#set document(title: "9.7 Higher Roots", 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")) == 9.7#h(0.6em)Higher Roots #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Simplify: #math.equation(block: false, alt: "y to the power 5 y to the power 4")[$y^(5) y^(4)$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "y to the power 9")[$y^(9)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "open parenthesis n squared close parenthesis to the power 6")[$attach(( n^(2) ), t: 6)$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "n to the power 12")[$n^(12)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction x to the power 8 over x cubed")[$frac(x^(8), x^(3))$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "x to the power 5")[$x^(5)$] ] ] === Simplify Expressions with Higher Roots Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots. Let’s review some vocabulary first. #math.equation(block: true, alt: "We write:, We say:; n squared, n squared; n cubed, n cubed; n to the power 4, n to the fourth; n to the power 5, n to the fifth")[$"We write:" & & #h(2em) "We say:" \ n^(2) & & #h(2em) n #h(0.2em) "squared" \ n^(3) & & #h(2em) n #h(0.2em) "cubed" \ n^(4) & & #h(2em) n #h(0.2em) "to the fourth" \ n^(5) & & #h(2em) n #h(0.2em) "to the fifth"$]The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube. It will be helpful to have a table of the powers of the integers from #math.equation(block: false, alt: "−5 to 5")[$−5 #h(0.2em) "to" #h(0.2em) 5$]. #figure(figph[This figure consists of two tables. The first table shows the results of raising the numbers 1, 2, 3, 4, 5, x, and x squared to the second, third, fourth, and fifth powers. The second table shows the results of raising the numbers negative one through negative five to the second, third, fourth, and fifth powers. The table first has five columns and nine rows. The second has five columns and seven rows. The columns in both tables are labeled, “Number,” “Square,” “Cube,” “Fourth power,” “Fifth power,” nothing, “Number,” “Square,” “Cube,” “Fourth power,” and “Fifth power.” In both tables, the next row reads: n, n squared, n cubed, n to the fourth power, n to the fifth power, nothing, n, n squared, n cubed, n to the fourth power, and n to the fifth power. In the first table, 1 squared, 1 cubed, 1 to the fourth power, and 1 to the fifth power are all shown to be 1. In the next row, 2 squared is 4, 2 cubed is 8, 2 to the fourth power is 16, and 2 to the fifth power is 32. In the next row, 3 squared is 9, 3 cubed is 27, 3 to the fourth power is 81, and 3 to the fifth power is 243. In the next row, 4 squared is 16, 4 cubed is 64, 4 to the fourth power is 246, and 4 to the fifth power is 1024. In the next row, 5 squared is 25, 5 cubed is 125, 5 to the fourth power is 625, and 5 to the fifth power is 3125. In the next row, x squared, x cubed, x to the fourth power, and x to the fifth power are listed. In the next row, x squared squared is x to the fourth power, x cubed squared is x to the fifth power, x squared to the fourth power is x to the eighth power, and x squared to the fifth power is x to the tenth power. In the second table, negative 1 squared is 1, negative 1 cubed is negative 1, negative 1 to the fourth power is 1, and negative 1 to the fifth power is negative 1. In the next row, negative 2 squared is 4, negative 2 cubed is negative 8, negative 2 to the fourth power is 16, and negative 2 to the fifth power is negative 32. In the next row, negative 4 squared is 16, negative 4 cubed is negative 64, negative 4 to the fourth power is 256, and negative 4 to the fifth power is negative 1024. In the next row, negative 5 squared is 25, negative 5 cubed is negative 125, negative 5 to the fourth power is 625, and negative 5 to the fifth power is negative 3125.], alt: "This figure consists of two tables. The first table shows the results of raising the numbers 1, 2, 3, 4, 5, x, and x squared to the second, third, fourth, and fifth powers. The second table shows the results of raising the numbers negative one through negative five to the second, third, fourth, and fifth powers. The table first has five columns and nine rows. The second has five columns and seven rows. The columns in both tables are labeled, “Number,” “Square,” “Cube,” “Fourth power,” “Fifth power,” nothing, “Number,” “Square,” “Cube,” “Fourth power,” and “Fifth power.” In both tables, the next row reads: n, n squared, n cubed, n to the fourth power, n to the fifth power, nothing, n, n squared, n cubed, n to the fourth power, and n to the fifth power. In the first table, 1 squared, 1 cubed, 1 to the fourth power, and 1 to the fifth power are all shown to be 1. In the next row, 2 squared is 4, 2 cubed is 8, 2 to the fourth power is 16, and 2 to the fifth power is 32. In the next row, 3 squared is 9, 3 cubed is 27, 3 to the fourth power is 81, and 3 to the fifth power is 243. In the next row, 4 squared is 16, 4 cubed is 64, 4 to the fourth power is 246, and 4 to the fifth power is 1024. In the next row, 5 squared is 25, 5 cubed is 125, 5 to the fourth power is 625, and 5 to the fifth power is 3125. In the next row, x squared, x cubed, x to the fourth power, and x to the fifth power are listed. In the next row, x squared squared is x to the fourth power, x cubed squared is x to the fifth power, x squared to the fourth power is x to the eighth power, and x squared to the fifth power is x to the tenth power. In the second table, negative 1 squared is 1, negative 1 cubed is negative 1, negative 1 to the fourth power is 1, and negative 1 to the fifth power is negative 1. In the next row, negative 2 squared is 4, negative 2 cubed is negative 8, negative 2 to the fourth power is 16, and negative 2 to the fifth power is negative 32. In the next row, negative 4 squared is 16, negative 4 cubed is negative 64, negative 4 to the fourth power is 256, and negative 4 to the fifth power is negative 1024. In the next row, negative 5 squared is 25, negative 5 cubed is negative 125, negative 5 to the fourth power is 625, and negative 5 to the fifth power is negative 3125.", caption: [First through fifth powers of integers from #math.equation(block: false, alt: "−5")[$−5$] to #math.equation(block: false, alt: "5 .")[$5 .$]]) Notice the signs in . All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of #math.equation(block: false, alt: "−2")[$−2$] below to help you see this. #figure(figph[This figure has five columns and two rows. The first row labels each column: n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row reads: negative 2, 4, negative 8, 16, and negative 32.], alt: "This figure has five columns and two rows. The first row labels each column: n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row reads: negative 2, 4, negative 8, 16, and negative 32.", caption: none) Earlier in this chapter we defined the square root of a number. #math.equation(block: true, alt: "If n squared equals m , then n is a square root of m .")[$"If" #h(0.2em) n^(2) = m , #h(0.2em) "then" #h(0.2em) n #h(0.2em) "is a square root of" #h(0.2em) m .$]And we have used the notation #math.equation(block: false, alt: "the square root of m")[$sqrt(m)$] to denote the #strong[principal square root]. So #math.equation(block: false, alt: "the square root of m greater than or equal to 0")[$sqrt(m) ≥ 0$] always. We will now extend the definition to higher roots. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[#emph[n]th Root of a Number] If #math.equation(block: false, alt: "b to the power n equals a")[$b^(n) = a$], then #math.equation(block: false, alt: "b")[$b$] is an #strong[#emph[n]th root of a number] #math.equation(block: false, alt: "a")[$a$]. The principal #emph[n]th root of #math.equation(block: false, alt: "a")[$a$] is written #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$]. #emph[n] is called the #strong[index] of the radical. ] We do not write the index for a square root. Just like we use the word ‘cubed’ for #math.equation(block: false, alt: "b cubed")[$b^(3)$], we use the term ‘cube root’ for #math.equation(block: false, alt: "the cube root of a")[$root(3, a)$]. We refer to to help us find higher roots. #math.equation(block: true, alt: "4 cubed, equals, 64, the cube root of 64, equals, 4; 3 to the power 4, equals, 81, the 4th root of 81, equals, 3; open parenthesis −2 close parenthesis to the power 5, equals, −32, the 5th root of −32, equals, −2")[$4^(3) & = & 64 & & #h(4em) root(3, 64) & = & 4 \ 3^(4) & = & 81 & & #h(4em) root(4, 81) & = & 3 \ attach(( −2 ), t: 5) & = & −32 & & #h(4em) root(5, −32) & = & −2$]Could we have an even root of a negative number? No. We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$]] When #math.equation(block: false, alt: "n")[$n$] is an even number and - #math.equation(block: false, alt: "a greater than or equal to 0")[$a ≥ 0$], then #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number - #math.equation(block: false, alt: "a less than 0")[$a < 0$], then #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is not a real number When #math.equation(block: false, alt: "n")[$n$] is an odd number, #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number for all values of #math.equation(block: false, alt: "a")[$a$]. ] #examplebox("Example 1")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 8")[$root(3, 8)$] ⓑ #math.equation(block: false, alt: "the 4th root of 81")[$root(4, 81)$] ⓒ #math.equation(block: false, alt: "the 5th root of 32")[$root(5, 32)$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of 8")[$root(3, 8)$]]), [Since #math.equation(block: false, alt: "open parenthesis 2 close parenthesis cubed equals 8 .")[$attach(( 2 ), t: 3) = 8 .