]]), [What do they mean?], [#math.equation(block: false, alt: "the fraction x times x times x times x times x over x times x")[$frac(x · x · x · x · x, x · x)$]], [], [#math.equation(block: false, alt: "the fraction x times x over x times x times x")[$frac(x · x, x · x · x)$]], [Use the Equivalent Fractions Property.], [#math.equation(block: false, alt: "the fraction x times x times x times x times x over x times x")[$frac(cancel(x) · cancel(x) · x · x · x, cancel(x) · cancel(x))$]], [], [#math.equation(block: false, alt: "the fraction x times x times 1 over x times x times x")[$frac(cancel(x) · cancel(x) · 1, cancel(x) · cancel(x) · x)$]], [Simplify.], [#math.equation(block: false, alt: "x cubed")[$x^(3)$]], [], [#math.equation(block: false, alt: "the fraction 1 over x")[$frac(1, x)$]], )) Notice, in each case the bases were the same and we subtracted exponents. When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1. We write: #math.equation(block: true, alt: "the fraction x to the power 5 over x squared, the fraction x squared over x cubed; x to the power 5 minus 2, the fraction 1 over x to the power 3 minus 2; x cubed, the fraction 1 over x")[$frac(x^(5), x^(2)) & & & frac(x^(2), x^(3)) \ x^(5 − 2) & & & frac(1, x^(3 − 2)) \ x^(3) & & & frac(1, x)$]This leads to the #emph[Quotient Property for Exponents]. #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")[$frac(a^(m), a^(n)) = a^(m − n) , m > n #h(0.5em) "and" #h(0.5em) frac(a^(m), a^(n)) = frac(1, a^(n − m)) , n > m$] ] A couple of examples with numbers may help to verify this property. #math.equation(block: true, alt: "the fraction 3 to the power 4 over 3 squared, equals, 3 to the power 4 minus 2, the fraction 5 squared over 5 cubed, equals, the fraction 1 over 5 to the power 3 minus 2; the fraction 81 over 9, equals, 3 squared, the fraction 25 over 125, equals, the fraction 1 over 5 to the power 1; 9, equals, 9 ✓, the fraction 1 over 5, equals, the fraction 1 over 5 ✓")[$frac(3^(4), 3^(2)) & = & 3^(4 − 2) & & & #h(4em) frac(5^(2), 5^(3)) & = & frac(1, 5^(3 − 2)) \ frac(81, 9) & = & 3^(2) & & & #h(4em) frac(25, 125) & = & frac(1, 5^(1)) \ 9 & = & 9 ✓ & & & #h(4em) frac(1, 5) & = & frac(1, 5) ✓$]#examplebox("Example 1")[][ Simplify: ⓐ #math.equation(block: false, alt: "the fraction x to the power 9 over x to the power 7")[$frac(x^(9), x^(7))$] ⓑ #math.equation(block: false, alt: "the fraction 3 to the power 10 over 3 squared .")[$frac(3^(10), 3^(2)) .$] #solutionbox[ To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator. + ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([Since 9 \> 7, there are more factors of #emph[x] in the numerator.], [#figure(figph[x to the ninth power divided by x to the seventh power.], alt: "x to the ninth power divided by x to the seventh power.", 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[x to the power of 9 minus 7.], alt: "x to the power of 9 minus 7.", caption: none)], [Simplify.], [#figure(figph[A close-up image shows the mathematical expression 'x squared'.], alt: "A close-up image shows the mathematical expression 'x squared'.", caption: none)], )) + ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([Since 10 \> 2, there are more factors of #emph[x] in the numerator.], [#figure(figph[3 to the tenth power divided by 3 squared.], alt: "3 to the tenth power divided by 3 squared.", 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[3 to the power of 10 minus 2.], alt: "3 to the power of 10 minus 2.", caption: none)], [Simplify.], [#figure(figph[3 to the eighth power.], alt: "3 to the eighth power.", caption: none)], )) #linebreak() Notice that when the larger exponent is in the numerator, we are left with factors in the numerator. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction x to the power 15 over x to the power 10")[$frac(x^(15), x^(10))$] ⓑ #math.equation(block: false, alt: "the fraction 6 to the power 14 over 6 to the power 5 .")[$frac(6^(14), 6^(5)) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "x to the power 5")[$x^(5)$] ⓑ #math.equation(block: false, alt: "6 to the power 9")[$6^(9)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction y to the power 43 over y to the power 37")[$frac(y^(43), y^(37))$] ⓑ #math.equation(block: false, alt: "the fraction 10 to the power 15 over 10 to the power 7 .")[$frac(10^(15), 10^(7)) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "y to the power 6")[$y^(6)$] ⓑ #math.equation(block: false, alt: "10 to the power 8")[$10^(8)$] ] ] #examplebox("Example 2")[][ Simplify: ⓐ #math.equation(block: false, alt: "the fraction b to the power 8 over b to the power 12")[$frac(b^(8), b^(12))$] ⓑ #math.equation(block: false, alt: "the fraction 7 cubed over 7 to the power 5 .")[$frac(7^(3), 7^(5)) .$] #solutionbox[ To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator. + ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([Since 12 \> 8, there are more factors of #emph[b] in the denominator.], [#figure(figph[b to the eighth power divided b to the twelfth power.], alt: "b to the eighth power divided b to the twelfth power.", 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 the fraction 1 over a to the power n minus m")[$frac(a^(m), a^(n)) = frac(1, a^(n − m))$].], [#figure(figph[1 divided by b to the power of 12 minus 8.], alt: "1 divided by b to the power of 12 minus 8.", caption: none)], [Simplify.], [#figure(figph[1 divided by b to the fourth power.], alt: "1 divided by b to the fourth power.", caption: none)], )) + ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([Since 5 \> 3, there are more factors of 3 in the denominator.], [#figure(figph[7 cubed divided by 7 to the fifth power.], alt: "7 cubed divided by 7 to the fifth power.", 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 the fraction 1 over a to the power n minus m")[$frac(a^(m), a^(n)) = frac(1, a^(n − m))$].], [#figure(figph[1 divided by 7 to the power of 5 minus 3.], alt: "1 divided by 7 to the power of 5 minus 3.", caption: none)], [Simplify.], [#figure(figph[1 divided by 7 squared.], alt: "1 divided by 7 squared.", caption: none)], [Simplify.], [#figure(figph[The fraction one forty-ninth is displayed, represented as 1 over 49 with a horizontal line separating the numerator and the denominator.], alt: "The fraction one forty-ninth is displayed, represented as 1 over 49 with a horizontal line separating the numerator and the denominator.", caption: none)], )) #linebreak() Notice that when the larger exponent is in the denominator, we are left with factors in the denominator. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction x to the power 18 over x to the power 22")[$frac(x^(18), x^(22))$] ⓑ #math.equation(block: false, alt: "the fraction 12 to the power 15 over 12 to the power 30 .")[$frac(12^(15), 12^(30)) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over x to the power 4")[$frac(1, x^(4))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 12 to the power 15")[$frac(1, 12^(15))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction m to the power 7 over m to the power 15")[$frac(m^(7), m^(15))$] ⓑ #math.equation(block: false, alt: "the fraction 9 to the power 8 over 9 to the power 19 .")[$frac(9^(8), 9^(19)) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over m to the power 8")[$frac(1, m^(8))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 9 to the power 11")[$frac(1, 9^(11))$] ] ] Notice the difference in the two previous examples: - If we start with more factors in the numerator, we will end up with factors in the numerator. - If we start with more factors in the denominator, we will end up with factors in the denominator. The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator. #examplebox("Example 3")[][ Simplify: ⓐ #math.equation(block: false, alt: "the fraction a to the power 5 over a to the power 9")[$frac(a^(5), a^(9))$] ⓑ #math.equation(block: false, alt: "the fraction x to the power 11 over x to the power 7 .")[$frac(x^(11), x^(7)) .$] #solutionbox[ + ⓐ Is the exponent of #math.equation(block: false, alt: "a")[$a$] larger in the numerator or denominator? Since 9 \> 5, there are more #math.equation(block: false, alt: "a ' s")[$a ' "s"$] in the denominator and so we will end up with factors in the denominator. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[a to the fifth power divided by a to the ninth power.], alt: "a to the fifth power divided by a to the ninth power.", 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 the fraction 1 over a to the power n minus m")[$frac(a^(m), a^(n)) = frac(1, a^(n − m))$].], [#figure(figph[1 divided by a to the power of 9 minus 5.], alt: "1 divided by a to the power of 9 minus 5.", caption: none)], [Simplify.], [#figure(figph[1 divided by a to the fourth power.], alt: "1 divided by a to the fourth power.", caption: none)], )) + ⓑ Notice there are more factors of #math.equation(block: false, alt: "x")[$x$] in the numerator, since 11 \> 7. So we will end up with factors in the numerator. