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== 9.8#h(0.6em)Rational Exponents

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Add: #math.equation(block: false, alt: "the fraction 7 over 15 plus the fraction 5 over 12")[$frac(7, 15) + frac(5, 12)$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 53 over 60")[$frac(53, 60)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 4 x squared y to the power 5 close parenthesis cubed")[$attach(( 4 x^(2) y^(5) ), t: 3)$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "64 x to the power 6 y to the power 15")[$64 x^(6) y^(15)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "5 to the power −3")[$5^(−3)$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 125")[$frac(1, 125)$]
]
]


=== Simplify Expressions with #math.equation(block: false, alt: "a to the power the fraction 1 over n")[$a^(frac(1, n))$]

Rational exponents are another way of writing expressions with radicals. When we use #strong[rational exponents], we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that #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")[$attach(( a^(m) ), t: n) = a^(m · n)$] when #emph[m] and #emph[n] are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number #emph[p] such that #math.equation(block: false, alt: "open parenthesis 8 to the power p close parenthesis cubed equals 8")[$attach(( 8^(p) ), t: 3) = 8$]. We will use the Power Property of Exponents to find the value of #emph[p].

#figure(table(
  columns: 2, align: left, inset: 6pt,
  [], [#math.equation(block: false, alt: "open parenthesis 8 to the power p close parenthesis cubed, equals, 8")[$#h(1.4em) attach(( 8^(p) ), t: 3) & = & 8$]],
  [Multiply the exponents on the left.], [#math.equation(block: false, alt: "8 to the power 3 p, equals, 8")[$#h(2em) 8^(3 p) & = & 8$]],
  [Write the exponent 1 on the right.], [#math.equation(block: false, alt: "8 to the power 3 p, equals, 8 to the power 1")[$#h(2em) 8^(3 p) & = & 8^(1)$]],
  [The exponents must be equal.], [#math.equation(block: false, alt: "3 p, equals, 1")[$#h(2.3em) 3 p & = & 1$]],
  [Solve for #math.equation(block: false, alt: "p .")[$p .$]], [#math.equation(block: false, alt: "p, equals, the fraction 1 over 3")[$#h(3em) p & = & frac(1, 3)$]],
  [], [#math.equation(block: false, alt: "So open parenthesis 8 to the power the fraction 1 over 3 close parenthesis cubed, equals, 8 .")[$"So" #h(0.2em) attach(( 8^(frac(1, 3)) ), t: 3) & = & 8 .$]],
  [But we know also #math.equation(block: false, alt: "open parenthesis the cube root of 8 close parenthesis cubed equals 8")[$attach(( root(3, 8) ), t: 3) = 8$]. Then it must be that #math.equation(block: false, alt: "8 to the power the fraction 1 over 3 equals the cube root of 8")[$8^(frac(1, 3)) = root(3, 8)$].], [],
))

But we know also #math.equation(block: false, alt: "open parenthesis the cube root of 8 close parenthesis cubed equals 8")[$attach(( root(3, 8) ), t: 3) = 8$]. Then it must be that #math.equation(block: false, alt: "8 to the power the fraction 1 over 3 equals the cube root of 8")[$8^(frac(1, 3)) = root(3, 8)$].

This same logic can be used for any positive integer exponent #emph[n] to show that #math.equation(block: false, alt: "a to the power the fraction 1 over n equals the nth root of a")[$a^(frac(1, n)) = root(n, a)$].

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Rational Exponent #math.equation(block: false, alt: "a to the power the fraction 1 over n")[$a^(frac(1, n))$]]

If #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$] is a real number and #math.equation(block: false, alt: "n greater than or equal to 2")[$n ≥ 2$], #math.equation(block: false, alt: "a to the power the fraction 1 over n equals the nth root of a")[$a^(frac(1, n)) = root(n, a)$].
]

There will be times when working with expressions will be easier if you use #strong[rational exponents] and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

#examplebox("Example 1")[][
Write as a radical expression: ⓐ #math.equation(block: false, alt: "x to the power the fraction 1 over 2")[$x^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "y to the power the fraction 1 over 3")[$y^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "z to the power the fraction 1 over 4")[$z^(frac(1, 4))$].

#solutionbox[
We want to write each expression in the form #math.equation(block: false, alt: "the nth root of a")[$root(n, a)$].

ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "x to the power the fraction 1 over 2")[$x^(frac(1, 2))$]]),
  [The denominator of the exponent is 2, so the index of the radical is 2. We do not show the index when it is 2.], [#math.equation(block: false, alt: "the square root of x")[$sqrt(x)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "y to the power the fraction 1 over 3")[$y^(frac(1, 3))$]]),
  [The denominator of the exponent is 3, so the index is 3.], [#math.equation(block: false, alt: "the cube root of y")[$root(3, y)$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "z to the power the fraction 1 over 4")[$z^(frac(1, 4))$]]),
  [The denominator of the exponent is 4, so the index is 4.], [#math.equation(block: false, alt: "the 4th root of z")[$root(4, z)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a radical expression: ⓐ #math.equation(block: false, alt: "t to the power the fraction 1 over 2")[$t^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "m to the power the fraction 1 over 3")[$m^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "r to the power the fraction 1 over 4")[$r^(frac(1, 4))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "the square root of t")[$sqrt(t)$] ⓑ #math.equation(block: false, alt: "the cube root of m")[$root(3, m)$] ⓒ #math.equation(block: false, alt: "the 4th root of r")[$root(4, r)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write as a radial expression: ⓐ #math.equation(block: false, alt: "b to the power the fraction 1 over 2")[$b^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "z to the power the fraction 1 over 3")[$z^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "p to the power the fraction 1 over 4")[$p^(frac(1, 4))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "the square root of b")[$sqrt(b)$] ⓑ #math.equation(block: false, alt: "the cube root of z")[$root(3, z)$] ⓒ #math.equation(block: false, alt: "the 4th root of p")[$root(4, p)$]
]
]

#examplebox("Example 2")[][
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the square root of x")[$sqrt(x)$] ⓑ #math.equation(block: false, alt: "the cube root of y")[$root(3, y)$] ⓒ #math.equation(block: false, alt: "the 4th root of z")[$root(4, z)$].

#solutionbox[
We want to write each radical in the form #math.equation(block: false, alt: "a to the power the fraction 1 over n")[$a^(frac(1, n))$].

ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of x")[$sqrt(x)$]]),
  [No index is shown, so it is 2. #linebreak() 
The denominator of the exponent will be 2.], [#math.equation(block: false, alt: "x to the power the fraction 1 over 2")[$x^(frac(1, 2))$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the cube root of y")[$root(3, y)$]]),
  [The index is 3, so the denominator of the exponent is 3.], [#math.equation(block: false, alt: "y to the power the fraction 1 over 3")[$y^(frac(1, 3))$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the 4th root of z")[$root(4, z)$]]),
  [The index is 4, so the denominator of the exponent is 4.], [#math.equation(block: false, alt: "z to the power the fraction 1 over 4")[$z^(frac(1, 4))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the square root of s")[$sqrt(s)$] ⓑ #math.equation(block: false, alt: "the cube root of x")[$root(3, x)$] ⓒ #math.equation(block: false, alt: "the 4th root of b")[$root(4, b)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "s to the power the fraction 1 over 2")[$s^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "x to the power the fraction 1 over 3")[$x^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "b to the power the fraction 1 over 4")[$b^(frac(1, 4))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the square root of v")[$sqrt(v)$] ⓑ #math.equation(block: false, alt: "the cube root of p")[$root(3, p)$] ⓒ #math.equation(block: false, alt: "the 4th root of p")[$root(4, p)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "v to the power the fraction 1 over 2")[$v^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "p to the power the fraction 1 over 3")[$p^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "p to the power the fraction 1 over 4")[$p^(frac(1, 4))$]
]
]

#examplebox("Example 3")[][
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the square root of 5 y")[$sqrt(5 y)$] ⓑ #math.equation(block: false, alt: "the cube root of 4 x")[$root(3, 4 x)$] ⓒ #math.equation(block: false, alt: "3 the 4th root of 5 z")[$3 root(4, 5 z)$].

#solutionbox[
We want to write each radical in the form #math.equation(block: false, alt: "a to the power the fraction 1 over n")[$a^(frac(1, n))$].

ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 5 y")[$sqrt(5 y)$]]),
  [No index is shown, so it is 2. #linebreak() 
The denominator of the exponent will be 2.], [#math.equation(block: false, alt: "open parenthesis 5 y close parenthesis to the power the fraction 1 over 2")[$attach(( 5 y ), t: frac(1, 2))$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the cube root of 4 x")[$root(3, 4 x)$]]),
  [The index is 3, so the denominator of the exponent is 3.], [#math.equation(block: false, alt: "open parenthesis 4 x close parenthesis to the power the fraction 1 over 3")[$attach(( 4 x ), t: frac(1, 3))$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 the 4th root of 5 z")[$3 root(4, 5 z)$]]),
  [The index is 4, so the denominator of the exponent is 4.], [#math.equation(block: false, alt: "3 open parenthesis 5 z close parenthesis to the power the fraction 1 over 4")[$3 attach(( 5 z ), t: frac(1, 4))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the square root of 10 m")[$sqrt(10 m)$] ⓑ #math.equation(block: false, alt: "the 5th root of 3 n")[$root(5, 3 n)$] ⓒ #math.equation(block: false, alt: "3 the 4th root of 6 y")[$3 root(4, 6 y)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis 10 m close parenthesis to the power the fraction 1 over 2")[$attach(( 10 m ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 n close parenthesis to the power the fraction 1 over 5")[$attach(( 3 n ), t: frac(1, 5))$] ⓒ #math.equation(block: false, alt: "open parenthesis 486 y close parenthesis to the power the fraction 1 over 4")[$attach(( 486 y ), t: frac(1, 4))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the 7th root of 3 k")[$root(7, 3 k)$] ⓑ #math.equation(block: false, alt: "the 4th root of 5 j")[$root(4, 5 j)$] ⓒ #math.equation(block: false, alt: "8 the cube root of 2 a")[$8 root(3, 2 a)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis 3 k close parenthesis to the power the fraction 1 over 7")[$attach(( 3 k ), t: frac(1, 7))$] ⓑ #math.equation(block: false, alt: "open parenthesis 5 j close parenthesis to the power the fraction 1 over 4")[$attach(( 5 j ), t: frac(1, 4))$] ⓒ #math.equation(block: false, alt: "open parenthesis 1024 a close parenthesis to the power the fraction 1 over 3")[$attach(( 1024 a ), t: frac(1, 3))$]
]
]