$]], [#math.equation(block: false, alt: "2")[$2$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of 81")[$root(4, 81)$]]), [Since #math.equation(block: false, alt: "open parenthesis 3 close parenthesis to the power 4 equals 81 .")[$attach(( 3 ), t: 4) = 81 .$]], [#math.equation(block: false, alt: "3")[$3$]], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 5th root of 32")[$root(5, 32)$]]), [Since #math.equation(block: false, alt: "open parenthesis 2 close parenthesis to the power 5 equals 32 .")[$attach(( 2 ), t: 5) = 32 .$]], [#math.equation(block: false, alt: "2")[$2$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 27")[$root(3, 27)$] ⓑ #math.equation(block: false, alt: "the 4th root of 256")[$root(4, 256)$] ⓒ #math.equation(block: false, alt: "the 5th root of 243")[$root(5, 243)$]. #solutionbox[ ⓐ 3 ⓑ 4 ⓒ 3 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 1000")[$root(3, 1000)$] ⓑ #math.equation(block: false, alt: "the 4th root of 16")[$root(4, 16)$] ⓒ #math.equation(block: false, alt: "the 5th root of 32")[$root(5, 32)$]. #solutionbox[ ⓐ 10 ⓑ 2 ⓒ 2 ] ] #examplebox("Example 2")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of −64")[$root(3, −64)$] ⓑ #math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$] ⓒ #math.equation(block: false, alt: "the 5th root of −243")[$root(5, −243)$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of −64")[$root(3, −64)$]]), [Since #math.equation(block: false, alt: "open parenthesis −4 close parenthesis cubed equals −64 .")[$attach(( −4 ), t: 3) = −64 .$]], [#math.equation(block: false, alt: "−4")[$−4$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$]]), [Think, #math.equation(block: false, alt: "open parenthesis ? close parenthesis to the power 4 equals −16 .")[$attach(( ? ), t: 4) = −16 .$] No real number raised to the fourth power is negative.], [Not a real number.], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 5th root of −243")[$root(5, −243)$]]), [Since #math.equation(block: false, alt: "open parenthesis −3 close parenthesis to the power 5 equals −243 .")[$attach(( −3 ), t: 5) = −243 .$]], [#math.equation(block: false, alt: "−3")[$−3$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of −125")[$root(3, −125)$] ⓑ #math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$] ⓒ #math.equation(block: false, alt: "the 5th root of −32")[$root(5, −32)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "−5")[$−5$] ⓑ not real ⓒ #math.equation(block: false, alt: "−2")[$−2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of −216")[$root(3, −216)$] ⓑ #math.equation(block: false, alt: "the 4th root of −81")[$root(4, −81)$] ⓒ #math.equation(block: false, alt: "the 5th root of −1024")[$root(5, −1024)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "−6")[$−6$] ⓑ not real ⓒ #math.equation(block: false, alt: "−4")[$−4$] ] ] When we worked with square roots that had variables in the radicand, we restricted the variables to non-negative values. Now we will remove this restriction. The odd root of a number can be either positive or negative. We have seen that #math.equation(block: false, alt: "the cube root of −64 equals −4")[$root(3, −64) = −4$]. But the even root of a non-negative number is always non-negative, because we take the #strong[principal #emph[n]th root]. Suppose we start with #math.equation(block: false, alt: "a equals −5")[$a = −5$]. #math.equation(block: true, alt: "open parenthesis −5 close parenthesis to the power 4 equals 625, the 4th root of 625 equals 5")[$attach(( −5 ), t: 4) = 625 & & #h(4em) root(4, 625) = 5$]How can we make sure the fourth root of −5 raised to the fourth power, #math.equation(block: false, alt: "open parenthesis −5 close parenthesis to the power 4")[$attach(( −5 ), t: 4)$] is 5? We will see in the following property. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Simplifying Odd and Even Roots] For any integer #math.equation(block: false, alt: "n greater than or equal to 2")[$n ≥ 2$], #math.equation(block: true, alt: "when n is odd, the nth root of a to the power n equals a; when n is even, the nth root of a to the power n equals | a |")[$"when" #h(0.2em) n #h(0.2em) "is odd" & & root(n, a^(n)) = a \ "when" #h(0.2em) n #h(0.2em) "is even" & & root(n, a^(n)) = | a |$]We must use the absolute value signs when we take an even root of an expression with a variable in the radical. ] #examplebox("Example 3")[][ Simplify: ⓐ #math.equation(block: false, alt: "the square root of x squared")[$sqrt(x^(2))$] ⓑ #math.equation(block: false, alt: "the cube root of n cubed")[$root(3, n^(3))$] ⓒ #math.equation(block: false, alt: "the 4th root of p to the power 4")[$root(4, p^(4))$] ⓓ #math.equation(block: false, alt: "the 5th root of y to the power 5")[$root(5, y^(5))$]. #solutionbox[ We use the absolute value to be sure to get the positive root. ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the square root of x squared")[$sqrt(x^(2))$]]), [Since #math.equation(block: false, alt: "open parenthesis x close parenthesis squared equals x squared")[$attach(( x ), t: 2) = x^(2)$] and we want the positive root.], [#math.equation(block: false, alt: "| x |")[$| x |$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of n cubed")[$root(3, n^(3))$]]), [Since #math.equation(block: false, alt: "open parenthesis n close parenthesis cubed equals n cubed .")[$attach(( n ), t: 3) = n^(3) .$] It is an odd root so there is no need for an absolute value sign.], [#math.equation(block: false, alt: "n")[$n$]], )) ⓒ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of p to the power 4")[$root(4, p^(4))$]]), [Since #math.equation(block: false, alt: "open parenthesis p close parenthesis to the power 4 equals p to the power 4")[$attach(( p ), t: 4) = p^(4)$] and we want the positive root.], [#math.equation(block: false, alt: "| p |")[$| p |$]], )) ⓓ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 5th root of y to the power 5")[$root(5, y^(5))$]]), [Since #math.equation(block: false, alt: "open parenthesis y close parenthesis to the power 5 equals y to the power 5 .")[$attach(( y ), t: 5) = y^(5) .$] It is an odd root so there is no need for an absolute value sign.], [#math.equation(block: false, alt: "y")[$y$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the square root of b squared")[$sqrt(b^(2))$] ⓑ #math.equation(block: false, alt: "the cube root of w cubed")[$root(3, w^(3))$] ⓒ #math.equation(block: false, alt: "the 4th root of m to the power 4")[$root(4, m^(4))$] ⓓ #math.equation(block: false, alt: "the 5th root of q to the power 5")[$root(5, q^(5))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "| b |")[$| b |$] ⓑ #math.equation(block: false, alt: "w")[$w$] ⓒ #math.equation(block: false, alt: "| m |")[$| m |$] ⓓ #math.equation(block: false, alt: "q")[$q$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the square root of y squared")[$sqrt(y^(2))$] ⓑ #math.equation(block: false, alt: "the cube root of p cubed")[$root(3, p^(3))$] ⓒ #math.equation(block: false, alt: "the 4th root of z to the power 4")[$root(4, z^(4))$] ⓓ #math.equation(block: false, alt: "the 5th root of q to the power 5")[$root(5, q^(5))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "| y |")[$| y |$] ⓑ #math.equation(block: false, alt: "p")[$p$] ⓒ #math.equation(block: false, alt: "| z |")[$| z |$] ⓓ #math.equation(block: false, alt: "q")[$q$] ] ] #examplebox("Example 4")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of y to the power 18")[$root(3, y^(18))$] ⓑ #math.equation(block: false, alt: "the 4th root of z to the power 8")[$root(4, z^(8))$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of y to the power 18")[$root(3, y^(18))$]]), [Since #math.equation(block: false, alt: "open parenthesis y to the power 6 close parenthesis cubed equals y to the power 18 .")[$attach(( y^(6) ), t: 3) = y^(18) .$]], [#math.equation(block: false, alt: "the cube root of open parenthesis y to the power 6 close parenthesis cubed")[$root(3, attach(( y^(6) ), t: 3))$]], [], [#math.equation(block: false, alt: "y to the power 6")[$y^(6)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of z to the power 8")[$root(4, z^(8))$]]), [Since #math.equation(block: false, alt: "open parenthesis z squared close parenthesis to the power 4 equals z to the power 8 .")[$attach(( z^(2) ), t: 4) = z^(8) .$]], [#math.equation(block: false, alt: "the 4th root of open parenthesis z squared close parenthesis to the power 4")[$root(4, attach(( z^(2) ), t: 4))$]], [Since #math.equation(block: false, alt: "z squared")[$z^(2)$] is positive, we do not need an absolute value sign.], [#math.equation(block: false, alt: "z squared")[$z^(2)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the 4th root of u to the power 12")[$root(4, u^(12))$] ⓑ #math.equation(block: false, alt: "the cube root of v to the power 15")[$root(3, v^(15))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "u cubed")[$u^(3)$] ⓑ #math.equation(block: false, alt: "v to the power 5")[$v^(5)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the 5th root of c to the power 20")[$root(5, c^(20))$] ⓑ #math.equation(block: false, alt: "the 6th root of d to the power 24")[$root(6, d^(24))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "c to the power 4")[$c^(4)$] ⓑ #math.equation(block: false, alt: "d to the power 4")[$d^(4)$] ] ] #examplebox("Example 5")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 64 p to the power 6")[$root(3, 64 p^(6))$] ⓑ #math.equation(block: false, alt: "the 4th root of 16 q to the power 12")[$root(4, 16 q^(12))$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of 64 p to the power 6")[$root(3, 64 p^(6))$]]), [Rewrite #math.equation(block: false, alt: "64 p to the power 6 as open parenthesis 4 p squared close parenthesis cubed .")