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[x to the eleventh power divided by x to the seventh power.], alt: "x to the eleventh power divided by x to the seventh power.", 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 the fraction 1 over a to the power n minus m")[$frac(a^(m), a^(n)) = frac(1, a^(n − m))$].], [#figure(figph[x to the power of 11 minus 7.], alt: "x to the power of 11 minus 7.", caption: none)], [Simplify.], [#figure(figph[x to the fourth power.], alt: "x to the fourth power.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction b to the power 19 over b to the power 11")[$frac(b^(19), b^(11))$] ⓑ #math.equation(block: false, alt: "the fraction z to the power 5 over z to the power 11 .")[$frac(z^(5), z^(11)) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "b to the power 8")[$b^(8)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over z to the power 6")[$frac(1, z^(6))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction p to the power 9 over p to the power 17")[$frac(p^(9), p^(17))$] ⓑ #math.equation(block: false, alt: "the fraction w to the power 13 over w to the power 9 .")[$frac(w^(13), w^(9)) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over p to the power 8")[$frac(1, p^(8))$] ⓑ #math.equation(block: false, alt: "w to the power 4")[$w^(4)$] ] ] === Simplify Expressions with an Exponent of Zero A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like #math.equation(block: false, alt: "the fraction a to the power m over a to the power m")[$frac(a^(m), a^(m))$]. From your earlier work with fractions, you know that: #math.equation(block: true, alt: "the fraction 2 over 2 equals 1 the fraction 17 over 17 equals 1 the fraction −43 over −43 equals 1")[$frac(2, 2) = 1 #h(2em) frac(17, 17) = 1 #h(2em) frac(−43, −43) = 1$]In words, a number divided by itself is 1. So, #math.equation(block: false, alt: "the fraction x over x equals 1")[$frac(x, x) = 1$], for any #math.equation(block: false, alt: "x open parenthesis x not equal to 0 close parenthesis")[$x #h(0.2em) ( x ≠ 0 )$], since any number divided by itself is 1. The Quotient Property for Exponents shows us how to simplify #math.equation(block: false, alt: "the fraction a to the power m over a to the power n")[$frac(a^(m), a^(n))$] when #math.equation(block: false, alt: "m greater than n")[$m > n$] and when #math.equation(block: false, alt: "n less than m")[$n < m$] by subtracting exponents. What if #math.equation(block: false, alt: "m equals n")[$m = n$]? Consider #math.equation(block: false, alt: "the fraction 8 over 8")[$frac(8, 8)$], which we know is 1. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 8 over 8 equals 1")[$frac(8, 8) = 1$]]), [Write #math.equation(block: false, alt: "8")[$8$] as #math.equation(block: false, alt: "2 cubed")[$2^(3)$].], [#math.equation(block: false, alt: "the fraction 2 cubed over 2 cubed equals 1")[$frac(2^(3), 2^(3)) = 1$]], [Subtract exponents.], [#math.equation(block: false, alt: "2 to the power 3 minus 3 equals 1")[$2^(3 − 3) = 1$]], [Simplify.], [#math.equation(block: false, alt: "2 to the power 0 equals 1")[$2^(0) = 1$]], )) Now we will simplify #math.equation(block: false, alt: "the fraction a to the power m over a to the power m")[$frac(a^(m), a^(m))$] in two ways to lead us to the definition of the zero exponent. In general, for #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$]: #figure(figph[This figure is divided into two columns. At the top of the figure, the left and right columns both contain a to the m power divided by a to the m power. In the next row, the left column contains a to the m minus m power. The right column contains the fraction m factors of a divided by m factors of a, represented in the numerator and denominator by a times a followed by an ellipsis. All the as in the numerator and denominator are canceled out. In the bottom row, the left column contains a to the zero power. The right column contains 1.], alt: "This figure is divided into two columns. At the top of the figure, the left and right columns both contain a to the m power divided by a to the m power. In the next row, the left column contains a to the m minus m power. The right column contains the fraction m factors of a divided by m factors of a, represented in the numerator and denominator by a times a followed by an ellipsis. All the as in the numerator and denominator are canceled out. In the bottom row, the left column contains a to the zero power. The right column contains 1.", caption: none) We see #math.equation(block: false, alt: "the fraction a to the power m over a to the power m")[$frac(a^(m), a^(m))$] simplifies to #math.equation(block: false, alt: "a to the power 0")[$a^(0)$] and to 1. So #math.equation(block: false, alt: "a to the power 0 equals 1")[$a^(0) = 1$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Zero Exponent] If #math.equation(block: false, alt: "a")[$a$] is a non-zero number, then #math.equation(block: false, alt: "a to the power 0 equals 1")[$a^(0) = 1$]. Any nonzero number raised to the zero power is 1. ] In this text, we assume any variable that we raise to the zero power is not zero. #examplebox("Example 4")[][ Simplify: ⓐ #math.equation(block: false, alt: "9 to the power 0")[$9^(0)$] ⓑ #math.equation(block: false, alt: "n to the power 0 .")[$n^(0) .$] #solutionbox[ The definition says any non-zero number raised to the zero power is 1. #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ #linebreak() Use the definition of the zero exponent.], [#math.equation(block: false, alt: "9 to the power 0; 1")[$9^(0) \ 1$]]), [ⓑ #linebreak() Use the definition of the zero exponent.], [#math.equation(block: false, alt: "n to the power 0; 1")[$n^(0) \ 1$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "15 to the power 0")[$15^(0)$] ⓑ #math.equation(block: false, alt: "m to the power 0 .")[$m^(0) .$] #solutionbox[ ⓐ 1 ⓑ 1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "k to the power 0")[$k^(0)$] ⓑ #math.equation(block: false, alt: "29 to the power 0 .")[$29^(0) .$] #solutionbox[ ⓐ 1 ⓑ 1 ] ] Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents. What about raising an expression to the zero power? Let’s look at #math.equation(block: false, alt: "open parenthesis 2 x close parenthesis to the power 0")[$attach(( 2 x ), t: 0)$]. We can use the product to a power rule to rewrite this expression. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis 2 x close parenthesis to the power 0")[$attach(( 2 x ), t: 0)$]]), [Use the product to a power rule.], [#math.equation(block: false, alt: "2 to the power 0 x to the power 0")[$2^(0) x^(0)$]], [Use the zero exponent property.], [#math.equation(block: false, alt: "1 times 1")[$1 · 1$]], [Simplify.], [#math.equation(block: false, alt: "1")[$1$]], )) This tells us that any nonzero expression raised to the zero power is one. #examplebox("Example 5")[][ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 5 b close parenthesis to the power 0")[$attach(( 5 b ), t: 0)$] ⓑ #math.equation(block: false, alt: "open parenthesis −4 a squared b close parenthesis to the power 0 .")[$attach(( −4 a^(2) b ), t: 0) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], [#math.equation(block: false, alt: "open parenthesis 5 b close parenthesis to the power 0")[$attach(( 5 b ), t: 0)$]]), [Use the definition of the zero exponent.], [#math.equation(block: false, alt: "1")[$1$]], [ⓑ], [#math.equation(block: false, alt: "open parenthesis −4 a squared b close parenthesis to the power 0")[$attach(( −4 a^(2) b ), t: 0)$]], [Use the definition of the zero exponent.], [#math.equation(block: false, alt: "1")[$1$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 11 z close parenthesis to the power 0")[$attach(( 11 z ), t: 0)$] ⓑ #math.equation(block: false, alt: "open parenthesis −11 p q cubed close parenthesis to the power 0 .")[$attach(( −11 p q^(3) ), t: 0) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "1")[$1$] ⓑ #math.equation(block: false, alt: "1")[$1$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −6 d close parenthesis to the power 0")[$attach(( −6 d ), t: 0)$] ⓑ #math.equation(block: false, alt: "open parenthesis −8 m squared n cubed close parenthesis to the power 0 .")[$attach(( −8 m^(2) n^(3) ), t: 0) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "1")[$1$] ⓑ #math.equation(block: false, alt: "1")[$1$] ] ] === Simplify Expressions Using the Quotient to a Power Property Now we will look at an example that will lead us to the Quotient to a Power Property. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction x over y close parenthesis cubed")[$attach(( frac(x, y) ), t: 3)$]]), [This means:], [#math.equation(block: false, alt: "the fraction x over y times the fraction x over y times the fraction x over y")[$frac(x, y) · frac(x, y) · frac(x, y)$]], [Multiply the fractions.], [#math.equation(block: false, alt: "the fraction x times x times x over y times y times y")[$frac(x · x · x, y · y · y)$]], [Write with exponents.], [#math.equation(block: false, alt: "the fraction x cubed over y cubed")[$frac(x^(3), y^(3))$]], )) Notice that the exponent applies to both the numerator and the denominator. #figure(table( columns: 2, align: left, inset: 6pt, table.header([We write:], [#math.equation(block: false, alt: "open parenthesis the fraction x over y close parenthesis cubed")[$attach(( frac(x, y) ), t: 3)$]]), [], [#math.equation(block: false, alt: "the fraction x cubed over y cubed")[$frac(x^(3), y^(3))$]], )) This leads to the #emph[Quotient to a Power Property for Exponents]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Quotient to a Power Property for Exponents] If #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are real numbers, #math.equation(block: false, alt: "b not equal to 0")[$b ≠ 0$], and #math.equation(block: false, alt: "m")[$m$] is a counting number, then #math.equation(block: true, alt: "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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m))$]To raise a fraction to a power, raise the numerator and denominator to that power. ] An example with numbers may help you understand this property: #math.equation(block: true, alt: "open parenthesis the fraction 2 over 3 close parenthesis cubed, equals, the fraction 2 cubed over 3 cubed; the fraction 2 over 3 times the fraction 2 over 3 times the fraction 2 over 3, equals, the fraction 8 over 27; the fraction 8 over 27, equals, the fraction 8 over 27 ✓")[$attach(( frac(2, 3) ), t: 3) & = & frac(2^(3), 3^(3)) \ frac(2, 3) · frac(2, 3) · frac(2, 3) & = & frac(8, 27) \ frac(8, 27) & = & frac(8, 27) ✓$]#examplebox("Example 6")[][ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 3 over 7 close parenthesis squared")[$attach(( frac(3, 7) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction b over 3 close parenthesis to the power 4")[$attach(( frac(b, 3) ), t: 4)$] ⓒ #math.equation(block: false, alt: "open parenthesis the fraction k over j close parenthesis cubed .")[$attach(( frac(k, j) ), t: 3) .$] #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[3 sevenths squared.], alt: "3 sevenths squared.", caption: none)]), [Use the Quotient Property, #math.equation(block: false, alt: "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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m))$].], [#figure(figph[3 squared divided by 7 squared.], alt: "3 squared divided by 7 squared.", caption: none)], [Simplify.], [#figure(figph[9 forty-ninths.], alt: "9 forty-ninths.", caption: none)], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[b thirds to the fourth power.], alt: "b thirds to the fourth power.", caption: none)]), [Use the Quotient Property, #math.equation(block: false, alt: "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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m))$].], [#figure(figph[b to the fourth power divided by 3 to the fourth power.], alt: "b to the fourth power divided by 3 to the fourth power.", caption: none)], [Simplify.], [#figure(figph[b to the fourth power divided by 81.], alt: "b to the fourth power divided by 81.", caption: none)], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#figure(figph[k divided by j, in parentheses, cubed.], alt: "k divided by j, in parentheses, cubed.", caption: none)]), [Raise the numerator and denominator to the third power.], [#figure(figph[k cubed divided by j cubed.], alt: "k cubed divided by j cubed.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 5 over 8 close parenthesis squared")[$attach(( frac(5, 8) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction p over 10 close parenthesis to the power 4")[$attach(( frac(p, 10) ), t: 4)$] ⓒ #math.equation(block: false, alt: "open parenthesis the fraction m over n close parenthesis to the power 7 .")[$attach(( frac(m, n) ), t: 7) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 25 over 64")[$frac(25, 64)$] ⓑ #math.equation(block: false, alt: "the fraction p to the power 4 over 10,000")[$frac(p^(4), 10","000)$] ⓒ #math.equation(block: false, alt: "the fraction m to the power 7 over n to the power 7")[$frac(m^(7), n^(7))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 1 over 3 close parenthesis cubed")[$attach(( frac(1, 3) ), t: 3)$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction −2 over q close parenthesis cubed")[$attach(( frac(−2, q) ), t: 3)$] ⓒ #math.equation(block: false, alt: "open parenthesis the fraction w over x close parenthesis to the power 4 .")[$attach(( frac(w, x) ), t: 4) .$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over 27")[$frac(1, 27)$] ⓑ #math.equation(block: false, alt: "the fraction −8 over q cubed")[$frac(−8, q^(3))$] ⓒ #math.equation(block: false, alt: "the fraction w to the power 4 over x to the power 4")[$frac(w^(4), x^(4))$] ] ] === Simplify Expressions by Applying Several Properties We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents. #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 whole numbers, then #figure(table( columns: 2, align: left, inset: 6pt, table.header([#emph[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")[$a^(m) · a^(n) = a^(m + n)$]]), [#emph[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")[$attach(( a^(m) ), t: n) = a^(m · n)$]], [#emph[Product to a Power]], [#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")[$attach(( a b ), t: m) = a^(m) b^(m)$]], [#emph[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 , a not equal to 0 , m greater than n; 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 , a not equal to 0 , n greater than m")[$frac(a^(m), a^(n)) & = & a^(m − n) , a ≠ 0 , m > n \ frac(a^(m), a^(n)) & = & frac(1, a^(n − m)) , a ≠ 0 , n > m$]], [#emph[Zero Exponent Definition]], [#math.equation(block: false, alt: "a to the power o equals 1 , a not equal to 0")[$a^(o) = 1 , a ≠ 0$]], [#emph[Quotient to a Power Property]], [#math.equation(block: false, alt: "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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m)) , b ≠ 0$]], )) ] #examplebox("Example 7")[][ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis y to the power 4 close parenthesis squared over y to the power 6 .")[$frac(attach(( y^(4) ), t: 2), y^(6)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction open parenthesis y to the power 4 close parenthesis squared over y to the power 6")[$frac(attach(( y^(4) ), t: 2), y^(6))$]]), [Multiply the exponents in the numerator.], [#math.equation(block: false, alt: "the fraction y to the power 8 over y to the power 6")[$frac(y^(8), y^(6))$]], [Subtract the exponents.], [#math.equation(block: false, alt: "y squared")[$y^(2)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis m to the power 5 close parenthesis to the power 4 over m to the power 7 .")[$frac(attach(( m^(5) ), t: 4), m^(7)) .$] #solutionbox[ #math.equation(block: true, alt: "m to the power 13")[$m^(13)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis k squared close parenthesis to the power 6 over k to the power 7 .")[$frac(attach(( k^(2) ), t: 6), k^(7)) .$] #solutionbox[ #math.equation(block: true, alt: "k to the power 5")[$k^(5)$] ] ] #examplebox("Example 8")[][ Simplify: #math.equation(block: false, alt: "the fraction b to the power 12 over open parenthesis b squared close parenthesis to the power 6 .")[$frac(b^(12), attach(( b^(2) ), t: 6)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction b to the power 12 over open parenthesis b squared close parenthesis to the power 6")[$frac(b^(12), attach(( b^(2) ), t: 6))$]]), [Multiply the exponents in the numerator.], [#math.equation(block: false, alt: "the fraction b to the power 12 over b to the power 12")[$frac(b^(12), b^(12))$]], [Subtract the exponents.], [#math.equation(block: false, alt: "b to the power 0")[$b^(0)$]], [Simplify.], [#math.equation(block: false, alt: "1")[$1$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction n to the power 12 over open parenthesis n cubed close parenthesis to the power 4 .")[$frac(n^(12), attach(( n^(3) ), t: 4)) .$] #solutionbox[ 1 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x to the power 15 over open parenthesis x cubed close parenthesis to the power 5 .")[$frac(x^(15), attach(( x^(3) ), t: 5)) .