In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

#examplebox("Example 4")[][
Simplify: ⓐ #math.equation(block: false, alt: "25 to the power the fraction 1 over 2")[$25^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "64 to the power the fraction 1 over 3")[$64^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "256 to the power the fraction 1 over 4")[$256^(frac(1, 4))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "25 to the power the fraction 1 over 2")[$25^(frac(1, 2))$]]),
  [Rewrite as a square root.], [#math.equation(block: false, alt: "the square root of 25")[$sqrt(25)$]],
  [Simplify.], [#math.equation(block: false, alt: "5")[$5$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "64 to the power the fraction 1 over 3")[$64^(frac(1, 3))$]]),
  [Rewrite as a cube root.], [#math.equation(block: false, alt: "the cube root of 64")[$root(3, 64)$]],
  [Recognize 64 is a perfect cube.], [#math.equation(block: false, alt: "the cube root of 4 cubed")[$root(3, 4^(3))$]],
  [Simplify.], [#math.equation(block: false, alt: "4")[$4$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "256 to the power the fraction 1 over 4")[$256^(frac(1, 4))$]]),
  [Rewrite as a fourth root.], [#math.equation(block: false, alt: "the 4th root of 256")[$root(4, 256)$]],
  [Recognize 256 is a perfect fourth power.], [#math.equation(block: false, alt: "the 4th root of 4 to the power 4")[$root(4, 4^(4))$]],
  [Simplify.], [#math.equation(block: false, alt: "4")[$4$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "36 to the power the fraction 1 over 2")[$36^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "8 to the power the fraction 1 over 3")[$8^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "16 to the power the fraction 1 over 4")[$16^(frac(1, 4))$].

#solutionbox[
ⓐ 6 ⓑ 2 ⓒ 2
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "100 to the power the fraction 1 over 2")[$100^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "27 to the power the fraction 1 over 3")[$27^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "81 to the power the fraction 1 over 4")[$81^(frac(1, 4))$].

#solutionbox[
ⓐ 10 ⓑ 3 ⓒ 3
]
]

Be careful of the placement of the negative signs in the next example. We will need to use the property #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$] in one case.

#examplebox("Example 5")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −64 close parenthesis to the power the fraction 1 over 3")[$attach(( −64 ), t: frac(1, 3))$] ⓑ #math.equation(block: false, alt: "− 64 to the power the fraction 1 over 3")[$"−" 64^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "open parenthesis 64 close parenthesis to the power minus the fraction 1 over 3")[$attach(( 64 ), t: − frac(1, 3))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis −64 close parenthesis to the power the fraction 1 over 3")[$attach(( −64 ), t: frac(1, 3))$]]),
  [Rewrite as a cube root.], [#math.equation(block: false, alt: "the cube root of −64")[$root(3, −64)$]],
  [Rewrite #math.equation(block: false, alt: "−64")[$−64$] as a perfect cube.], [#math.equation(block: false, alt: "the cube root of open parenthesis −4 close parenthesis cubed")[$root(3, attach(( −4 ), t: 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "−4")[$−4$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− 64 to the power the fraction 1 over 3")[$"−" 64^(frac(1, 3))$]]),
  [The exponent applies only to the 64.], [#math.equation(block: false, alt: "− open parenthesis 64 to the power the fraction 1 over 3 close parenthesis")[$"−" ( 64^(frac(1, 3)) )$]],
  [Rewrite as a cube root.], [#math.equation(block: false, alt: "− the cube root of 64")[$"−" root(3, 64)$]],
  [Rewrite 64 as #math.equation(block: false, alt: "4 cubed .")[$4^(3) .$]], [#math.equation(block: false, alt: "− the cube root of 4 cubed")[$"−" root(3, 4^(3))$]],
  [Simplify.], [#math.equation(block: false, alt: "−4")[$−4$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 64 close parenthesis to the power minus the fraction 1 over 3")[$attach(( 64 ), t: − frac(1, 3))$]]),
  [Rewrite as a fraction with a positive exponent, using the property, #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n .")[$a^("−" n) = frac(1, a^(n)) .$]], [#math.equation(block: false, alt: "the fraction 1 over the cube root of 64")[$frac(1, root(3, 64))$]],
  [Write as a cube root.], [],
  [Rewrite 64 as #math.equation(block: false, alt: "4 cubed .")[$4^(3) .$]], [#math.equation(block: false, alt: "the fraction 1 over the cube root of 4 cubed")[$frac(1, root(3, 4^(3)))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −125 close parenthesis to the power the fraction 1 over 3")[$attach(( −125 ), t: frac(1, 3))$] ⓑ #math.equation(block: false, alt: "− 125 to the power the fraction 1 over 3")[$"−" 125^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "open parenthesis 125 close parenthesis to the power minus the fraction 1 over 3")[$attach(( 125 ), t: − frac(1, 3))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "−5")[$−5$] ⓑ #math.equation(block: false, alt: "−5")[$−5$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 5")[$frac(1, 5)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −32 close parenthesis to the power the fraction 1 over 5")[$attach(( −32 ), t: frac(1, 5))$] ⓑ #math.equation(block: false, alt: "− 32 to the power the fraction 1 over 5")[$"−" 32^(frac(1, 5))$] ⓒ #math.equation(block: false, alt: "open parenthesis 32 close parenthesis to the power minus the fraction 1 over 5")[$attach(( 32 ), t: − frac(1, 5))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "−2")[$−2$] ⓑ #math.equation(block: false, alt: "−2")[$−2$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]
]
]