[$64 p^(6) #h(0.2em) "as" #h(0.2em) attach(( 4 p^(2) ), t: 3) .$]], [#math.equation(block: false, alt: "the cube root of open parenthesis 4 p squared close parenthesis cubed")[$root(3, attach(( 4 p^(2) ), t: 3))$]], [Take the cube root.], [#math.equation(block: false, alt: "4 p squared")[$4 p^(2)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of 16 q to the power 12")[$root(4, 16 q^(12))$]]), [Rewrite the radicand as a fourth power.], [#math.equation(block: false, alt: "the 4th root of open parenthesis 2 q cubed close parenthesis to the power 4")[$root(4, attach(( 2 q^(3) ), t: 4))$]], [Take the fourth root.], [#math.equation(block: false, alt: "2 | q cubed |")[$2 | q^(3) |$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 27 x to the power 27")[$root(3, 27 x^(27))$] ⓑ #math.equation(block: false, alt: "the 4th root of 81 q to the power 28")[$root(4, 81 q^(28))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "3 x to the power 9")[$3 x^(9)$] ⓑ #math.equation(block: false, alt: "3 | q to the power 7 |")[$3 | q^(7) |$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 125 p to the power 9")[$root(3, 125 p^(9))$] ⓑ #math.equation(block: false, alt: "the 5th root of 243 q to the power 25")[$root(5, 243 q^(25))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "5 p cubed")[$5 p^(3)$] ⓑ #math.equation(block: false, alt: "3 q to the power 5")[$3 q^(5)$] ] ] === Use the Product Property to Simplify Expressions with Higher Roots We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An #emph[n]th root is considered simplified if it has no factors of #math.equation(block: false, alt: "m to the power n")[$m^(n)$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Simplified #emph[n]th Root] #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is considered simplified if #math.equation(block: false, alt: "a")[$a$] has no factors of #math.equation(block: false, alt: "m to the power n")[$m^(n)$]. ] We will generalize the Product Property of Square Roots to include any integer root #math.equation(block: false, alt: "n greater than or equal to 2")[$n ≥ 2$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Product Property of #emph[n]th Roots] #math.equation(block: true, alt: "the nth root of a b equals the nth root of a times the nth root of b, and, the nth root of a times the nth root of b equals the nth root of a b")[$root(n, a b) = root(n, a) · root(n, b) & "and" & root(n, a) · root(n, b) = root(n, a b)$]when #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] and #math.equation(block: false, alt: "the nth root of b")[$root(n, b)$] are real numbers and for any integer #math.equation(block: false, alt: "n greater than or equal to 2")[$n ≥ 2$] ] #examplebox("Example 6")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of x to the power 4")[$root(3, x^(4))$] ⓑ #math.equation(block: false, alt: "the 4th root of x to the power 7")[$root(4, x^(7))$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of x to the power 4")[$root(3, x^(4))$]]), [Rewrite the radicand as a product using the largest perfect cube factor.], [#math.equation(block: false, alt: "the cube root of x cubed times x")[$root(3, x^(3) · x)$]], [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the cube root of x cubed times the cube root of x")[$root(3, x^(3)) · root(3, x)$]], [Simplify.], [#math.equation(block: false, alt: "x the cube root of x")[$x root(3, x)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of x to the power 7")[$root(4, x^(7))$]]), [Rewrite the radicand as a product using the greatest perfect fourth power factor.], [#math.equation(block: false, alt: "the 4th root of x to the power 4 times x cubed")[$root(4, x^(4) · x^(3))$]], [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the 4th root of x to the power 4 times the 4th root of x cubed")[$root(4, x^(4)) · root(4, x^(3))$]], [Simplify.], [#math.equation(block: false, alt: "| x | the 4th root of x cubed")[$| x | root(4, x^(3))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the 4th root of y to the power 6")[$root(4, y^(6))$] ⓑ #math.equation(block: false, alt: "the cube root of z to the power 5")[$root(3, z^(5))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "| y | the 4th root of y squared")[$| y | root(4, y^(2))$] ⓑ #math.equation(block: false, alt: "z the cube root of z squared")[$z root(3, z^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the 5th root of p to the power 8")[$root(5, p^(8))$] ⓑ #math.equation(block: false, alt: "the 6th root of q to the power 13")[$root(6, q^(13))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "p the 5th root of p cubed")[$p root(5, p^(3))$] ⓑ #math.equation(block: false, alt: "q squared the 6th root of q")[$q^(2) root(6, q)$] ] ] #examplebox("Example 7")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 16")[$root(3, 16)$] ⓑ #math.equation(block: false, alt: "the 4th root of 243")[$root(4, 243)$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of 16")[$root(3, 16)$]]), [], [#math.equation(block: false, alt: "the cube root of 2 to the power 4")[$root(3, 2^(4))$]], [Rewrite the radicand as a product using the greatest perfect cube factor.], [#math.equation(block: false, alt: "the cube root of 2 cubed times 2")[$root(3, 2^(3) · 2)$]], [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the cube root of 2 cubed times the cube root of 2")[$root(3, 2^(3)) · root(3, 2)$]], [Simplify.], [#math.equation(block: false, alt: "2 the cube root of 2")[$2 root(3, 2)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of 243")[$root(4, 243)$]]), [], [#math.equation(block: false, alt: "the 4th root of 3 to the power 5")[$root(4, 3^(5))$]], [Rewrite the radicand as a product using the greatest perfect fourth power factor.], [#math.equation(block: false, alt: "the 4th root of 3 to the power 4 times 3")[$root(4, 3^(4) · 3)$]], [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the 4th root of 3 to the power 4 times the 4th root of 3")[$root(4, 3^(4)) · root(4, 3)$]], [Simplify.], [#math.equation(block: false, alt: "3 the 4th root of 3")[$3 root(4, 3)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 81")[$root(3, 81)$] ⓑ #math.equation(block: false, alt: "the 4th root of 64")[$root(4, 64)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "3 the cube root of 3")[$3 root(3, 3)$] ⓑ #math.equation(block: false, alt: "2 the 4th root of 4")[$2 root(4, 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 625")[$root(3, 625)$] ⓑ #math.equation(block: false, alt: "the 4th root of 729")[$root(4, 729)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "5 the cube root of 5")[$5 root(3, 5)$] ⓑ #math.equation(block: false, alt: "3 the 4th root of 9")[$3 root(4, 9)$] ] ] Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical. #examplebox("Example 8")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 24 x to the power 7")[$root(3, 24 x^(7))$] ⓑ #math.equation(block: false, alt: "the 4th root of 80 y to the power 14")[$root(4, 80 y^(14))$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of 24 x to the power 7")[$root(3, 24 x^(7))$]]), [Rewrite the radicand as a product using perfect cube factors.], [#math.equation(block: false, alt: "the cube root of 2 cubed x to the power 6 times 3 x")[$root(3, 2^(3) x^(6) · 3 x)$]], [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the cube root of 2 cubed x to the power 6 times the cube root of 3 x")[$root(3, 2^(3) x^(6)) · root(3, 3 x)$]], [Rewrite the first radicand as #math.equation(block: false, alt: "open parenthesis 2 x squared close parenthesis cubed .")[$attach(( 2 x^(2) ), t: 3) .$]], [#math.equation(block: false, alt: "the cube root of open parenthesis 2 x squared close parenthesis cubed times the cube root of 3 x")[$root(3, attach(( 2 x^(2) ), t: 3)) · root(3, 3 x)$]], [Simplify.], [#math.equation(block: false, alt: "2 x squared the cube root of 3 x")[$2 x^(2) root(3, 3 x)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of 80 y to the power 14")[$root(4, 80 y^(14))$]]), [Rewrite the radicand as a product using perfect fourth power factors.], [#math.equation(block: false, alt: "the 4th root of 2 to the power 4 y to the power 12 times 5 y squared")[$root(4, 2^(4) y^(12) · 5 y^(2))$]], [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the 4th root of 2 to the power 4 y to the power 12 times the 4th root of 5 y squared")[$root(4, 2^(4) y^(12)) · root(4, 5 y^(2))$]], [Rewrite the first radicand as #math.equation(block: false, alt: "open parenthesis 2 y cubed close parenthesis to the power 4 .")[$attach(( 2 y^(3) ), t: 4) .$]], [#math.equation(block: false, alt: "the 4th root of open parenthesis 2 y cubed close parenthesis to the power 4 times the 4th root of 5 y squared")[$root(4, attach(( 2 y^(3) ), t: 4)) · root(4, 5 y^(2))$]], [Simplify.], [#math.equation(block: false, alt: "2 | y cubed | the 4th root of 5 y squared")[$2 | y^(3) | root(4, 5 y^(2))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 54 p to the power 10")[$root(3, 54 p^(10))$] ⓑ #math.equation(block: false, alt: "the 4th root of 64 q to the power 10")[$root(4, 64 q^(10))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "3 p cubed the cube root of 2 p")[$3 p^(3) root(3, 2 p)$] ⓑ #math.equation(block: false, alt: "2 q squared the 4th root of 4 q squared")[$2 q^(2) root(4, 4 q^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 128 m to the power 11")[$root(3, 128 m^(11))$] ⓑ #math.equation(block: false, alt: "the 4th root of 162 n to the power 7")[$root(4, 162 n^(7))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "4 m cubed the cube root of 2 m squared")[$4 m^(3) root(3, 2 m^(2))$] ⓑ #math.equation(block: false, alt: "3 | n | the 4th root of 2 n cubed")[$3 | n | root(4, 2 n^(3))$] ] ] #examplebox("Example 9")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of −27")[$root(3, −27)$] ⓑ #math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of −27")[$root(3, −27)$]]), [Rewrite the radicand as a product using perfect cube factors.], [#math.equation(block: false, alt: "the cube root of open parenthesis −3 close parenthesis cubed")[$root(3, attach(( −3 ), t: 3))$]], [Take the cube root.], [#math.equation(block: false, alt: "−3")[$−3$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$]]), [There is no real number #math.equation(block: false, alt: "n")[$n$] where #math.equation(block: false, alt: "n to the power 4 equals −16 .")