$] #solutionbox[ 1 ] ] #examplebox("Example 9")[][ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction y to the power 9 over y to the power 4 close parenthesis squared .")[$attach(( frac(y^(9), y^(4)) ), t: 2) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction y to the power 9 over y to the power 4 close parenthesis squared")[$attach(( frac(y^(9), y^(4)) ), t: 2)$]]), [Remember parentheses come before exponents. #linebreak() Notice the bases are the same, so we can simplify #linebreak() inside the parentheses. Subtract the exponents.], [#math.equation(block: false, alt: "open parenthesis y to the power 5 close parenthesis squared")[$attach(( y^(5) ), t: 2)$]], [Multiply the exponents.], [#math.equation(block: false, alt: "y to the power 10")[$y^(10)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction r to the power 5 over r cubed close parenthesis to the power 4 .")[$attach(( frac(r^(5), r^(3)) ), t: 4) .$] #solutionbox[ #math.equation(block: true, alt: "r to the power 8")[$r^(8)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction v to the power 6 over v to the power 4 close parenthesis cubed .")[$attach(( frac(v^(6), v^(4)) ), t: 3) .$] #solutionbox[ #math.equation(block: true, alt: "v to the power 6")[$v^(6)$] ] ] #examplebox("Example 10")[][ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction j squared over k cubed close parenthesis to the power 4 .")[$attach(( frac(j^(2), k^(3)) ), t: 4) .$] #solutionbox[ Here we cannot simplify inside the parentheses first, since the bases are not the same. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction j squared over k cubed close parenthesis to the power 4")[$attach(( frac(j^(2), k^(3)) ), t: 4)$]]), [Raise the numerator and denominator to the fourth power #linebreak() using the Quotient to a Power Property, #math.equation(block: false, alt: "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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m))$].], [$$], [Use the Power Property and simplify.], [#math.equation(block: false, alt: "the fraction j to the power 8 over k to the power 12")[$frac(j^(8), k^(12))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction a cubed over b squared close parenthesis to the power 4 .")[$attach(( frac(a^(3), b^(2)) ), t: 4) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction a to the power 12 over b to the power 8")[$frac(a^(12), b^(8))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction q to the power 7 over r to the power 5 close parenthesis cubed .")[$attach(( frac(q^(7), r^(5)) ), t: 3) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction q to the power 21 over r to the power 15")[$frac(q^(21), r^(15))$] ] ] #examplebox("Example 11")[][ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 2 m squared over 5 n close parenthesis to the power 4 .")[$attach(( frac(2 m^(2), 5 n) ), t: 4) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction 2 m squared over 5 n close parenthesis to the power 4")[$attach(( frac(2 m^(2), 5 n) ), t: 4)$]]), [Raise the numberator and denominator to the fourth power, #linebreak() using the Quotient to a Power Property, #math.equation(block: false, alt: "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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m))$].], [#math.equation(block: false, alt: "the fraction open parenthesis 2 m squared close parenthesis to the power 4 over open parenthesis 5 n close parenthesis to the power 4")[$frac(attach(( 2 m^(2) ), t: 4), attach(( 5 n ), t: 4))$]], [Raise each factor to the fourth power.], [#math.equation(block: false, alt: "the fraction open parenthesis 2 m squared close parenthesis to the power 4 over open parenthesis 5 n close parenthesis to the power 4")[$frac(attach(( 2 m^(2) ), t: 4), attach(( 5 n ), t: 4))$]], [Use the Power Property and simplify.], [#math.equation(block: false, alt: "the fraction 16 m to the power 8 over 625 n to the power 4")[$frac(16 m^(8), 625 n^(4))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 7 x cubed over 9 y close parenthesis squared .")[$attach(( frac(7 x^(3), 9 y) ), t: 2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 49 x to the power 6 over 81 y squared")[$frac(49 x^(6), 81 y^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis the fraction 3 x to the power 4 over 7 y close parenthesis squared .")[$attach(( frac(3 x^(4), 7 y) ), t: 2) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 9 x to the power 8 over 49 y squared")[$frac(9 x^(8), 49 y^(2))$] ] ] #examplebox("Example 12")[][ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis x cubed close parenthesis to the power 4 open parenthesis x squared close parenthesis to the power 5 over open parenthesis x to the power 6 close parenthesis to the power 5 .")[$frac(attach(( x^(3) ), t: 4) attach(( x^(2) ), t: 5), attach(( x^(6) ), t: 5)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction open parenthesis x cubed close parenthesis to the power 4 open parenthesis x squared close parenthesis to the power 5 over open parenthesis x to the power 6 close parenthesis to the power 5")[$frac(attach(( x^(3) ), t: 4) attach(( x^(2) ), t: 5), attach(( x^(6) ), t: 5))$]]), [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: "the fraction open parenthesis x to the power 12 close parenthesis open parenthesis x to the power 10 close parenthesis over open parenthesis x to the power 30 close parenthesis")[$frac(( x^(12) ) ( x^(10) ), ( x^(30) ))$]], [Add the exponents in the numerator.], [#math.equation(block: false, alt: "the fraction x to the power 22 over x to the power 30")[$frac(x^(22), x^(30))$]], [Use the Quotient Property, #math.equation(block: false, alt: "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")[$frac(a^(m), a^(n)) = frac(1, a^(n − m))$].], [#math.equation(block: false, alt: "the fraction 1 over x to the power 8")[$frac(1, x^(8))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis a squared close parenthesis cubed open parenthesis a squared close parenthesis to the power 4 over open parenthesis a to the power 4 close parenthesis to the power 5 .")[$frac(attach(( a^(2) ), t: 3) attach(( a^(2) ), t: 4), attach(( a^(4) ), t: 5)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over a to the power 6")[$frac(1, a^(6))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis p cubed close parenthesis to the power 4 open parenthesis p to the power 5 close parenthesis cubed over open parenthesis p to the power 7 close parenthesis to the power 6 .")[$frac(attach(( p^(3) ), t: 4) attach(( p^(5) ), t: 3), attach(( p^(7) ), t: 6)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over p to the power 15")[$frac(1, p^(15))$] ] ] #examplebox("Example 13")[][ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis 10 p cubed close parenthesis squared over open parenthesis 5 p close parenthesis cubed open parenthesis 2 p to the power 5 close parenthesis to the power 4 .")[$frac(attach(( 10 p^(3) ), t: 2), attach(( 5 p ), t: 3) attach(( 2 p^(5) ), t: 4)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction open parenthesis 10 p cubed close parenthesis squared over open parenthesis 5 p close parenthesis cubed open parenthesis 2 p to the power 5 close parenthesis to the power 4")[$frac(attach(( 10 p^(3) ), t: 2), attach(( 5 p ), t: 3) attach(( 2 p^(5) ), t: 4))$]]), [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: "the fraction open parenthesis 10 close parenthesis squared open parenthesis p cubed close parenthesis squared over open parenthesis 5 close parenthesis cubed open parenthesis p close parenthesis cubed open parenthesis 2 close parenthesis to the power 4 open parenthesis p to the power 5 close parenthesis to the power 4")[$frac(attach(( 10 ), t: 2) attach(( p^(3) ), t: 2), attach(( 5 ), t: 3) attach(( p ), t: 3) attach(( 2 ), t: 4) attach(( p^(5) ), t: 4))$]], [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: "the fraction 100 p to the power 6 over 125 p cubed times 16 p to the power 20")[$frac(100 p^(6), 125 p^(3) · 16 p^(20))$]], [Add the exponents in the denominator.], [#math.equation(block: false, alt: "the fraction 100 p to the power 6 over 125 times 16 p to the power 23")[$frac(100 p^(6), 125 · 16 p^(23))$]], [Use the Quotient Property, #math.equation(block: false, alt: "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")[$frac(a^(m), a^(n)) = frac(1, a^(n − m))$].], [#math.equation(block: false, alt: "the fraction 100 over 125 times 16 p to the power 17")[$frac(100, 125 · 16 p^(17))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 20 p to the power 17")[$frac(1, 20 p^(17))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis 3 r cubed close parenthesis squared open parenthesis r cubed close parenthesis to the power 7 over open parenthesis r cubed close parenthesis cubed .")