#examplebox("Example 6")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −16 close parenthesis to the power the fraction 1 over 4")[$attach(( −16 ), t: frac(1, 4))$] ⓑ #math.equation(block: false, alt: "− 16 to the power the fraction 1 over 4")[$"−" 16^(frac(1, 4))$] ⓒ #math.equation(block: false, alt: "open parenthesis 16 close parenthesis to the power minus the fraction 1 over 4")[$attach(( 16 ), t: − frac(1, 4))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis −16 close parenthesis to the power the fraction 1 over 4")[$attach(( −16 ), t: frac(1, 4))$]]),
  [Rewrite as a fourth root.], [#math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$]],
  [There is no real number whose fourth power is #math.equation(block: false, alt: "−16 .")[$−16 .$]], [],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− 16 to the power the fraction 1 over 4")[$"−" 16^(frac(1, 4))$]]),
  [The exponent only applies to the 16. #linebreak() 
Rewrite as a fourth root.], [#math.equation(block: false, alt: "− the 4th root of 16")[$"−" root(4, 16)$]],
  [Rewrite #math.equation(block: false, alt: "16")[$16$] as #math.equation(block: false, alt: "2 to the power 4 .")[$2^(4) .$]], [#math.equation(block: false, alt: "− the 4th root of 2 to the power 4")[$"−" root(4, 2^(4))$]],
  [Simplify.], [#math.equation(block: false, alt: "−2")[$−2$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 16 close parenthesis to the power minus the fraction 1 over 4")[$attach(( 16 ), t: − frac(1, 4))$]]),
  [Rewrite using the property #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n .")[$a^("−" n) = frac(1, a^(n)) .$]], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis 16 close parenthesis to the power the fraction 1 over 4")[$frac(1, attach(( 16 ), t: frac(1, 4)))$]],
  [Rewrite as a fourth root.], [#math.equation(block: false, alt: "the fraction 1 over the 4th root of 16")[$frac(1, root(4, 16))$]],
  [Rewrite #math.equation(block: false, alt: "16")[$16$] as #math.equation(block: false, alt: "2 to the power 4 .")[$2^(4) .$]], [#math.equation(block: false, alt: "the fraction 1 over the 4th root of 2 to the power 4")[$frac(1, root(4, 2^(4)))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −64 close parenthesis to the power the fraction 1 over 2")[$attach(( −64 ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "− 64 to the power the fraction 1 over 2")[$"−" 64^(frac(1, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis 64 close parenthesis to the power minus the fraction 1 over 2")[$attach(( 64 ), t: − frac(1, 2))$].

#solutionbox[
ⓐ not real ⓑ #math.equation(block: false, alt: "−8")[$−8$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −256 close parenthesis to the power the fraction 1 over 4")[$attach(( −256 ), t: frac(1, 4))$] ⓑ #math.equation(block: false, alt: "− 256 to the power the fraction 1 over 4")[$"−" 256^(frac(1, 4))$] ⓒ #math.equation(block: false, alt: "open parenthesis 256 close parenthesis to the power minus the fraction 1 over 4")[$attach(( 256 ), t: − frac(1, 4))$].

#solutionbox[
ⓐ not real ⓑ #math.equation(block: false, alt: "−4")[$−4$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]
]
]


=== Simplify Expressions with #math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$]

Let’s work with the Power Property for Exponents some more.

Suppose we raise #math.equation(block: false, alt: "a to the power the fraction 1 over n")[$a^(frac(1, n))$] to the power #emph[m.]