[$n^(4) = −16 .$]], [Not a real number.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of −108")[$root(3, −108)$] ⓑ #math.equation(block: false, alt: "the 4th root of −48")[$root(4, −48)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "−3 the cube root of 4")[$−3 root(3, 4)$] ⓑ #math.equation(block: false, alt: "not real")[$"not real"$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of −625")[$root(3, −625)$] ⓑ #math.equation(block: false, alt: "the 4th root of −324")[$root(4, −324)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "−5 the cube root of 5")[$−5 root(3, 5)$] ⓑ #math.equation(block: false, alt: "not real")[$"not real"$] ] ] === Use the Quotient Property to Simplify Expressions with Higher Roots We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical. #examplebox("Example 10")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of the fraction a to the power 8 over a to the power 5")[$root(3, frac(a^(8), a^(5)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction a to the power 10 over a squared")[$root(4, frac(a^(10), a^(2)))$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of the fraction a to the power 8 over a to the power 5")[$root(3, frac(a^(8), a^(5)))$]]), [Simplify the fraction under the radical first.], [#math.equation(block: false, alt: "the cube root of a cubed")[$root(3, a^(3))$]], [Simplify.], [#math.equation(block: false, alt: "a")[$a$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of the fraction a to the power 10 over a squared")[$root(4, frac(a^(10), a^(2)))$]]), [Simplify the fraction under the radical first.], [#math.equation(block: false, alt: "the 4th root of a to the power 8")[$root(4, a^(8))$]], [Rewrite the radicand using perfect fourth power factors.], [#math.equation(block: false, alt: "the 4th root of open parenthesis a squared close parenthesis to the power 4")[$root(4, attach(( a^(2) ), t: 4))$]], [Simplify.], [#math.equation(block: false, alt: "a squared")[$a^(2)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the 4th root of the fraction x to the power 7 over x cubed")[$root(4, frac(x^(7), x^(3)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction y to the power 17 over y to the power 5")[$root(4, frac(y^(17), y^(5)))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "| x |")[$| x |$] ⓑ #math.equation(block: false, alt: "y cubed")[$y^(3)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of the fraction m to the power 13 over m to the power 7")[$root(3, frac(m^(13), m^(7)))$] ⓑ #math.equation(block: false, alt: "the 5th root of the fraction n to the power 12 over n squared")[$root(5, frac(n^(12), n^(2)))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "m squared")[$m^(2)$] ⓑ #math.equation(block: false, alt: "n squared")[$n^(2)$] ] ] Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Quotient Property of #emph[n]th Roots] #math.equation(block: true, alt: "the nth root of the fraction a over b equals the fraction the nth root of a over the nth root of b, and, the fraction the nth root of a over the nth root of b equals the nth root of the fraction a over b")[$root(n, frac(a, b)) = frac(root(n, a), root(n, b)) & "and" & frac(root(n, a), root(n, b)) = root(n, frac(a, b))$]when #math.equation(block: false, alt: "the nth root of a and the nth root of b are real numbers, b not equal to 0 , and for any integer n greater than or equal to 2")[$root(n, a) #h(0.2em) "and" #h(0.2em) root(n, b) #h(0.2em) "are real numbers," #h(0.2em) b ≠ 0 , #h(0.2em) "and for any integer" #h(0.2em) n ≥ 2$] ] #examplebox("Example 11")[][ Simplify: ⓐ #math.equation(block: false, alt: "the fraction the cube root of −108 over the cube root of 2")[$frac(root(3, −108), root(3, 2))$] ⓑ #math.equation(block: false, alt: "the fraction the 4th root of 96 x to the power 7 over the 4th root of 3 x squared")[$frac(root(4, 96 x^(7)), root(4, 3 x^(2)))$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction the cube root of −108 over the cube root of 2")[$frac(root(3, −108), root(3, 2))$]]), [Neither radicand is a perfect cube, so use the Quotient Property to write as one radical.], [#math.equation(block: false, alt: "the cube root of the fraction −108 over 2")[$root(3, frac(−108, 2))$]], [Simplify the fraction under the radical.], [#math.equation(block: false, alt: "the cube root of −54")[$root(3, −54)$]], [Rewrite the radicand as a product using perfect cube factors.], [#math.equation(block: false, alt: "the cube root of open parenthesis −3 close parenthesis cubed times 2")[$root(3, attach(( −3 ), t: 3) · 2)$]], [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the cube root of open parenthesis −3 close parenthesis cubed times the cube root of 2")[$root(3, attach(( −3 ), t: 3)) · root(3, 2)$]], [Simplify.], [#math.equation(block: false, alt: "−3 the cube root of 2")[$−3 root(3, 2)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the fraction the 4th root of 96 x to the power 7 over the 4th root of 3 x squared")[$frac(root(4, 96 x^(7)), root(4, 3 x^(2)))$]]), [Neither radicand is a perfect fourth power, so use the Quotient Property to write as one radical.], [#math.equation(block: false, alt: "the 4th root of the fraction 96 x to the power 7 over 3 x squared")[$root(4, frac(96 x^(7), 3 x^(2)))$]], [Simplify the fraction under the radical.], [#math.equation(block: false, alt: "the 4th root of 32 x to the power 5")[$root(4, 32 x^(5))$]], [Rewrite the radicand as a product using perfect fourth power factors.], [#math.equation(block: false, alt: "the 4th root of 2 to the power 4 x to the power 4 times 2 x")[$root(4, 2^(4) x^(4) · 2 x)$]], [Rewrite the radical as the product of two radicals.], [#math.equation(block: false, alt: "the 4th root of open parenthesis 2 x close parenthesis to the power 4 times the 4th root of 2 x")[$root(4, attach(( 2 x ), t: 4)) · root(4, 2 x)$]], [Simplify.], [#math.equation(block: false, alt: "2 | x | the 4th root of 2 x")[$2 | x | root(4, 2 x)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction the cube root of −532 over the cube root of 2")[$frac(root(3, −532), root(3, 2))$] ⓑ #math.equation(block: false, alt: "the fraction the 4th root of 486 m to the power 11 over the 4th root of 3 m to the power 5")[$frac(root(4, 486 m^(11)), root(4, 3 m^(5)))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "minus the cube root of 266")[$− root(3, 266)$] ⓑ #math.equation(block: false, alt: "3 | m | the 4th root of 2 m squared")[$3 | m | root(4, 2 m^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the fraction the cube root of −192 over the cube root of 3")[$frac(root(3, −192), root(3, 3))$] ⓑ #math.equation(block: false, alt: "the fraction the 4th root of 324 n to the power 7 over the 4th root of 2 n cubed")[$frac(root(4, 324 n^(7)), root(4, 2 n^(3)))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "−4")[$−4$] ⓑ #math.equation(block: false, alt: "3 | n | the 4th root of 2")[$3 | n | root(4, 2)$] ] ] If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals. #examplebox("Example 12")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of the fraction 24 x to the power 7 over y cubed")[$root(3, frac(24 x^(7), y^(3)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 48 x to the power 10 over y to the power 8")[$root(4, frac(48 x^(10), y^(8)))$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of the fraction 24 x to the power 7 over y cubed")[$root(3, frac(24 x^(7), y^(3)))$]]), [The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.], [#math.equation(block: false, alt: "the fraction the cube root of 24 x to the power 7 over the cube root of y cubed")[$frac(root(3, 24 x^(7)), root(3, y^(3)))$]], [Rewrite each radicand as a product using perfect cube factors.], [#math.equation(block: false, alt: "the fraction the cube root of 8 x to the power 6 times 3 x over the cube root of y cubed")[$frac(root(3, 8 x^(6) · 3 x), root(3, y^(3)))$]], [Rewrite the numerator as the product of two radicals.], [#math.equation(block: false, alt: "the fraction the cube root of open parenthesis 2 x squared close parenthesis cubed the cube root of 3 x over the cube root of y cubed")[$frac(root(3, attach(( 2 x^(2) ), t: 3)) #h(0.2em) root(3, 3 x), root(3, y^(3)))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 2 x squared the cube root of 3 x over y")[$frac(2 x^(2) root(3, 3 x), y)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of the fraction 48 x to the power 10 over y to the power 8")[$root(4, frac(48 x^(10), y^(8)))$]]), [The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals.], [#math.equation(block: false, alt: "the fraction the 4th root of 48 x to the power 10 over the 4th root of y to the power 8")[$frac(root(4, 48 x^(10)), root(4, y^(8)))$]], [Rewrite each radicand as a product using perfect fourth power factors.], [#math.equation(block: false, alt: "the fraction the 4th root of 16 x to the power 8 times 3 x squared over the 4th root of y to the power 8")[$frac(root(4, 16 x^(8) · 3 x^(2)), root(4, y^(8)))$]], [Rewrite the numerator