[$frac(attach(( 3 r^(3) ), t: 2) attach(( r^(3) ), t: 7), attach(( r^(3) ), t: 3)) .$] #solutionbox[ #math.equation(block: true, alt: "9 r to the power 18")[$9 r^(18)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction open parenthesis 2 x to the power 4 close parenthesis to the power 5 over open parenthesis 4 x cubed close parenthesis squared open parenthesis x cubed close parenthesis to the power 5 .")[$frac(attach(( 2 x^(4) ), t: 5), attach(( 4 x^(3) ), t: 2) attach(( x^(3) ), t: 5)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 over x")[$frac(2, x)$] ] ] === Divide Monomials You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials. #examplebox("Example 14")[][ Find the quotient: #math.equation(block: false, alt: "56 x to the power 7 divided by 8 x cubed .")[$56 x^(7) ÷ 8 x^(3) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "56 x to the power 7 divided by 8 x cubed")[$56 x^(7) ÷ 8 x^(3)$]]), [Rewrite as a fraction.], [#math.equation(block: false, alt: "the fraction 56 x to the power 7 over 8 x cubed")[$frac(56 x^(7), 8 x^(3))$]], [Use fraction multiplication.], [#math.equation(block: false, alt: "the fraction 56 over 8 times the fraction x to the power 7 over x cubed")[$frac(56, 8) ⋅ frac(x^(7), x^(3))$]], [Simplify and use the Quotient Property.], [#math.equation(block: false, alt: "7 x to the power 4")[$7 x^(4)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "42 y to the power 9 divided by 6 y cubed .")[$42 y^(9) ÷ 6 y^(3) .$] #solutionbox[ #math.equation(block: true, alt: "7 y to the power 6")[$7 y^(6)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "48 z to the power 8 divided by 8 z squared .")[$48 z^(8) ÷ 8 z^(2) .$] #solutionbox[ #math.equation(block: true, alt: "6 z to the power 6")[$6 z^(6)$] ] ] #examplebox("Example 15")[][ Find the quotient: #math.equation(block: false, alt: "the fraction 45 a squared b cubed over −5 a b to the power 5 .")[$frac(45 a^(2) b^(3), −5 a b^(5)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 45 a squared b cubed over −5 a b to the power 5")[$frac(45 a^(2) b^(3), −5 a b^(5))$]]), [Use fraction multiplication.], [#math.equation(block: false, alt: "the fraction 45 over −5 times the fraction a squared over a times the fraction b cubed over b to the power 5")[$frac(45, −5) · frac(a^(2), a) · frac(b^(3), b^(5))$]], [Simplify and use the Quotient Property.], [#math.equation(block: false, alt: "−9 times a times the fraction 1 over b squared")[$−9 · a · frac(1, b^(2))$]], [Multiply.], [#math.equation(block: false, alt: "minus the fraction 9 a over b squared")[$− frac(9 a, b^(2))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "the fraction −72 a to the power 7 b cubed over 8 a to the power 12 b to the power 4 .")[$frac(−72 a^(7) b^(3), 8 a^(12) b^(4)) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 9 over a to the power 5 b")[$− frac(9, a^(5) b)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "the fraction −63 c to the power 8 d cubed over 7 c to the power 12 d squared .")[$frac(−63 c^(8) d^(3), 7 c^(12) d^(2)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction −9 d over c to the power 4")[$frac(−9 d, c^(4))$] ] ] #examplebox("Example 16")[][ Find the quotient: #math.equation(block: false, alt: "the fraction 24 a to the power 5 b cubed over 48 a b to the power 4 .")[$frac(24 a^(5) b^(3), 48 a b^(4)) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 24 a to the power 5 b cubed over 48 a b to the power 4")[$frac(24 a^(5) b^(3), 48 a b^(4))$]]), [Use fraction multiplication.], [#math.equation(block: false, alt: "the fraction 24 over 48 times the fraction a to the power 5 over a times the fraction b cubed over b to the power 4")[$frac(24, 48) · frac(a^(5), a) · frac(b^(3), b^(4))$]], [Simplify and use the Quotient Property.], [#math.equation(block: false, alt: "the fraction 1 over 2 times a to the power 4 times the fraction 1 over b")[$frac(1, 2) · a^(4) · frac(1, b)$]], [Multiply.], [#math.equation(block: false, alt: "the fraction a to the power 4 over 2 b")[$frac(a^(4), 2 b)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "the fraction 16 a to the power 7 b to the power 6 over 24 a b to the power 8 .")[$frac(16 a^(7) b^(6), 24 a b^(8)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 a to the power 6 over 3 b squared")[$frac(2 a^(6), 3 b^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "the fraction 27 p to the power 4 q to the power 7 over −45 p to the power 12 q .")[$frac(27 p^(4) q^(7), −45 p^(12) q) .$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 q to the power 6 over 5 p to the power 8")[$− frac(3 q^(6), 5 p^(8))$] ] ] Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step. #examplebox("Example 17")[][ Find the quotient: #math.equation(block: false, alt: "the fraction 14 x to the power 7 y to the power 12 over 21 x to the power 11 y to the power 6 .")[$frac(14 x^(7) y^(12), 21 x^(11) y^(6)) .$] #solutionbox[ Be very careful to simplify #math.equation(block: false, alt: "the fraction 14 over 21")[$frac(14, 21)$] by dividing out a common factor, and to simplify the variables by subtracting their exponents. #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction 14 x to the power 7 y to the power 12 over 21 x to the power 11 y to the power 6")[$frac(14 x^(7) y^(12), 21 x^(11) y^(6))$]]), [Simplify and use the Quotient Property.], [#math.equation(block: false, alt: "the fraction 2 y to the power 6 over 3 x to the power 4")[$frac(2 y^(6), 3 x^(4))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "the fraction 28 x to the power 5 y to the power 14 over 49 x to the power 9 y to the power 12 .")[$frac(28 x^(5) y^(14), 49 x^(9) y^(12)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 y squared over 7 x to the power 4")[$frac(4 y^(2), 7 x^(4))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "the fraction 30 m to the power 5 n to the power 11 over 48 m to the power 10 n to the power 14 .")[$frac(30 m^(5) n^(11), 48 m^(10) n^(14)) .$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 8 m to the power 5 n cubed")[$frac(5, 8 m^(5) n^(3))$] ] ] In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction. This follows the order of operations. Remember, a fraction bar is a grouping symbol. #examplebox("Example 18")[][ Find the quotient: #math.equation(block: false, alt: "the fraction open parenthesis 6 x squared y cubed close parenthesis open parenthesis 5 x cubed y squared close parenthesis over open parenthesis 3 x to the power 4 y to the power 5 close parenthesis .")[$frac(( 6 x^(2) y^(3) ) ( 5 x^(3) y^(2) ), ( 3 x^(4) y^(5) )) .$] #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction open parenthesis 6 x squared y cubed close parenthesis open parenthesis 5 x cubed y squared close parenthesis over open parenthesis 3 x to the power 4 y to the power 5 close parenthesis")[$frac(( 6 x^(2) y^(3) ) ( 5 x^(3) y^(2) ), ( 3 x^(4) y^(5) ))$]]), [Simplify the numerator.], [#math.equation(block: false, alt: "the fraction 30 x to the power 5 y to the power 5 over 3 x to the power 4 y to the power 5")[$frac(30 x^(5) y^(5), 3 x^(4) y^(5))$]], [Simplify.], [#math.equation(block: false, alt: "10 x")[$10 x$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "the fraction open parenthesis 6 a to the power 4 b to the power 5 close parenthesis open parenthesis 4 a squared b to the power 5 close parenthesis over 12 a to the power 5 b to the power 8 .")[$frac(( 6 a^(4) b^(5) ) ( 4 a^(2) b^(5) ), 12 a^(5) b^(8)) .$] #solutionbox[ #math.equation(block: true, alt: "2 a b squared")[$2 a b^(2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the quotient: #math.equation(block: false, alt: "the fraction open parenthesis −12 x to the power 6 y to the power 9 close parenthesis open parenthesis −4 x to the power 5 y to the power 8 close parenthesis over −12 x to the power 10 y to the power 12 .")[$frac(( −12 x^(6) y^(9) ) ( −4 x^(5) y^(8) ), −12 x^(10) y^(12)) .