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis a to the power the fraction 1 over n close parenthesis to the power m")[$attach(( a^(frac(1, n)) ), t: m)$]]),
  [Multiply the exponents.], [#math.equation(block: false, alt: "a to the power the fraction 1 over n to the power times m")[$a^(frac(1, n)) · m$]],
  [Simplify.], [#math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$]],
  [So #math.equation(block: false, alt: "a to the power the fraction m over n equals open parenthesis the nth root of a close parenthesis to the power m .")[$a^(frac(m, n)) = attach(( root(n, a) ), t: m) .$]], [],
))

Now suppose we take #math.equation(block: false, alt: "a to the power m")[$a^(m)$] to the #math.equation(block: false, alt: "the fraction 1 over n")[$frac(1, n)$] power.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power the fraction 1 over n")[$attach(( a^(m) ), t: frac(1, n))$]]),
  [Multiply the exponents.], [#math.equation(block: false, alt: "a to the power m times the fraction 1 over n")[$a^(m · frac(1, n))$]],
  [Simplify.], [#math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$]],
  [So #math.equation(block: false, alt: "a to the power the fraction m over n equals the nth root of a to the power m")[$a^(frac(m, n)) = root(n, a^(m))$] also.], [],
))

Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Rational Exponent #math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$]]

For any positive integers #emph[m] and #emph[n],

#math.equation(block: true, alt: "a to the power the fraction m over n equals open parenthesis the nth root of a close parenthesis to the power m; a to the power the fraction m over n equals the nth root of a to the power m")[$a^(frac(m, n)) = attach(( root(n, a) ), t: m) \ a^(frac(m, n)) = root(n, a^(m))$]
]

#examplebox("Example 7")[][
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the square root of y cubed")[$sqrt(y^(3))$] ⓑ #math.equation(block: false, alt: "the cube root of x squared")[$root(3, x^(2))$] ⓒ #math.equation(block: false, alt: "the 4th root of z cubed")[$root(4, z^(3))$].

#solutionbox[
We want to use #math.equation(block: false, alt: "a to the power the fraction m over n equals the nth root of a to the power m")[$a^(frac(m, n)) = root(n, a^(m))$] to write each radical in the form #math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$].

+ ⓐ  #linebreak() 
#figure(figph[This figure says, “The numerator of the exponent is the exponent of y, 3.” It then shows the square root of y cubed. The figure then says, “The denominator of the exponent is the index of the radical, 2.” It then shows y to the 3/2 power.], alt: "This figure says, “The numerator of the exponent is the exponent of y, 3.” It then shows the square root of y cubed. The figure then says, “The denominator of the exponent is the index of the radical, 2.” It then shows y to the 3/2 power.", caption: none)
+ ⓑ  #linebreak() 
#figure(figph[This figure says, “The numerator of the exponent is the exponent of x, 2.” It then shows the cubed root of x squared. The figure then reads, “The denominator of the exponent is the index of the radical, 3.” It then shows y to the 2/3 power.], alt: "This figure says, “The numerator of the exponent is the exponent of x, 2.” It then shows the cubed root of x squared. The figure then reads, “The denominator of the exponent is the index of the radical, 3.” It then shows y to the 2/3 power.", caption: none)
+ ⓒ  #linebreak() 
#figure(figph[This figure reads, “The numerator of the exponent is the exponent of z, 3.” It then shows the fourth root of z cubed. The figure then reads, “The denominator of the exponent is the index of the radical, 4.” It then shows z to the 3/4 power.], alt: "This figure reads, “The numerator of the exponent is the exponent of z, 3.” It then shows the fourth root of z cubed. The figure then reads, “The denominator of the exponent is the index of the radical, 4.” It then shows z to the 3/4 power.", caption: none)
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the square root of x to the power 5")[$sqrt(x^(5))$] ⓑ #math.equation(block: false, alt: "the 4th root of z cubed")[$root(4, z^(3))$] ⓒ #math.equation(block: false, alt: "the 5th root of y squared")[$root(5, y^(2))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "x to the power the fraction 5 over 2")[$x^(frac(5, 2))$] ⓑ #math.equation(block: false, alt: "z to the power the fraction 3 over 4")[$z^(frac(3, 4))$] ⓒ #math.equation(block: false, alt: "y to the power the fraction 2 over 5")[$y^(frac(2, 5))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the 5th root of a squared")[$root(5, a^(2))$] ⓑ #math.equation(block: false, alt: "the cube root of b to the power 7")[$root(3, b^(7))$] ⓒ #math.equation(block: false, alt: "the 4th root of m to the power 5")[$root(4, m^(5))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "a to the power the fraction 2 over 5")[$a^(frac(2, 5))$] ⓑ #math.equation(block: false, alt: "b to the power the fraction 7 over 3")[$b^(frac(7, 3))$] ⓒ #math.equation(block: false, alt: "m to the power the fraction 5 over 4")[$m^(frac(5, 4))$]
]
]

#examplebox("Example 8")[][
Simplify: ⓐ #math.equation(block: false, alt: "9 to the power the fraction 3 over 2")[$9^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "125 to the power the fraction 2 over 3")[$125^(frac(2, 3))$] ⓒ #math.equation(block: false, alt: "81 to the power the fraction 3 over 4")[$81^(frac(3, 4))$].

#solutionbox[
We will rewrite each expression as a radical first using the property, #math.equation(block: false, alt: "a to the power the fraction m over n equals open parenthesis the nth root of a close parenthesis to the power m")[$a^(frac(m, n)) = attach(( root(n, a) ), t: m)$]. This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "9 to the power the fraction 3 over 2")[$9^(frac(3, 2))$]]),
  [The power of the radical is the numerator of the exponent, 3. Since the denominator of the exponent is 2, this is a square root.], [#math.equation(block: false, alt: "open parenthesis the square root of 9 close parenthesis cubed")[$attach(( sqrt(9) ), t: 3)$]],
  [Simplify.], [#math.equation(block: false, alt: "open parenthesis 3 close parenthesis cubed")[$attach(( 3 ), t: 3)$]],
  [], [#math.equation(block: false, alt: "27")[$27$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "125 to the power the fraction 2 over 3")[$125^(frac(2, 3))$]]),
  [The power of the radical is the numerator of the exponent, 2. The index of the radical is the denominator of the exponent, 3.], [#math.equation(block: false, alt: "open parenthesis the cube root of 125 close parenthesis squared")[$attach(( root(3, 125) ), t: 2)$]],
  [Simplify.], [#math.equation(block: false, alt: "open parenthesis 5 close parenthesis squared")[$attach(( 5 ), t: 2)$]],
  [], [#math.equation(block: false, alt: "25")[$25$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "81 to the power the fraction 3 over 4")[$81^(frac(3, 4))$]]),
  [The power of the radical is the numerator of the exponent, 3. The index of the radical is the denominator of the exponent, 4.], [#math.equation(block: false, alt: "open parenthesis the 4th root of 81 close parenthesis cubed")[$attach(( root(4, 81) ), t: 3)$]],
  [Simplify.], [#math.equation(block: false, alt: "open parenthesis 3 close parenthesis cubed")[$attach(( 3 ), t: 3)$]],
  [], [#math.equation(block: false, alt: "27")[$27$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "4 to the power the fraction 3 over 2")[$4^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "27 to the power the fraction 2 over 3")[$27^(frac(2, 3))$] ⓒ #math.equation(block: false, alt: "625 to the power the fraction 3 over 4")[$625^(frac(3, 4))$].

#solutionbox[
ⓐ 8 ⓑ 9 ⓒ 125
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "8 to the power the fraction 5 over 3")[$8^(frac(5, 3))$] ⓑ #math.equation(block: false, alt: "81 to the power the fraction 3 over 2")[$81^(frac(3, 2))$] ⓒ #math.equation(block: false, alt: "16 to the power the fraction 3 over 4")[$16^(frac(3, 4))$].

#solutionbox[
ⓐ 32 ⓑ 729 ⓒ 8
]
]

Remember that #math.equation(block: false, alt: "b to the power − p equals the fraction 1 over b to the power p")[$b^("−" p) = frac(1, b^(p))$]. The negative sign in the exponent does not change the sign of the expression.

#examplebox("Example 9")[][
Simplify: ⓐ #math.equation(block: false, alt: "16 to the power minus the fraction 3 over 2")[$16^(− frac(3, 2))$] ⓑ #math.equation(block: false, alt: "32 to the power minus the fraction 2 over 5")[$32^(− frac(2, 5))$] ⓒ #math.equation(block: false, alt: "4 to the power minus the fraction 5 over 2")[$4^(− frac(5, 2))$].

#solutionbox[
We will rewrite each expression first using #math.equation(block: false, alt: "b to the power − p equals the fraction 1 over b to the power p")[$b^("−" p) = frac(1, b^(p))$] and then change to radical form.

ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "16 to the power minus the fraction 3 over 2")[$16^(− frac(3, 2))$]]),
  [Rewrite using #math.equation(block: false, alt: "b to the power − p equals the fraction 1 over b to the power p .")[$b^("−" p) = frac(1, b^(p)) .$]], [#math.equation(block: false, alt: "the fraction 1 over 16 to the power the fraction 3 over 2")[$frac(1, 16^(frac(3, 2)))$]],
  [Change to radical form. The power of the radical is the numerator of the exponent, 3. #linebreak() 
The index is the denominator of the exponent, 2.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis the square root of 16 close parenthesis cubed")[$frac(1, attach(( sqrt(16) ), t: 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 4 cubed")[$frac(1, 4^(3))$]],
  [], [#math.equation(block: false, alt: "the fraction 1 over 64")[$frac(1, 64)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "32 to the power minus the fraction 2 over 5")[$32^(− frac(2, 5))$]]),
  [Rewrite using #math.equation(block: false, alt: "b to the power − p equals the fraction 1 over b to the power p .")[$b^("−" p) = frac(1, b^(p)) .$]], [#math.equation(block: false, alt: "the fraction 1 over 32 to the power the fraction 2 over 5")[$frac(1, 32^(frac(2, 5)))$]],
  [Change to radical form.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis the 5th root of 32 close parenthesis squared")[$frac(1, attach(( root(5, 32) ), t: 2))$]],
  [Rewrite the radicand as a power.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis the 5th root of 2 to the power 5 close parenthesis squared")[$frac(1, attach(( root(5, 2^(5)) ), t: 2))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 2 squared")[$frac(1, 2^(2))$]],
  [], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 to the power minus the fraction 5 over 2")[$4^(− frac(5, 2))$]]),
  [Rewrite using #math.equation(block: false, alt: "b to the power − p equals the fraction 1 over b to the power p .")[$b^("−" p) = frac(1, b^(p)) .$]], [#math.equation(block: false, alt: "the fraction 1 over 4 to the power the fraction 5 over 2")[$frac(1, 4^(frac(5, 2)))$]],
  [Change to radical form.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis the square root of 4 close parenthesis to the power 5")[$frac(1, attach(( sqrt(4) ), t: 5))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 2 to the power 5")[$frac(1, 2^(5))$]],
  [], [#math.equation(block: false, alt: "the fraction 1 over 32")[$frac(1, 32)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "8 to the power minus the fraction 5 over 3")[$8^(− frac(5, 3))$] ⓑ #math.equation(block: false, alt: "81 to the power minus the fraction 3 over 2")[$81^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "16 to the power minus the fraction 3 over 4")[$16^(− frac(3, 4))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 1 over 32")[$frac(1, 32)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 729")[$frac(1, 729)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "4 to the power minus the fraction 3 over 2")[$4^(− frac(3, 2))$] ⓑ #math.equation(block: false, alt: "27 to the power minus the fraction 2 over 3")[$27^(− frac(2, 3))$] ⓒ #math.equation(block: false, alt: "625 to the power minus the fraction 3 over 4")[$625^(− frac(3, 4))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 9")[$frac(1, 9)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 125")[$frac(1, 125)$]
]
]

#examplebox("Example 10")[][
Simplify: ⓐ #math.equation(block: false, alt: "− 25 to the power the fraction 3 over 2")[$"−" 25^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "− 25 to the power minus the fraction 3 over 2")[$"−" 25^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −25 close parenthesis to the power the fraction 3 over 2")[$attach(( −25 ), t: frac(3, 2))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− 25 to the power the fraction 3 over 2")[$"−" 25^(frac(3, 2))$]]),
  [Rewrite in radical form.], [#math.equation(block: false, alt: "− open parenthesis the square root of 25 close parenthesis cubed")[$"−" attach(( sqrt(25) ), t: 3)$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "− open parenthesis 5 close parenthesis cubed")[$"−" attach(( 5 ), t: 3)$]],
  [Simplify.], [#math.equation(block: false, alt: "−125")[$−125$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "− 25 to the power minus the fraction 3 over 2")[$"−" 25^(− frac(3, 2))$]]),
  [Rewrite using #math.equation(block: false, alt: "b to the power − p equals the fraction 1 over b to the power p .")[$b^("−" p) = frac(1, b^(p)) .$]], [#math.equation(block: false, alt: "− open parenthesis the fraction 1 over 25 to the power the fraction 3 over 2 close parenthesis")[$"−" ( frac(1, 25^(frac(3, 2))) )$]],
  [Rewrite in radical form.], [#math.equation(block: false, alt: "− open parenthesis the fraction 1 over open parenthesis the square root of 25 close parenthesis cubed close parenthesis")[$"−" ( frac(1, attach(( sqrt(25) ), t: 3)) )$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "− open parenthesis the fraction 1 over open parenthesis 5 close parenthesis cubed close parenthesis")[$"−" ( frac(1, attach(( 5 ), t: 3)) )$]],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 1 over 125")[$− frac(1, 125)$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis −25 close parenthesis to the power the fraction 3 over 2")[$attach(( −25 ), t: frac(3, 2))$]]),
  [Rewrite in radical form.], [#math.equation(block: false, alt: "open parenthesis the square root of −25 close parenthesis cubed")[$attach(( sqrt(−25) ), t: 3)$]],
  [There is no real number whose square root is #math.equation(block: false, alt: "−25 .")[$−25 .$]], [Not a real number.],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "−16 to the power the fraction 3 over 2")[$−16^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "−16 to the power minus the fraction 3 over 2")[$−16^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −16 close parenthesis to the power minus the fraction 3 over 2")[$attach(( −16 ), t: − frac(3, 2))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "−64")[$−64$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 64")[$− frac(1, 64)$] ⓒ not a real number
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "−81 to the power the fraction 3 over 2")[$−81^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "−81 to the power minus the fraction 3 over 2")[$−81^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −81 close parenthesis to the power minus the fraction 3 over 2")[$attach(( −81 ), t: − frac(3, 2))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "−729")[$−729$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 729")[$− frac(1, 729)$] ⓒ not a real number
]
]