as the product of two radicals.], [#math.equation(block: false, alt: "the fraction the 4th root of open parenthesis 2 x squared close parenthesis to the power 4 the 4th root of 3 x squared over the 4th root of open parenthesis y squared close parenthesis to the power 4")[$frac(root(4, attach(( 2 x^(2) ), t: 4)) #h(0.2em) root(4, 3 x^(2)), root(4, attach(( y^(2) ), t: 4)))$]], [Simplify.], [#math.equation(block: false, alt: "the fraction 2 x squared the 4th root of 3 x squared over y squared")[$frac(2 x^(2) root(4, 3 x^(2)), y^(2))$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of the fraction 108 c to the power 10 over d to the power 6")[$root(3, frac(108 c^(10), d^(6)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 80 x to the power 10 over y to the power 5")[$root(4, frac(80 x^(10), y^(5)))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 3 c cubed the cube root of 4 c over d squared")[$frac(3 c^(3) root(3, 4 c), d^(2))$] ⓑ #math.equation(block: false, alt: "the fraction 2 x squared over | y | the 4th root of the fraction 5 x squared over y")[$frac(2 x^(2), | y |) #h(0.2em) root(4, frac(5 x^(2), y))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of the fraction 40 r cubed over s")[$root(3, frac(40 r^(3), s))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 162 m to the power 14 over n to the power 12")[$root(4, frac(162 m^(14), n^(12)))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "2 r the cube root of the fraction 5 over s")[$2 r root(3, frac(5, s))$] ⓑ #math.equation(block: false, alt: "the fraction 3 m cubed the 4th root of 2 m squared over | n cubed |")[$frac(3 m^(3) root(4, 2 m^(2)), | n^(3) |)$] ] ] === Add and Subtract Higher Roots We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of #strong[like radicals]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Like Radicals] Radicals with the same index and same radicand are called #strong[like radicals]. ] Like radicals have the same index and the same radicand. - #math.equation(block: false, alt: "9 the 4th root of 42 x")[$9 root(4, 42 x)$] and #math.equation(block: false, alt: "−2 the 4th root of 42 x")[$−2 root(4, 42 x)$] are like radicals. - #math.equation(block: false, alt: "5 the cube root of 125 x")[$5 root(3, 125 x)$] and #math.equation(block: false, alt: "6 the cube root of 125 y")[$6 root(3, 125 y)$] are not like radicals. The radicands are different. - #math.equation(block: false, alt: "2 the 5th root of 1000 q")[$2 root(5, 1000 q)$] and #math.equation(block: false, alt: "−4 the 4th root of 1000 q")[$−4 root(4, 1000 q)$] are not like radicals. The indices are different. We add and subtract like radicals in the same way we add and subtract like terms. We can add #math.equation(block: false, alt: "9 the 4th root of 42 x plus open parenthesis −2 the 4th root of 42 x close parenthesis")[$9 root(4, 42 x) + ( −2 root(4, 42 x) )$] and the result is #math.equation(block: false, alt: "7 the 4th root of 42 x")[$7 root(4, 42 x)$]. #examplebox("Example 13")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 4 x plus the cube root of 4 x")[$root(3, 4 x) + root(3, 4 x)$] ⓑ #math.equation(block: false, alt: "4 the 4th root of 8 minus 2 the 4th root of 8")[$4 root(4, 8) − 2 root(4, 8)$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of 4 x plus the cube root of 4 x")[$root(3, 4 x) + root(3, 4 x)$]]), [The radicals are like, so we add the coefficients.], [#math.equation(block: false, alt: "2 the cube root of 4 x")[$2 root(3, 4 x)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "4 the 4th root of 8 minus 2 the 4th root of 8")[$4 root(4, 8) − 2 root(4, 8)$]]), [The radicals are like, so we subtract the coefficients.], [#math.equation(block: false, alt: "2 the 4th root of 8")[$2 root(4, 8)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the 5th root of 3 x plus the 5th root of 3 x")[$root(5, 3 x) + root(5, 3 x)$] ⓑ #math.equation(block: false, alt: "3 the cube root of 9 minus the cube root of 9")[$3 root(3, 9) − root(3, 9)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "2 the 5th root of 3 x")[$2 root(5, 3 x)$] ⓑ #math.equation(block: false, alt: "2 the cube root of 9")[$2 root(3, 9)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the 4th root of 10 y plus the 4th root of 10 y")[$root(4, 10 y) + root(4, 10 y)$] ⓑ #math.equation(block: false, alt: "5 the 6th root of 32 minus 3 the 6th root of 32")[$5 root(6, 32) − 3 root(6, 32)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "2 the 4th root of 10 y")[$2 root(4, 10 y)$] ⓑ #math.equation(block: false, alt: "2 the 6th root of 32")[$2 root(6, 32)$] ] ] When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals. #examplebox("Example 14")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 54 minus the cube root of 16")[$root(3, 54) − root(3, 16)$] ⓑ #math.equation(block: false, alt: "the 4th root of 48 plus the 4th root of 243")[$root(4, 48) + root(4, 243)$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of 54 minus the cube root of 16")[$root(3, 54) − root(3, 16)$]]), [Rewrite each radicand using perfect cube factors.], [#math.equation(block: false, alt: "the cube root of 27 times the cube root of 2 minus the cube root of 8 times the cube root of 2")[$root(3, 27) · root(3, 2) − root(3, 8) · root(3, 2)$]], [Rewrite the perfect cubes.], [#math.equation(block: false, alt: "the cube root of open parenthesis 3 close parenthesis cubed the cube root of 2 minus the cube root of open parenthesis 2 close parenthesis cubed the cube root of 2")[$root(3, attach(( 3 ), t: 3)) #h(0.2em) root(3, 2) − root(3, attach(( 2 ), t: 3)) #h(0.2em) root(3, 2)$]], [Simplify the radicals where possible.], [#math.equation(block: false, alt: "3 the cube root of 2 minus 2 the cube root of 2")[$3 root(3, 2) − 2 root(3, 2)$]], [Combine like radicals.], [#math.equation(block: false, alt: "the cube root of 2")[$root(3, 2)$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of 48 plus the 4th root of 243")[$root(4, 48) + root(4, 243)$]]), [Rewrite using perfect fourth power factors.], [#math.equation(block: false, alt: "the 4th root of 16 times the 4th root of 3 plus the 4th root of 81 times the 4th root of 3")[$root(4, 16) · root(4, 3) + root(4, 81) · root(4, 3)$]], [Rewrite the perfect fourth powers.], [#math.equation(block: false, alt: "the 4th root of open parenthesis 2 close parenthesis to the power 4 the 4th root of 3 plus the 4th root of open parenthesis 3 close parenthesis to the power 4 the 4th root of 3")[$root(4, attach(( 2 ), t: 4)) #h(0.2em) root(4, 3) + root(4, attach(( 3 ), t: 4)) #h(0.2em) root(4, 3)$]], [Simplify the radicals where possible.], [#math.equation(block: false, alt: "2 the 4th root of 3 plus 3 the 4th root of 3")[$2 root(4, 3) + 3 root(4, 3)$]], [Combine like radicals.], [#math.equation(block: false, alt: "5 the 4th root of 3")[$5 root(4, 3)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 192 minus the cube root of 81")[$root(3, 192) − root(3, 81)$] ⓑ #math.equation(block: false, alt: "the 4th root of 32 plus the 4th root of 512")[$root(4, 32) + root(4, 512)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "the cube root of 3")[$root(3, 3)$] ⓑ #math.equation(block: false, alt: "6 the 4th root of 2")[$6 root(4, 2)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 128 minus the cube root of 250")[$root(3, 128) − root(3, 250)$] ⓑ #math.equation(block: false, alt: "the 5th root of 64 plus the 5th root of 486")[$root(5, 64) + root(5, 486)$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "− the cube root of 2")[$"−" root(3, 2)$] ⓑ #math.equation(block: false, alt: "5 the 5th root of 2")[$5 root(5, 2)$] ] ] #examplebox("Example 15")[][ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 24 x to the power 4 minus the cube root of −81 x to the power 7")[$root(3, 24 x^(4)) − root(3, −81 x^(7))$] ⓑ #math.equation(block: false, alt: "the 4th root of 162 y to the power 9 plus the 4th root of 512 y to the power 5")[$root(4, 162 y^(9)) + root(4, 512 y^(5))$]. #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the cube root of 24 x to the power 4 minus the cube root of −81 x to the power 7")[$root(3, 24 x^(4)) − root(3, −81 x^(7))$]]), [Rewrite each radicand using perfect cube factors.], [#math.equation(block: false, alt: "the cube root of 8 x cubed times the cube root of 3 x minus the cube root of −27 x to the power 6 times the cube root of 3 x")[$root(3, 8 x^(3)) · root(3, 3 x) − root(3, −27 x^(6)) · root(3, 3 x)$]], [Rewrite the perfect cubes.], [#math.equation(block: false, alt: "the cube root of open parenthesis 2 x close parenthesis cubed the cube root of 3 x minus the cube root of open parenthesis −3 x squared close parenthesis cubed the cube root of 3 x")[$root(3, attach(( 2 x ), t: 3)) #h(0.2em) root(3, 3 x) − root(3, attach(( −3 x^(2) ), t: 3)) #h(0.2em) root(3, 3 x)$]], [Simplify the radicals where possible.], [#math.equation(block: false, alt: "2 x the cube root of 3 x minus open parenthesis −3 x squared the cube root of 3 x close parenthesis")[$2 x root(3, 3 x) − ( −3 x^(2) root(3, 3 x) )$]], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "the 4th root of 162 y to the power 9 plus the 4th root of 512 y to the power 5")[$root(4, 162 y^(9)) + root(4, 512 y^(5))$]]), [Rewrite each radicand using perfect fourth power factors.], [#math.equation(block: false, alt: "the 4th root of 81 y to the power 8 times the 4th root of 2 y plus the 4th root of 256 y to the power 4 times the 4th root of 2 y")[$root(4, 81 y^(8)) · root(4, 2 y) + root(4, 256 y^(4)) · root(4, 2 y)$]], [Rewrite the perfect fourth powers.], [#math.equation(block: false, alt: "the 4th root of open parenthesis 3 y squared close parenthesis to the power 4 times the 4th root of 2 y