$] #solutionbox[ #math.equation(block: true, alt: "−4 x y to the power 5")[$−4 x y^(5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access these online resources for additional instruction and practice with dividing monomials: - #link("https://openstax.org/l/25RationalExp")[Rational Expressions] - #link("https://openstax.org/l/25DivideMono")[Dividing Monomials] - #link("https://openstax.org/l/25DivideMono2")[Dividing Monomials 2] ] === Key Concepts - #strong[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 , n")[$m , n$] are whole numbers, then: #linebreak() #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 , 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 m minus n , n greater than m")[$frac(a^(m), a^(n)) = a^(m − n) , m > n #h(0.5em) "and" #h(0.5em) frac(a^(m), a^(n)) = frac(1, a^(m − n)) , n > m$] - #strong[Zero Exponent] - If #math.equation(block: false, alt: "a")[$a$] is a non-zero number, then #math.equation(block: false, alt: "a to the power 0 equals 1")[$a^(0) = 1$]. #linebreak() #linebreak() - #strong[Quotient to a Power Property for Exponents]: - If #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are real numbers, #math.equation(block: false, alt: "b not equal to 0 ,")[$b ≠ 0 ,$] and #math.equation(block: false, alt: "m")[$m$] is a counting number, then: #linebreak() #math.equation(block: false, alt: "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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m))$] - To raise a fraction to a power, raise the numerator and denominator to that power. #linebreak() #linebreak() - #strong[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 whole numbers, then #linebreak() #math.equation(block: false, 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 , m greater than n; 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 , a not equal to 0 , n greater than m; Zero Exponent Definition, a to the power o, 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")[$"Product Property" & & a^(m) · a^(n) & = & a^(m + n) \ "Power Property" & & attach(( a^(m) ), t: n) & = & a^(m · n) \ "Product to a Power" & & attach(( a b ), t: m) & = & a^(m) b^(m) \ "Quotient Property" & & frac(a^(m), a^(n)) & = & a^(m − n) , a ≠ 0 , m > n \ & & frac(a^(m), a^(n)) & = & frac(1, a^(n − m)) , a ≠ 0 , n > m \ "Zero Exponent Definition" & & a^(o) & = & 1 , a ≠ 0 \ "Quotient to a Power Property" & & attach(( frac(a, b) ), t: m) & = & frac(a^(m), b^(m)) , b ≠ 0$] ==== Practice Makes Perfect #strong[Simplify Expressions Using the Quotient Property for Exponents] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "the fraction x to the power 18 over x cubed")[$frac(x^(18), x^(3))$] ⓑ #math.equation(block: false, alt: "the fraction 5 to the power 12 over 5 cubed")[$frac(5^(12), 5^(3))$] ⓐ #math.equation(block: false, alt: "the fraction y to the power 20 over y to the power 10")[$frac(y^(20), y^(10))$] ⓑ #math.equation(block: false, alt: "the fraction 7 to the power 16 over 7 squared")[$frac(7^(16), 7^(2))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "y to the power 10")[$y^(10)$] ⓑ #math.equation(block: false, alt: "7 to the power 14")[$7^(14)$] ] ⓐ #math.equation(block: false, alt: "the fraction p to the power 21 over p to the power 7")[$frac(p^(21), p^(7))$] ⓑ #math.equation(block: false, alt: "the fraction 4 to the power 16 over 4 to the power 4")[$frac(4^(16), 4^(4))$] ⓐ #math.equation(block: false, alt: "the fraction u to the power 24 over u cubed")[$frac(u^(24), u^(3))$] ⓑ #math.equation(block: false, alt: "the fraction 9 to the power 15 over 9 to the power 5")[$frac(9^(15), 9^(5))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "u to the power 21")[$u^(21)$] ⓑ #math.equation(block: false, alt: "9 to the power 10")[$9^(10)$] ] ⓐ #math.equation(block: false, alt: "the fraction q to the power 18 over q to the power 36")[$frac(q^(18), q^(36))$] ⓑ #math.equation(block: false, alt: "the fraction 10 squared over 10 cubed")[$frac(10^(2), 10^(3))$] ⓐ #math.equation(block: false, alt: "the fraction t to the power 10 over t to the power 40")[$frac(t^(10), t^(40))$] ⓑ #math.equation(block: false, alt: "the fraction 8 cubed over 8 to the power 5")[$frac(8^(3), 8^(5))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over t to the power 30")[$frac(1, t^(30))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 64")[$frac(1, 64)$] ] ⓐ #math.equation(block: false, alt: "the fraction b over b to the power 9")[$frac(b, b^(9))$] ⓑ #math.equation(block: false, alt: "the fraction 4 over 4 to the power 6")[$frac(4, 4^(6))$] ⓐ #math.equation(block: false, alt: "the fraction x over x to the power 7")[$frac(x, x^(7))$] ⓑ #math.equation(block: false, alt: "the fraction 10 over 10 cubed")[$frac(10, 10^(3))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 1 over x to the power 6")[$frac(1, x^(6))$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 100")[$frac(1, 100)$] ] #strong[Simplify Expressions with Zero Exponents] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "20 to the power 0")[$20^(0)$] ⓑ #math.equation(block: false, alt: "b to the power 0")[$b^(0)$] ⓐ #math.equation(block: false, alt: "13 to the power 0")[$13^(0)$] ⓑ #math.equation(block: false, alt: "k to the power 0")[$k^(0)$] #solutionbox[ ⓐ 1 ⓑ 1 ] ⓐ #math.equation(block: false, alt: "− 27 to the power 0")[$"−" 27^(0)$] ⓑ #math.equation(block: false, alt: "− open parenthesis 27 to the power 0 close parenthesis")[$"−" ( 27^(0) )$] ⓐ #math.equation(block: false, alt: "− 15 to the power 0")[$"−" 15^(0)$] ⓑ #math.equation(block: false, alt: "− open parenthesis 15 to the power 0 close parenthesis")[$"−" ( 15^(0) )$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−1")[$−1$] ⓑ #math.equation(block: false, alt: "−1")[$−1$] ] ⓐ #math.equation(block: false, alt: "open parenthesis 25 x close parenthesis to the power 0")[$attach(( 25 x ), t: 0)$] ⓑ #math.equation(block: false, alt: "25 x to the power 0")[$25 x^(0)$] ⓐ #math.equation(block: false, alt: "open parenthesis 6 y close parenthesis to the power 0")[$attach(( 6 y ), t: 0)$] ⓑ #math.equation(block: false, alt: "6 y to the power 0")[$6 y^(0)$] #solutionbox[ ⓐ 1 ⓑ 6 ] ⓐ #math.equation(block: false, alt: "open parenthesis 12 x close parenthesis to the power 0")[$attach(( 12 x ), t: 0)$] ⓑ #math.equation(block: false, alt: "open parenthesis −56 p to the power 4 q cubed close parenthesis to the power 0")[$attach(( −56 p^(4) q^(3) ), t: 0)$] ⓐ #math.equation(block: false, alt: "7 y to the power 0")[$7 y^(0)$]#math.equation(block: false, alt: "open parenthesis 17 y close parenthesis to the power 0")[$attach(( 17 y ), t: 0)$] ⓑ #math.equation(block: false, alt: "open parenthesis −93 c to the power 7 d to the power 15 close parenthesis to the power 0")[$attach(( −93 c^(7) d^(15) ), t: 0)$] #solutionbox[ ⓐ 7 ⓑ 1 ] ⓐ #math.equation(block: false, alt: "12 n to the power 0 minus 18 m to the power 0")[$12 n^(0) − 18 m^(0)$] ⓑ #math.equation(block: false, alt: "open parenthesis 12 n close parenthesis to the power 0 minus open parenthesis 18 m close parenthesis to the power 0")[$attach(( 12 n ), t: 0) − attach(( 18 m ), t: 0)$] ⓐ #math.equation(block: false, alt: "15 r to the power 0 minus 22 s to the power 0")[$15 r^(0) − 22 s^(0)$] ⓑ #math.equation(block: false, alt: "open parenthesis 15 r close parenthesis to the power 0 minus open parenthesis 22 s close parenthesis to the power 0")[$attach(( 15 r ), t: 0) − attach(( 22 s ), t: 0)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−7")[$−7$] ⓑ 0 ] #strong[Simplify Expressions Using the Quotient to a Power Property] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 3 over 4 close parenthesis cubed")[$attach(( frac(3, 4) ), t: 3)$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction p over 2 close parenthesis to the power 5")[$attach(( frac(p, 2) ), t: 5)$] ⓒ #math.equation(block: false, alt: "open parenthesis the fraction x over y close parenthesis to the power 6")[$attach(( frac(x, y) ), t: 6)$] ⓐ #math.equation(block: false, alt: "open parenthesis the fraction 2 over 5 close parenthesis squared")[$attach(( frac(2, 5) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction x over 3 close parenthesis to the power 4")[$attach(( frac(x, 3) ), t: 4)$] ⓒ #math.equation(block: false, alt: "open parenthesis the fraction a over b close parenthesis to the power 5")[$attach(( frac(a, b) ), t: 5)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 4 over 25")[$frac(4, 25)$] ⓑ #math.equation(block: false, alt: "the fraction x to the power 4 over 81")[$frac(x^(4), 81)$] ⓒ #math.equation(block: false, alt: "the fraction a to the power 5 over b to the power 5")[$frac(a^(5), b^(5))$] ] ⓐ #math.equation(block: false, alt: "open parenthesis the fraction a over 3 b close parenthesis to the power 4")[$attach(( frac(a, 3 b) ), t: 4)$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 5 over 4 m close parenthesis squared")[$attach(( frac(5, 4 m) ), t: 2)$] ⓐ #math.equation(block: false, alt: "open parenthesis the fraction x over 2 y close parenthesis cubed")[$attach(( frac(x, 2 y) ), t: 3)$] ⓑ #math.equation(block: false, alt: "open parenthesis the fraction 10 over 3 q close parenthesis to the power 4")[$attach(( frac(10, 3 q) ), t: 4)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction x cubed over 8 y cubed")[$frac(x^(3), 8 y^(3))$] ⓑ #math.equation(block: false, alt: "the fraction 10,000 over 81 q to the power 4")[$frac(10","000, 81 q^(4))$] ] #strong[Simplify Expressions by Applying Several Properties] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction open parenthesis a squared close parenthesis cubed over a to the power 4")[$frac(attach(( a^(2) ), t: 3), a^(4))$] #math.equation(block: true, alt: "the fraction open parenthesis p cubed close parenthesis to the power 4 over p to the power 5")[$frac(attach(( p^(3) ), t: 4), p^(5))$] #solutionbox[ #math.equation(block: true, alt: "p to the power 7")[$p^(7)$] ] #math.equation(block: true, alt: "the fraction open parenthesis y cubed close parenthesis to the power 4 over y to the power 10")[$frac(attach(( y^(3) ), t: 4), y^(10))$] #math.equation(block: true, alt: "the fraction open parenthesis x to the power 4 close parenthesis to the power 5 over x to the power 15")[$frac(attach(( x^(4) ), t: 5), x^(15))$] #solutionbox[ #math.equation(block: true, alt: "x to the power 5")[$x^(5)$] ] #math.equation(block: true, alt: "the fraction u to the power 6 over open parenthesis u cubed close parenthesis squared")[$frac(u^(6), attach(( u^(3) ), t: 2))$] #math.equation(block: true, alt: "the fraction v to the power 20 over open parenthesis v to the power 4 close parenthesis to the power 5")[$frac(v^(20), attach(( v^(4) ), t: 5))$] #solutionbox[ 1 ] #math.equation(block: true, alt: "the fraction m to the power 12 over open parenthesis m to the power 8 close parenthesis cubed")[$frac(m^(12), attach(( m^(8) ), t: 3))$] #math.equation(block: true, alt: "the fraction n to the power 8 over open parenthesis n to the power 6 close parenthesis to the power 4")[$frac(n^(8), attach(( n^(6) ), t: 4))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over n to the power 16")[$frac(1, n^(16))$] ] #math.equation(block: true, alt: "open parenthesis the fraction p to the power 9 over p cubed close parenthesis to the power 5")[$attach(( frac(p^(9), p^(3)) ), t: 5)$] #math.equation(block: true, alt: "open parenthesis the fraction q to the power 8 over q squared close parenthesis cubed")[$attach(( frac(q^(8), q^(2)) ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "q to the power 18")[$q^(18)$] ] #math.equation(block: true, alt: "open parenthesis the fraction r squared over r to the power 6 close parenthesis cubed")[$attach(( frac(r^(2), r^(6)) ), t: 3)$] #math.equation(block: true, alt: "open parenthesis the fraction m to the power 4 over m to the power 7 close parenthesis to the power 4")[$attach(( frac(m^(4), m^(7)) ), t: 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over m to the power 12")[$frac(1, m^(12))$] ] #math.equation(block: true, alt: "open parenthesis the fraction p over r to the power 11 close parenthesis squared")[$attach(( frac(p, r^(11)) ), t: 2)$] #math.equation(block: true, alt: "open parenthesis the fraction a over b to the power 6 close parenthesis cubed")[$attach(( frac(a, b^(6)) ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction a cubed over b to the power 18")[$frac(a^(3), b^(18))$] ] #math.equation(block: true, alt: "open parenthesis the fraction w to the power 5 over x cubed close parenthesis to the power 8")[$attach(( frac(w^(5), x^(3)) ), t: 8)$] #math.equation(block: true, alt: "open parenthesis the fraction y to the power 4 over z to the power 10 close parenthesis to the power 5")[$attach(( frac(y^(4), z^(10)) ), t: 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction y to the power 20 over z to the power 50")[$frac(y^(20), z^(50))$] ] #math.equation(block: true, alt: "open parenthesis the fraction 2 j cubed over 3 k close parenthesis to the power 4")[$attach(( frac(2 j^(3), 3 k) ), t: 4)$] #math.equation(block: true, alt: "open parenthesis the fraction 3 m to the power 5 over 5 n close parenthesis cubed")[$attach(( frac(3 m^(5), 5 n) ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 27 m to the power 15 over 125 n cubed")[$frac(27 m^(15), 125 n^(3))$] ] #math.equation(block: true, alt: "open parenthesis the fraction 3 c squared over 4 d to the power 6 close parenthesis cubed")[$attach(( frac(3 c^(2), 4 d^(6)) ), t: 3)$] #math.equation(block: true, alt: "open parenthesis the fraction 5 u to the power 7 over 2 v cubed close parenthesis to the power 4")[$attach(( frac(5 u^(7), 2 v^(3)) ), t: 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 625 u to the power 28 over 16 v to the power to the power 12")[$frac(625 u^(28), 16 v^(12))$] ] #math.equation(block: true, alt: "open parenthesis the fraction k squared k to the power 8 over k cubed close parenthesis squared")[$attach(( frac(k^(2) k^(8), k^(3)) ), t: 2)$] #math.equation(block: true, alt: "open parenthesis the fraction j squared j to the power 5 over j to the power 4 close parenthesis cubed")[$attach(( frac(j^(2) j^(5), j^(4)) ), t: 3)$] #solutionbox[ #math.equation(block: true, alt: "j to the power 9")[$j^(9)$] ] #math.equation(block: true, alt: "the fraction open parenthesis t squared close parenthesis to the power 5 open parenthesis t to the power 4 close parenthesis squared over open parenthesis t cubed close parenthesis to the power 7")[$frac(attach(( t^(2) ), t: 5) attach(( t^(4) ), t: 2), attach(( t^(3) ), t: 7))$] #math.equation(block: true, alt: "the fraction open parenthesis q cubed close parenthesis to the power 6 open parenthesis q squared close parenthesis cubed over open parenthesis q to the power 4 close parenthesis to the power 8")[$frac(attach(( q^(3) ), t: 6) attach(( q^(2) ), t: 3), attach(( q^(4) ), t: 8))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over q to the power 8")[$frac(1, q^(8))$] ] #math.equation(block: true, alt: "the fraction open parenthesis −2 p squared close parenthesis to the power 4 open parenthesis 3 p to the power 4 close parenthesis squared over open parenthesis −6 p cubed close parenthesis squared")[$frac(attach(( −2 p^(2) ), t: 4) attach(( 3 p^(4) ), t: 2), attach(( −6 p^(3) ), t: 2))$] #math.equation(block: true, alt: "the fraction open parenthesis −2 k cubed close parenthesis squared open parenthesis 6 k squared close parenthesis to the power 4 over open parenthesis 9 k to the power 4 close parenthesis squared")[$frac(attach(( −2 k^(3) ), t: 2) attach(( 6 k^(2) ), t: 4), attach(( 9 k^(4) ), t: 2))$] #solutionbox[ #math.equation(block: true, alt: "64 k to the power 6")[$64 k^(6)$] ] #math.equation(block: true, alt: "the fraction open parenthesis −4 m cubed close parenthesis squared open parenthesis 5 m to the power 4 close parenthesis cubed over open parenthesis −10 m to the power 6 close parenthesis cubed")[$frac(attach(( −4 m^(3) ), t: 2) attach(( 5 m^(4) ), t: 3), attach(( −10 m^(6) ), t: 3))$] #math.equation(block: true, alt: "the fraction open parenthesis −10 n squared close parenthesis cubed open parenthesis 4 n to the power 5 close parenthesis squared over open parenthesis 2 n to the power 8 close parenthesis squared")[$frac(attach(( −10 n^(2) ), t: 3) attach(( 4 n^(5) ), t: 2), attach(( 2 n^(8) ), t: 2))$] #solutionbox[ #math.equation(block: true, alt: "−4,000")[$−4,000$] ] #strong[Divide Monomials] In the following exercises, divide the monomials. #math.equation(block: true, alt: "56 b to the power 8 divided by 7 b squared")[$56 b^(8) ÷ 7 b^(2)$] #math.equation(block: true, alt: "63 v to the power 10 divided by 9 v squared")[$63 v^(10) ÷ 9 v^(2)$] #solutionbox[ #math.equation(block: true, alt: "7 v to the power 8")[$7 v^(8)$] ] #math.equation(block: true, alt: "−88 y to the power 15 divided by 8 y cubed")[$−88 y^(15) ÷ 8 y^(3)$] #math.equation(block: true, alt: "−72 u to the power 12 divided by 1 2 u to the power 4")[$−72 u^(12) ÷ 1 2 u^(4)$] #solutionbox[ #math.equation(block: true, alt: "−6 u to the power 8")[$−6 u^(8)$] ] #math.equation(block: true, alt: "the fraction 45 a to the power 6 b to the power 8 over −15 a to the power 10 b squared")[$frac(45 a^(6) b^(8), −15 a^(10) b^(2))$] #math.equation(block: true, alt: "the fraction 54 x to the power 9 y cubed over −18 x to the power 6 y to the power 15")[$frac(54 x^(9) y^(3), −18 x^(6) y^(15))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 3 x cubed over y to the power 12")[$− frac(3 x^(3), y^(12))$] ] #math.equation(block: true, alt: "the fraction 15 r to the power 4 s to the power 9 over 18 r to the power 9 s squared")[$frac(15 r^(4) s^(9), 18 r^(9) s^(2))$] #math.equation(block: true, alt: "the fraction 20 m to the power 8 n to the power 4 over 30 m to the power 5 n to the power 9")[$frac(20 m^(8) n^(4), 30 m^(5) n^(9))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 m cubed over 3 n to the power 5")[$frac(2 m^(3), 3 n^(5))$] ] #math.equation(block: true, alt: "the fraction 18 a to the power 4 b to the power 8 over −27 a to the power 9 b to the power 5")[$frac(18 a^(4) b^(8), −27 a^(9) b^(5))$] #math.equation(block: true, alt: "the fraction 45 x to the power 5 y to the power 9 over −60 x to the power 8 y to the power 6")[$frac(45 x^(5) y^(9), −60 x^(8) y^(6))$] #solutionbox[ #math.equation(block: true, alt: "the fraction −3 y cubed over 4 x cubed")[$frac(−3 y^(3), 4 x^(3))$] ] #math.equation(block: true, alt: "the fraction 64 q to the power 11 r to the power 9 s cubed over 48 q to the power 6 r to the power 8 s to the power 5")[$frac(64 q^(11) r^(9) s^(3), 48 q^(6) r^(8) s^(5))$] #math.equation(block: true, alt: "the fraction 65 a to the power 10 b to the power 8 c to the power 5 over 42 a to the power 7 b to the power 6 c to the power 8")[$frac(65 a^(10) b^(8) c^(5), 42 a^(7) b^(6) c^(8))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 65 a cubed b squared over 42 c cubed")[$frac(65 a^(3) b^(2), 42 c^(3))$] ] #math.equation(block: true, alt: "the fraction open parenthesis 10 m to the power 5 n to the power 4 close parenthesis open parenthesis 5 m cubed n to the power 6 close parenthesis over 25 m to the power 7 n to the power 5")[$frac(( 10 m^(5) n^(4) ) ( 5 m^(3) n^(6) ), 25 m^(7) n^(5))$] #math.equation(block: true, alt: "the fraction open parenthesis −18 p to the power 4 q to the power 7 close parenthesis open parenthesis −6 p cubed q to the power 8 close parenthesis over −36 p to the power 12 q to the power 10")[$frac(( −18 p^(4) q^(7) ) ( −6 p^(3) q^(8) ), −36 p^(12) q^(10))$] #solutionbox[ #math.equation(block: true, alt: "the fraction −3 q to the power 5 over p to the power 5")[$frac(−3 q^(5), p^(5))$] ] #math.equation(block: true, alt: "the fraction open parenthesis 6 a to the power 4 b cubed close parenthesis open parenthesis 4 a b to the power 5 close parenthesis over open parenthesis 12 a squared b close parenthesis open parenthesis a cubed b close parenthesis")[$frac(( 6 a^(4) b^(3) ) ( 4 a b^(5) ), ( 12 a^(2) b ) ( a^(3) b ))$] #math.equation(block: true, alt: "the fraction open parenthesis 4 u squared v to the power 5 close parenthesis open parenthesis 15 u cubed v close parenthesis over open parenthesis 12 u cubed v close parenthesis open parenthesis u to the power 4 v close parenthesis")[$frac(( 4 u^(2) v^(5) ) ( 15 u^(3) v ), ( 12 u^(3) v ) ( u^(4) v ))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 v to the power 4 over u squared")[$frac(5 v^(4), u^(2))$] ] #strong[Mixed Practice] ⓐ #math.equation(block: false, alt: "24 a to the power 5 plus 2 a to the power 5")[$24 a^(5) + 2 a^(5)$] ⓑ #math.equation(block: false, alt: "24 a to the power 5 minus 2 a to the power 5")[$24 a^(5) − 2 a^(5)$] ⓒ #math.equation(block: false, alt: "24 a to the power 5 times 2 a to the power 5")[$24 a^(5) · 2 a^(5)$] ⓓ #math.equation(block: false, alt: "24 a to the power 5 divided by 2 a to the power 5")[$24 a^(5) ÷ 2 a^(5)$] ⓐ #math.equation(block: false, alt: "15 n to the power 10 plus 3 n to the power 10")[$15 n^(10) + 3 n^(10)$] ⓑ #math.equation(block: false, alt: "15 n to the power 10 minus 3 n to the power 10")[$15 n^(10) − 3 n^(10)$] ⓒ #math.equation(block: false, alt: "15 n to the power 10 times 3 n to the power 10")[$15 n^(10) · 3 n^(10)$] ⓓ #math.equation(block: false, alt: "15 n to the power 10 divided by 3 n to the power 10")[$15 n^(10) ÷ 3 n^(10)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "18 n to the power 10")[$18 n^(10)$] ⓑ #math.equation(block: false, alt: "12 n to the power 10")[$12 n^(10)$] ⓒ #math.equation(block: false, alt: "45 n to the power 20")[$45 n^(20)$] ⓓ 5 ] ⓐ #math.equation(block: false, alt: "p to the power 4 times p to the power 6")[$p^(4) · p^(6)$] ⓑ #math.equation(block: false, alt: "open parenthesis p to the power 4 close parenthesis to the power 6")[$attach(( p^(4) ), t: 6)$] ⓐ #math.equation(block: false, alt: "q to the power 5 times q cubed")[$q^(5) · q^(3)$] ⓑ #math.equation(block: false, alt: "open parenthesis q to the power 5 close parenthesis cubed")[$attach(( q^(5) ), t: 3)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "q to the power 8")[$q^(8)$] ⓑ #math.equation(block: false, alt: "q to the power 15")[$q^(15)$] ] ⓐ #math.equation(block: false, alt: "the fraction y cubed over y")[$frac(y^(3), y)$] ⓑ #math.equation(block: false, alt: "the fraction y over y cubed")[$frac(y, y^(3))$] ⓐ #math.equation(block: false, alt: "the fraction z to the power 6 over z to the power 5")[$frac(z^(6), z^(5))$] ⓑ #math.equation(block: false, alt: "the fraction z to the power 5 over z to the power 6")[$frac(z^(5), z^(6))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "z")[$z$] ⓑ #math.equation(block: false, alt: "the fraction 1 over z")[$frac(1, z)$] ] #math.equation(block: true, alt: "open parenthesis 8 x to the power 5 close parenthesis open parenthesis 9 x close parenthesis divided by 6 x cubed")[$( 8 x^(5) ) ( 9 x ) ÷ 6 x^(3)$] #math.equation(block: true, alt: "open parenthesis 4 y close parenthesis open parenthesis 12 y to the power 7 close parenthesis divided by 8 y squared")[$( 4 y ) ( 12 y^(7) ) ÷ 8 y^(2)$] #solutionbox[ #math.equation(block: true, alt: "6 y to the power 6")[$6 y^(6)$] ] #math.equation(block: true, alt: "the fraction 27 a to the power 7 over 3 a cubed plus the fraction 54 a to the power 9 over 9 a to the power 5")[$frac(27 a^(7), 3 a^(3)) + frac(54 a^(9), 9 a^(5))$] #math.equation(block: true, alt: "the fraction 32 c to the power 11 over 4 c to the power 5 plus the fraction 42 c to the power 9 over 6 c cubed")[$frac(32 c^(11), 4 c^(5)) + frac(42 c^(9), 6 c^(3))$] #solutionbox[ #math.equation(block: true, alt: "15 c to the power 6")[$15 c^(6)$] ] #math.equation(block: true, alt: "the fraction 32 y to the power 5 over 8 y squared minus the fraction 60 y to the power 10 over 5 y to the power 7")[$frac(32 y^(5), 8 y^(2)) − frac(60 y^(10), 5 y^(7))$] #math.equation(block: true, alt: "the fraction 48 x to the power 6 over 6 x to the power 4 minus the fraction 35 x to the power 9 over 7 x to the power 7")[$frac(48 x^(6), 6 x^(4)) − frac(35 x^(9), 7 x^(7))$] #solutionbox[ #math.equation(block: true, alt: "3 x squared")[$3 x^(2)$] ] #math.equation(block: true, alt: "the fraction 63 r to the power 6 s cubed over 9 r to the power 4 s squared minus the fraction 72 r squared s squared over 6 s")[$frac(63 r^(6) s^(3), 9 r^(4) s^(2)) − frac(72 r^(2) s^(2), 6 s)$] #math.equation(block: true, alt: "the fraction 56 y to the power 4 z to the power 5 over 7 y cubed z cubed minus the fraction 45 y squared z squared over 5 y")[$frac(56 y^(4) z^(5), 7 y^(3) z^(3)) − frac(45 y^(2) z^(2), 5 y)$] #solutionbox[ #math.equation(block: true, alt: "− y z squared")[$"−" y z^(2)$] ] ==== Everyday Math #strong[Memory] One megabyte is approximately #math.equation(block: false, alt: "10 to the power 6")[$10^(6)$] bytes. One gigabyte is approximately #math.equation(block: false, alt: "10 to the power 9")[$10^(9)$] bytes. How many megabytes are in one gigabyte? #strong[Memory] One gigabyte is approximately #math.equation(block: false, alt: "10 to the power 9")[$10^(9)$] bytes. One terabyte is approximately #math.equation(block: false, alt: "10 to the power 12")[$10^(12)$] bytes. How many gigabytes are in one terabyte? #solutionbox[ #math.equation(block: true, alt: "10 cubed")[$10^(3)$] ] ==== Writing Exercises Jennifer thinks the quotient #math.equation(block: false, alt: "the fraction a to the power 24 over a to the power 6")[$frac(a^(24), a^(6))$] simplifies to #math.equation(block: false, alt: "a to the power 4")[$a^(4)$]. What is wrong with her reasoning? Maurice simplifies the quotient #math.equation(block: false, alt: "the fraction d to the power 7 over d")[$frac(d^(7), d)$] by writing #math.equation(block: false, alt: "the fraction d to the power 7 over d equals 7")[$frac(cancel(d)^(7), cancel(d)) = 7$]. What is wrong with his reasoning? #solutionbox[ Answers will vary. ] When Drake simplified #math.equation(block: false, alt: "− 3 to the power 0")[$"−" 3^(0)$] and #math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power 0")[$attach(( −3 ), t: 0)$] he got the same answer. Explain how using the Order of Operations correctly gives different answers. Robert thinks #math.equation(block: false, alt: "x to the power 0")[$x^(0)$] simplifies to 0. What would you say to convince Robert he is wrong? #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 “simplify expressions using the Quotient Property for Exponents,” “simplify expressions with zero exponents,” “simplify expressions using the Quotient to a Power Property,” “simplify expressions by applying several properties,” and “divide monomials.” 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 “simplify expressions using the Quotient Property for Exponents,” “simplify expressions with zero exponents,” “simplify expressions using the Quotient to a Power Property,” “simplify expressions by applying several properties,” and “divide monomials.” The rest of the cells are blank.", caption: none) ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?