plus the 4th root of open parenthesis 4 y close parenthesis to the power 4 times the 4th root of 2 y")[$root(4, attach(( 3 y^(2) ), t: 4)) · root(4, 2 y) + root(4, attach(( 4 y ), t: 4)) · root(4, 2 y)$]], [Simplify the radicals where possible.], [#math.equation(block: false, alt: "3 y squared the 4th root of 2 y plus 4 | y | the 4th root of 2 y")[$3 y^(2) root(4, 2 y) + 4 | y | root(4, 2 y)$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 32 y to the power 5 minus the cube root of −108 y to the power 8")[$root(3, 32 y^(5)) − root(3, −108 y^(8))$] ⓑ #math.equation(block: false, alt: "the 4th root of 243 r to the power 11 plus the 4th root of 768 r to the power 10")[$root(4, 243 r^(11)) + root(4, 768 r^(10))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "2 y the cube root of 4 y squared plus 3 y squared the cube root of 4 y squared")[$2 y root(3, 4 y^(2)) + 3 y^(2) root(3, 4 y^(2))$] ⓑ #math.equation(block: false, alt: "3 r squared the 4th root of 3 r cubed plus 4 r squared the 4th root of 3 r squared")[$3 r^(2) root(4, 3 r^(3)) + 4 r^(2) root(4, 3 r^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: ⓐ #math.equation(block: false, alt: "the cube root of 40 z to the power 7 minus the cube root of −135 z to the power 4")[$root(3, 40 z^(7)) − root(3, −135 z^(4))$] ⓑ #math.equation(block: false, alt: "the 4th root of 80 s to the power 13 plus the 4th root of 1280 s to the power 6")[$root(4, 80 s^(13)) + root(4, 1280 s^(6))$]. #solutionbox[ ⓐ #math.equation(block: false, alt: "2 z squared the cube root of 5 z plus 3 z the cube root of 5 z")[$2 z^(2) root(3, 5 z) + 3 z root(3, 5 z)$] ⓑ #math.equation(block: false, alt: "2 | s cubed | the 4th root of 5 s plus 4 | s | the 4th root of 5 s squared")[$2 | s^(3) | root(4, 5 s) + 4 | s | root(4, 5 s^(2))$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Access these online resources for additional instruction and practice with simplifying higher roots. - #link("https://openstax.org/l/25SimplifyHR")[Simplifying Higher Roots] - #link("https://openstax.org/l/25AddSubtrHR")[Add/Subtract Roots with Higher Indices] ] === Key Concepts - #strong[Properties of] - #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] when #math.equation(block: false, alt: "n")[$n$] is an even number and - #math.equation(block: false, alt: "a greater than or equal to 0")[$a ≥ 0$], then #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number - #math.equation(block: false, alt: "a less than 0")[$a < 0$], then #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is not a real number - When #math.equation(block: false, alt: "n")[$n$] is an odd number, #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number for all values of #emph[a]. - For any integer #math.equation(block: false, alt: "n greater than or equal to 2")[$n ≥ 2$], when #emph[n] is odd #math.equation(block: false, alt: "the nth root of a to the power n equals a")[$root(n, a^(n)) = a$] - For any integer #math.equation(block: false, alt: "n greater than or equal to 2")[$n ≥ 2$], when #emph[n] is even #math.equation(block: false, alt: "the nth root of a to the power n equals | a |")[$root(n, a^(n)) = | a |$] - #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is considered simplified if #emph[a] has no factors of #math.equation(block: false, alt: "m to the power n")[$m^(n)$]. - #strong[Product Property of #emph[n]th Roots] #linebreak() #math.equation(block: true, alt: "the nth root of a b equals the nth root of a times the nth root of b and the nth root of a times the nth root of b equals the nth root of a b")[$root(n, a b) = root(n, a) · root(n, b) #h(0.5em) "and" #h(0.5em) root(n, a) · root(n, b) = root(n, a b)$] - #strong[Quotient Property of #emph[n]th Roots] #linebreak() #math.equation(block: true, alt: "the nth root of the fraction a over b equals the fraction the nth root of a over the nth root of b and the fraction the nth root of a over the nth root of b equals the nth root of the fraction a over b")[$root(n, frac(a, b)) = frac(root(n, a), root(n, b)) #h(0.5em) "and" #h(0.5em) frac(root(n, a), root(n, b)) = root(n, frac(a, b))$] - To combine like radicals, simply add or subtract the coefficients while keeping the radical the same. ==== Practice Makes Perfect #strong[Simplify Expressions with Higher Roots] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "the cube root of 216")[$root(3, 216)$] ⓑ #math.equation(block: false, alt: "the 4th root of 256")[$root(4, 256)$] ⓒ #math.equation(block: false, alt: "the 5th root of 32")[$root(5, 32)$] ⓐ #math.equation(block: false, alt: "the cube root of 27")[$root(3, 27)$] ⓑ #math.equation(block: false, alt: "the 4th root of 16")[$root(4, 16)$] ⓒ #math.equation(block: false, alt: "the 5th root of 243")[$root(5, 243)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "3")[$3$] ⓑ #math.equation(block: false, alt: "2")[$2$] ⓒ #math.equation(block: false, alt: "3")[$3$] ] ⓐ #math.equation(block: false, alt: "the cube root of 512")[$root(3, 512)$] ⓑ #math.equation(block: false, alt: "the 4th root of 81")[$root(4, 81)$] ⓒ #math.equation(block: false, alt: "the 5th root of 1")[$root(5, 1)$] ⓐ #math.equation(block: false, alt: "the cube root of 125")[$root(3, 125)$] ⓑ #math.equation(block: false, alt: "the 4th root of 1296")[$root(4, 1296)$] ⓒ #math.equation(block: false, alt: "the 5th root of 1024")[$root(5, 1024)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "5")[$5$] ⓑ #math.equation(block: false, alt: "6")[$6$] ⓒ #math.equation(block: false, alt: "4")[$4$] ] ⓐ #math.equation(block: false, alt: "the cube root of −8")[$root(3, −8)$] ⓑ #math.equation(block: false, alt: "the 4th root of −81")[$root(4, −81)$] ⓒ #math.equation(block: false, alt: "the 5th root of −32")[$root(5, −32)$] ⓐ #math.equation(block: false, alt: "the cube root of −64")[$root(3, −64)$] ⓑ #math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$] ⓒ #math.equation(block: false, alt: "the 5th root of −243")[$root(5, −243)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−4")[$−4$] ⓑ #math.equation(block: false, alt: "not real")[$"not real"$] ⓒ #math.equation(block: false, alt: "−3")[$−3$] ] ⓐ #math.equation(block: false, alt: "the cube root of −125")[$root(3, −125)$] ⓑ #math.equation(block: false, alt: "the 4th root of −1296")[$root(4, −1296)$] ⓒ #math.equation(block: false, alt: "the 5th root of −1024")[$root(5, −1024)$] ⓐ #math.equation(block: false, alt: "the cube root of −512")[$root(3, −512)$] ⓑ #math.equation(block: false, alt: "the 4th root of −81")[$root(4, −81)$] ⓒ #math.equation(block: false, alt: "the 5th root of −1")[$root(5, −1)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−8")[$−8$] ⓑ not a real number ⓒ #math.equation(block: false, alt: "−1")[$−1$] ] ⓐ #math.equation(block: false, alt: "the 5th root of u to the power 5")[$root(5, u^(5))$] ⓑ #math.equation(block: false, alt: "the 8th root of v to the power 8")[$root(8, v^(8))$] + ⓐ #math.equation(block: false, alt: "the cube root of a cubed")[$root(3, a^(3))$] + ⓑ #figure(figph[A mathematical expression featuring the 12th root of b raised to the power of 12. This expression simplifies to 'b'.], alt: "A mathematical expression featuring the 12th root of b raised to the power of 12. This expression simplifies to 'b'.", caption: none) #solutionbox[ ⓐ #math.equation(block: false, alt: "a")[$a$] ⓑ #math.equation(block: false, alt: "| b |")[$| b |$] ] ⓐ #math.equation(block: false, alt: "the 4th root of y to the power 4")[$root(4, y^(4))$] ⓑ #math.equation(block: false, alt: "the 7th root of m to the power 7")[$root(7, m^(7))$] ⓐ #math.equation(block: false, alt: "the 8th root of k to the power 8")[$root(8, k^(8))$] ⓑ #math.equation(block: false, alt: "the 6th root of p to the power 6")[$root(6, p^(6))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "| k |")[$| k |$] ⓑ #math.equation(block: false, alt: "| p |")[$| p |$] ] ⓐ #math.equation(block: false, alt: "the cube root of x to the power 9")[$root(3, x^(9))$] ⓑ #math.equation(block: false, alt: "the 4th root of y to the power 12")[$root(4, y^(12))$] ⓐ #math.equation(block: false, alt: "the 5th root of a to the power 10")[$root(5, a^(10))$] ⓑ #math.equation(block: false, alt: "the cube root of b to the power 27")[$root(3, b^(27))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "a squared")[$a^(2)$] ⓑ #math.equation(block: false, alt: "b to the power 9")[$b^(9)$] ] ⓐ #math.equation(block: false, alt: "the 4th root of m to the power 8")[$root(4, m^(8))$] ⓑ #math.equation(block: false, alt: "the 5th root of n to the power 20")[$root(5, n^(20))$] ⓐ #math.equation(block: false, alt: "the 6th root of r to the power 12")[$root(6, r^(12))$] ⓑ #math.equation(block: false, alt: "the cube root of s to the power 30")[$root(3, s^(30))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "r squared")[$r^(2)$] ⓑ #math.equation(block: false, alt: "s to the power 10")[$s^(10)$] ] ⓐ #math.equation(block: false, alt: "the 4th root of 16 x to the power 8")[$root(4, 16 x^(8))$] ⓑ #math.equation(block: false, alt: "the 6th root of 64 y to the power 12")[$root(6, 64 y^(12))$] ⓐ #math.equation(block: false, alt: "the cube root of −8 c to the power 9")[$root(3, −8 c^(9))$] ⓑ #math.equation(block: false, alt: "the cube root of 125 d to the power 15")[$root(3, 125 d^(15))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−2 c cubed")[$−2 c^(3)$] ⓑ #math.equation(block: false, alt: "5 d to the power 5")[$5 d^(5)$] ] ⓐ #math.equation(block: false, alt: "the cube root of 216 a to the power 6")[$root(3, 216 a^(6))$] ⓑ #math.equation(block: false, alt: "the 5th root of 32 b to the power 20")[$root(5, 32 b^(20))$] ⓐ #math.equation(block: false, alt: "the 7th root of 128 r to the power 14")[$root(7, 128 r^(14))$] ⓑ #math.equation(block: false, alt: "the 4th root of 81 s to the power 24")[$root(4, 81 s^(24))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "2 r squared")[$2 r^(2)$] ⓑ #math.equation(block: false, alt: "3 s to the power 6")[$3 s^(6)$] ] #strong[Use the Product Property to Simplify Expressions with Higher Roots] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "the cube root of r to the power 5")[$root(3, r^(5))$] ⓑ #math.equation(block: false, alt: "the 4th root of s to the power 10")[$root(4, s^(10))$] ⓐ #math.equation(block: false, alt: "the 5th root of u to the power 7")[$root(5, u^(7))$] ⓑ #math.equation(block: false, alt: "the 6th root of v to the power 11")[$root(6, v^(11))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "u the 5th root of u squared")[$u root(5, u^(2))$] ⓑ #math.equation(block: false, alt: "v the 6th root of v to the power 5")[$v root(6, v^(5))$] ] ⓐ #math.equation(block: false, alt: "the 4th root of m to the power 5")[$root(4, m^(5))$] ⓑ #math.equation(block: false, alt: "the 8th root of n to the power 10")[$root(8, n^(10))$] ⓐ #math.equation(block: false, alt: "the 5th root of p to the power 8")[$root(5, p^(8))$] ⓑ #math.equation(block: false, alt: "the cube root of q to the power 8")[$root(3, q^(8))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "p the 5th root of p cubed")[$p root(5, p^(3))$] ⓑ #math.equation(block: false, alt: "q squared the cube root of q squared")[$q^(2) root(3, q^(2))$] ] ⓐ #math.equation(block: false, alt: "the 4th root of 32")[$root(4, 32)$] ⓑ #math.equation(block: false, alt: "the 7th root of 64")[$root(7, 64)$] ⓐ #math.equation(block: false, alt: "the cube root of 625")[$root(3, 625)$] ⓑ #math.equation(block: false, alt: "the 6th root of 128")[$root(6, 128)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "5 the cube root of 5")[$5 root(3, 5)$] ⓑ #math.equation(block: false, alt: "2 the 6th root of 2")[$2 root(6, 2)$] ] ⓐ #math.equation(block: false, alt: "the 5th root of 64")[$root(5, 64)$] ⓑ #math.equation(block: false, alt: "the cube root of 256")[$root(3, 256)$] ⓐ #math.equation(block: false, alt: "the 4th root of 3125")[$root(4, 3125)$] ⓑ #math.equation(block: false, alt: "the cube root of 81")[$root(3, 81)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "5 the 4th root of 5")[$5 root(4, 5)$] ⓑ #math.equation(block: false, alt: "3 the cube root of 3")[$3 root(3, 3)$] ] ⓐ #math.equation(block: false, alt: "the cube root of 108 x to the power 5")[$root(3, 108 x^(5))$] ⓑ #math.equation(block: false, alt: "the 4th root of 48 y to the power 6")[$root(4, 48 y^(6))$] ⓐ #math.equation(block: false, alt: "the 5th root of 96 a to the power 7")[$root(5, 96 a^(7))$] ⓑ #math.equation(block: false, alt: "the cube root of 375 b to the power 4")[$root(3, 375 b^(4))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "2 a the 5th root of 3 a squared")[$2 a root(5, 3 a^(2))$] ⓑ #math.equation(block: false, alt: "5 b the cube root of 3 b")[$5 b root(3, 3 b)$] ] ⓐ #math.equation(block: false, alt: "the 4th root of 405 m to the power 10")[$root(4, 405 m^(10))$] ⓑ #math.equation(block: false, alt: "the 5th root of 160 n to the power 8")[$root(5, 160 n^(8))$] ⓐ #math.equation(block: false, alt: "the cube root of 512 p to the power 5")[$root(3, 512 p^(5))$] ⓑ #math.equation(block: false, alt: "the 4th root of 324 q to the power 7")[$root(4, 324 q^(7))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "8 p the cube root of p squared")[$8 p root(3, p^(2))$] ⓑ #math.equation(block: false, alt: "3 q the 4th root of 4 q cubed")[$3 q root(4, 4 q^(3))$] ] ⓐ #math.equation(block: false, alt: "the cube root of −864")[$root(3, −864)$] ⓑ #math.equation(block: false, alt: "the 4th root of −256")[$root(4, −256)$] ⓐ #math.equation(block: false, alt: "the 5th root of −486")[$root(5, −486)$] ⓑ #math.equation(block: false, alt: "the 6th root of −64")[$root(6, −64)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−3 the 5th root of 2")[$−3 root(5, 2)$] ⓑ #math.equation(block: false, alt: "not real")[$"not real"$] ] ⓐ #math.equation(block: false, alt: "the 5th root of −32")[$root(5, −32)$] ⓑ #math.equation(block: false, alt: "the 8th root of −1")[$root(8, −1)$] ⓐ #math.equation(block: false, alt: "the cube root of −8")[$root(3, −8)$] ⓑ #math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−2")[$−2$] ⓑ #math.equation(block: false, alt: "not real")[$"not real"$] ] #strong[Use the Quotient Property to Simplify Expressions with Higher Roots] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "the cube root of the fraction p to the power 11 over p squared")[$root(3, frac(p^(11), p^(2)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction q to the power 17 over q to the power 13")[$root(4, frac(q^(17), q^(13)))$] ⓐ #math.equation(block: false, alt: "the 5th root of the fraction d to the power 12 over d to the power 7")[$root(5, frac(d^(12), d^(7)))$] ⓑ #math.equation(block: false, alt: "the 8th root of the fraction m to the power 12 over m to the power 4")[$root(8, frac(m^(12), m^(4)))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "d")[$d$] ⓑ #math.equation(block: false, alt: "| m |")[$| m |$] ] ⓐ #math.equation(block: false, alt: "the 5th root of the fraction u to the power 21 over u to the power 11")[$root(5, frac(u^(21), u^(11)))$] ⓑ #math.equation(block: false, alt: "the 6th root of the fraction v to the power 30 over v to the power 12")[$root(6, frac(v^(30), v^(12)))$] ⓐ #math.equation(block: false, alt: "the cube root of the fraction r to the power 14 over r to the power 5")[$root(3, frac(r^(14), r^(5)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction c to the power 21 over c to the power 9")[$root(4, frac(c^(21), c^(9)))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "r cubed")[$r^(3)$] ⓑ #math.equation(block: false, alt: "| c cubed |")[$| c^(3) |$] ] ⓐ #math.equation(block: false, alt: "the fraction the 4th root of 64 over the 4th root of 2")[$frac(root(4, 64), root(4, 2))$] ⓑ #math.equation(block: false, alt: "the fraction the 5th root of 128 x to the power 8 over the 5th root of 2 x squared")[$frac(root(5, 128 x^(8)), root(5, 2 x^(2)))$] ⓐ #math.equation(block: false, alt: "the fraction the cube root of −625 over the cube root of 5")[$frac(root(3, −625), root(3, 5))$] ⓑ #math.equation(block: false, alt: "the fraction the 4th root of 80 m to the power 7 over the 4th root of 5 m")[$frac(root(4, 80 m^(7)), root(4, 5 m))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "−5")[$−5$] ⓑ #math.equation(block: false, alt: "2 m the 4th root of m squared")[$2 m root(4, m^(2))$] ] ⓐ #math.equation(block: false, alt: "the cube root of the fraction 1250 over 2")[$root(3, frac(1250, 2))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 486 y to the power 9 over 2 y cubed")[$root(4, frac(486 y^(9), 2 y^(3)))$] ⓐ #math.equation(block: false, alt: "the cube root of the fraction 162 over 6")[$root(3, frac(162, 6))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 160 r to the power 10 over 5 r cubed")[$root(4, frac(160 r^(10), 5 r^(3)))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "3 the cube root of 6")[$3 root(3, 6)$] ⓑ #math.equation(block: false, alt: "2 | r | the 4th root of 2 r cubed")[$2 | r | root(4, 2 r^(3))$] ] ⓐ #math.equation(block: false, alt: "the cube root of the fraction 54 a to the power 8 over b cubed")[$root(3, frac(54 a^(8), b^(3)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 64 c to the power 5 over d squared")[$root(4, frac(64 c^(5), d^(2)))$] ⓐ #math.equation(block: false, alt: "the 5th root of the fraction 96 r to the power 11 over s cubed")[$root(5, frac(96 r^(11), s^(3)))$] ⓑ #math.equation(block: false, alt: "the 6th root of the fraction 128 u to the power 7 over v cubed")[$root(6, frac(128 u^(7), v^(3)))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "2 r squared the 5th root of the fraction 3 r over s cubed")[$2 r^(2) root(5, frac(3 r, s^(3)))$] ⓑ #math.equation(block: false, alt: "2 u the 6th root of the fraction 2 u over v cubed")[$2 u root(6, frac(2 u, v^(3)))$] ] ⓐ #math.equation(block: false, alt: "the cube root of the fraction 81 s to the power 8 over t cubed")[$root(3, frac(81 s^(8), t^(3)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 64 p to the power 15 over q to the power 12")[$root(4, frac(64 p^(15), q^(12)))$] ⓐ #math.equation(block: false, alt: "the cube root of the fraction 625 u to the power 10 over v cubed")[$root(3, frac(625 u^(10), v^(3)))$] ⓑ #math.equation(block: false, alt: "the 4th root of the fraction 729 c to the power 21 over d to the power 8")[$root(4, frac(729 c^(21), d^(8)))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "the fraction 5 u cubed the cube root of 5 u over v")[$frac(5 u^(3) root(3, 5 u), v)$] ⓑ #math.equation(block: false, alt: "the fraction 3 c to the power 5 the 4th root of 9 c over d squared")[$frac(3 c^(5) root(4, 9 c), d^(2))$] ] #strong[Add and Subtract Higher Roots] In the following exercises, simplify. ⓐ #math.equation(block: false, alt: "the 7th root of 8 p plus the 7th root of 8 p")[$root(7, 8 p) + root(7, 8 p)$] ⓑ #math.equation(block: false, alt: "3 the cube root of 25 minus the cube root of 25")[$3 root(3, 25) − root(3, 25)$] ⓐ #math.equation(block: false, alt: "the cube root of 15 q plus the cube root of 15 q")[$root(3, 15 q) + root(3, 15 q)$] ⓑ #math.equation(block: false, alt: "2 the 4th root of 27 minus 6 the 4th root of 27")[$2 root(4, 27) − 6 root(4, 27)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "2 the cube root of 15 q")[$2 root(3, 15 q)$] ⓑ #math.equation(block: false, alt: "−4 the 4th root of 27")[$−4 root(4, 27)$] ] ⓐ #math.equation(block: false, alt: "3 the 5th root of 9 x plus 7 the 5th root of 9 x")[$3 root(5, 9 x) + 7 root(5, 9 x)$] ⓑ #math.equation(block: false, alt: "8 the 7th root of 3 q minus 2 the 7th root of 3 q")[$8 root(7, 3 q) − 2 root(7, 3 q)$] ⓐ #figure(figph[An algebraic expression showing the sum of 23 times the 12th root of 4y and 19 times the 12th root of 4y. The expression is 23(12th root of 4y) + 19(12th root of 4y).], alt: "An algebraic expression showing the sum of 23 times the 12th root of 4y and 19 times the 12th root of 4y. The expression is 23(12th root of 4y) + 19(12th root of 4y).", caption: none) ⓑ #figure(figph[The mathematical expression 31 times the 10th root of 5z minus 17 times the 10th root of 5z is displayed.], alt: "The mathematical expression 31 times the 10th root of 5z minus 17 times the 10th root of 5z is displayed.", caption: none) #solutionbox[ ⓐ #figure(figph[The image displays the mathematical expression 42 times the 12th root of 4y.], alt: "The image displays the mathematical expression 42 times the 12th root of 4y.", caption: none) ⓑ #figure(figph[The image shows the mathematical expression 14 times the 10th root of 5z.], alt: "The image shows the mathematical expression 14 times the 10th root of 5z.", caption: none) ] ⓐ #math.equation(block: false, alt: "the cube root of 81 minus the cube root of 192")[$root(3, 81) − root(3, 192)$] ⓑ #math.equation(block: false, alt: "the 4th root of 512 minus the 4th root of 32")[$root(4, 512) − root(4, 32)$] ⓐ #math.equation(block: false, alt: "the cube root of 250 minus the cube root of 54")[$root(3, 250) − root(3, 54)$] ⓑ #math.equation(block: false, alt: "the 4th root of 243 minus the 4th root of 1875")[$root(4, 243) − root(4, 1875)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "2 the cube root of 2")[$2 root(3, 2)$] ⓑ #math.equation(block: false, alt: "−2 the 4th root of 3")[$−2 root(4, 3)$] ] ⓐ #math.equation(block: false, alt: "the cube root of 128 plus the cube root of 250")[$root(3, 128) + root(3, 250)$] ⓑ #math.equation(block: false, alt: "the 5th root of 729 plus the 5th root of 96")[$root(5, 729) + root(5, 96)$] ⓐ #math.equation(block: false, alt: "the 4th root of 243 plus the 4th root of 1250")[$root(4, 243) + root(4, 1250)$] ⓑ #math.equation(block: false, alt: "the cube root of 2000 plus the cube root of 54")[$root(3, 2000) + root(3, 54)$] #solutionbox[ ⓐ #math.equation(block: false, alt: "3 the 4th root of 3 plus 5 the 4th root of 2")[$3 root(4, 3) + 5 root(4, 2)$] ⓑ #math.equation(block: false, alt: "13 the cube root of 2")[$13 root(3, 2)$] ] ⓐ #math.equation(block: false, alt: "the cube root of 64 a to the power 10 minus the cube root of −216 a to the power 12")[$root(3, 64 a^(10)) − root(3, −216 a^(12))$] ⓑ #math.equation(block: false, alt: "the 4th root of 486 u to the power 7 plus the 4th root of 768 u cubed")[$root(4, 486 u^(7)) + root(4, 768 u^(3))$] ⓐ #math.equation(block: false, alt: "the cube root of 80 b to the power 5 minus the cube root of −270 b cubed")[$root(3, 80 b^(5)) − root(3, −270 b^(3))$] ⓑ #math.equation(block: false, alt: "the 4th root of 160 v to the power 10 minus the 4th root of 1280 v cubed")[$root(4, 160 v^(10)) − root(4, 1280 v^(3))$] #solutionbox[ ⓐ #math.equation(block: false, alt: "2 b the cube root of 10 b squared plus 3 b the cube root of 10")[$2 b root(3, 10 b^(2)) + 3 b root(3, 10)$] ⓑ #math.equation(block: false, alt: "2 v squared the 4th root of 10 v squared minus 4 the 4th root of 5 v cubed")[$2 v^(2) root(4, 10 v^(2)) − 4 root(4, 5 v^(3))$] ] #strong[Mixed Practice] In the following exercises, simplify. #math.equation(block: true, alt: "the 4th root of 16")[$root(4, 16)$] #math.equation(block: true, alt: "the 6th root of 64")[$root(6, 64)$] #solutionbox[ #math.equation(block: true, alt: "2")[$2$] ] #math.equation(block: true, alt: "the cube root of a cubed")[$root(3, a^(3))$] #figure(figph[A mathematical expression featuring the 12th root of b raised to the power of 12. This expression simplifies to 'b'.], alt: "A mathematical expression featuring the 12th root of b raised to the power of 12. This expression simplifies to 'b'.", caption: none) #solutionbox[ #math.equation(block: true, alt: "| b |")[$| b |$] ] #math.equation(block: true, alt: "the cube root of −8 c to the power 9")[$root(3, −8 c^(9))$] #math.equation(block: true, alt: "the cube root of 125 d to the power 15")[$root(3, 125 d^(15))$] #solutionbox[ #math.equation(block: true, alt: "5 d to the power 5")[$5 d^(5)$] ] #math.equation(block: true, alt: "the cube root of r to the power 5")[$root(3, r^(5))$] #math.equation(block: true, alt: "the 4th root of s to the power 10")[$root(4, s^(10))$] #solutionbox[ #math.equation(block: true, alt: "s squared the 4th root of s squared")[$s^(2) root(4, s^(2))$] ] #math.equation(block: true, alt: "the cube root of 108 x to the power 5")[$root(3, 108 x^(5))$] #math.equation(block: true, alt: "the 4th root of 48 y to the power 6")[$root(4, 48 y^(6))$] #solutionbox[ #math.equation(block: true, alt: "2 y the 4th root of 3 y squared")[$2 y root(4, 3 y^(2))$] ] #math.equation(block: true, alt: "the 5th root of −486")[$root(5, −486)$] #math.equation(block: true, alt: "the 6th root of −64")[$root(6, −64)$] #solutionbox[ #math.equation(block: true, alt: "not real")[$"not real"$] ] #math.equation(block: true, alt: "the fraction the 4th root of 64 over the 4th root of 2")[$frac(root(4, 64), root(4, 2))$] #math.equation(block: true, alt: "the fraction the 5th root of 128 x to the power 8 over the 5th root of 2 x squared")[$frac(root(5, 128 x^(8)), root(5, 2 x^(2)))$] #solutionbox[ #math.equation(block: true, alt: "2 x the 5th root of 2 x")[$2 x root(5, 2 x)$] ] #math.equation(block: true, alt: "the 5th root of the fraction 96 r to the power 11 over s cubed")[$root(5, frac(96 r^(11), s^(3)))$] #math.equation(block: true, alt: "the 6th root of the fraction 128 u to the power 7 over v cubed")[$root(6, frac(128 u^(7), v^(3)))$] #solutionbox[ #math.equation(block: true, alt: "2 u the 6th root of the fraction 2 u over v cubed")[$2 u root(6, frac(2 u, v^(3)))$] ] #math.equation(block: true, alt: "the cube root of 81 minus the cube root of 192")[$root(3, 81) − root(3, 192)$] #math.equation(block: true, alt: "the 4th root of 512 minus the 4th root of 32")[$root(4, 512) − root(4, 32)$] #solutionbox[ #math.equation(block: true, alt: "2 the 4th root of 2")[$2 root(4, 2)$] ] #math.equation(block: true, alt: "the cube root of 64 a to the power 10 minus the cube root of −216 a to the power 12")[$root(3, 64 a^(10)) − root(3, −216 a^(12))$] #math.equation(block: true, alt: "the 4th root of 486 u to the power 7 plus the 4th root of 768 u cubed")[$root(4, 486 u^(7)) + root(4, 768 u^(3))$] #solutionbox[ #math.equation(block: true, alt: "3 │ u │ the 4th root of 6 u cubed plus 4 the 4th root of 3 u cubed")[$3 │ u │ root(4, 6 u^(3)) + 4 root(4, 3 u^(3))$] ] ==== Everyday Math #strong[Population growth] The expression #math.equation(block: false, alt: "10 times x to the power n")[$10 · x^(n)$] models the growth of a mold population after #math.equation(block: false, alt: "n")[$n$] generations. There were 10 spores at the start, and each had #math.equation(block: false, alt: "x")[$x$] offspring. So #math.equation(block: false, alt: "10 times x to the power 5")[$10 · x^(5)$] is the number of offspring at the fifth generation. At the fifth generation there were 10,240 offspring. Simplify the expression #math.equation(block: false, alt: "the 5th root of the fraction 10,240 over 10")[$root(5, frac(10","240, 10))$] to determine the number of offspring of each spore. #strong[Spread of a virus] The expression #math.equation(block: false, alt: "3 times x to the power n")[$3 · x^(n)$] models the spread of a virus after #math.equation(block: false, alt: "n")[$n$] cycles. There were three people originally infected with the virus, and each of them infected #math.equation(block: false, alt: "x")[$x$] people. So #math.equation(block: false, alt: "3 times x to the power 4")[$3 · x^(4)$] is the number of people infected on the fourth cycle. At the fourth cycle 1875 people were infected. Simplify the expression #math.equation(block: false, alt: "the 4th root of the fraction 1875 over 3")[$root(4, frac(1875, 3))$] to determine the number of people each person infected. #solutionbox[ #math.equation(block: true, alt: "5")[$5$] ] ==== Writing Exercises Explain how you know that #math.equation(block: false, alt: "the 5th root of x to the power 10 equals x squared")[$root(5, x^(10)) = x^(2)$] . Explain why #math.equation(block: false, alt: "the 4th root of −64")[$root(4, −64)$] is not a real number but #math.equation(block: false, alt: "the cube root of −64")[$root(3, −64)$] is. #solutionbox[ Answers may vary. ] ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This table has four columns and five rows. The first row labels each column: “I can…,” “Confidentaly,” “With some help,” and “No – I don’t get it!” The rows under the “I can…,” column read, “simplify expressions with hither roots.,” “use the product property to simplify expressions with higher roots.,” “use the quotient property to simplify expressions with higher roots.,” and “add and subtract higher roots.” The rest of the rows under the columns are empty.], alt: "This table has four columns and five rows. The first row labels each column: “I can…,” “Confidentaly,” “With some help,” and “No – I don’t get it!” The rows under the “I can…,” column read, “simplify expressions with hither roots.,” “use the product property to simplify expressions with higher roots.,” “use the quotient property to simplify expressions with higher roots.,” and “add and subtract higher roots.” The rest of the rows under the columns are empty.", caption: none) ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?