=== Use the Laws of Exponents to Simplify Expressions with Rational Exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Summary of Exponent Properties]

If #math.equation(block: false, alt: "a , b")[$a , b$] are real numbers and #math.equation(block: false, alt: "m , n")[$m , n$] are rational numbers, then

#math.equation(block: true, alt: "Product Property, a to the power m times a to the power n equals a to the power m plus n; Power Property, open parenthesis a to the power m close parenthesis to the power n equals a to the power m times n; Product to a Power, open parenthesis a b close parenthesis to the power m equals a to the power m b to the power m; Quotient Property, the fraction a to the power m over a to the power n equals a to the power m minus n , a not equal to 0 , 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 0 equals 1 , a not equal to 0; Quotient to a Power Property, open parenthesis the fraction a over b close parenthesis to the power m equals the fraction")[$"Product Property" &  &  & a^(m) · a^(n) = a^(m + n) \ "Power Property" &  &  & attach(( a^(m) ), t: n) = a^(m · n) \ "Product to a Power" &  &  & attach(( a b ), t: m) = a^(m) b^(m) \ "Quotient Property" &  &  & frac(a^(m), a^(n)) = a^(m − n) , #h(0.2em) #h(0.2em) a ≠ 0 , #h(0.2em) #h(0.2em) m > n \  &  &  & frac(a^(m), a^(n)) = frac(1, a^(n − m)) , #h(0.2em) #h(0.2em) a ≠ 0 , #h(0.2em) #h(0.2em) n > m \ "Zero Exponent Definition" &  &  & a^(0) = 1 , #h(0.2em) #h(0.2em) a ≠ 0 \ "Quotient to a Power Property" &  &  & attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m)) , #h(0.2em) #h(0.2em) b ≠ 0$]
]

When we multiply the same base, we add the exponents.

#examplebox("Example 11")[][
Simplify: ⓐ #math.equation(block: false, alt: "2 to the power the fraction 1 over 2 times 2 to the power the fraction 5 over 2")[$2^(frac(1, 2)) · 2^(frac(5, 2))$] ⓑ #math.equation(block: false, alt: "x to the power the fraction 2 over 3 times x to the power the fraction 4 over 3")[$x^(frac(2, 3)) · x^(frac(4, 3))$] ⓒ #math.equation(block: false, alt: "z to the power the fraction 3 over 4 times z to the power the fraction 5 over 4")[$z^(frac(3, 4)) · z^(frac(5, 4))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 to the power the fraction 1 over 2 times 2 to the power the fraction 5 over 2")[$2^(frac(1, 2)) · 2^(frac(5, 2))$]]),
  [The bases are the same, so we add the exponents.], [#math.equation(block: false, alt: "2 to the power the fraction 1 over 2 plus the fraction 5 over 2")[$2^(frac(1, 2) + frac(5, 2))$]],
  [Add the fractions.], [#math.equation(block: false, alt: "2 to the power the fraction 6 over 2")[$2^(frac(6, 2))$]],
  [Simplify the exponent.], [#math.equation(block: false, alt: "2 cubed")[$2^(3)$]],
  [Simplify.], [#math.equation(block: false, alt: "8")[$8$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "x to the power the fraction 2 over 3 times x to the power the fraction 4 over 3")[$x^(frac(2, 3)) · x^(frac(4, 3))$]]),
  [The bases are the same, so we add the exponents.], [#math.equation(block: false, alt: "x to the power the fraction 2 over 3 plus the fraction 4 over 3")[$x^(frac(2, 3) + frac(4, 3))$]],
  [Add the fractions.], [#math.equation(block: false, alt: "x to the power the fraction 6 over 3")[$x^(frac(6, 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "x squared")[$x^(2)$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "z to the power the fraction 3 over 4 times z to the power the fraction 5 over 4")[$z^(frac(3, 4)) · z^(frac(5, 4))$]]),
  [The bases are the same, so we add the exponents.], [#math.equation(block: false, alt: "z to the power the fraction 3 over 4 plus the fraction 5 over 4")[$z^(frac(3, 4) + frac(5, 4))$]],
  [Add the fractions.], [#math.equation(block: false, alt: "z to the power the fraction 8 over 4")[$z^(frac(8, 4))$]],
  [Simplify.], [#math.equation(block: false, alt: "z squared")[$z^(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "3 to the power the fraction 2 over 3 times 3 to the power the fraction 4 over 3")[$3^(frac(2, 3)) · 3^(frac(4, 3))$] ⓑ #math.equation(block: false, alt: "y to the power the fraction 1 over 3 times y to the power the fraction 8 over 3")[$y^(frac(1, 3)) · y^(frac(8, 3))$] ⓒ #math.equation(block: false, alt: "m to the power the fraction 1 over 4 times m to the power the fraction 3 over 4")[$m^(frac(1, 4)) · m^(frac(3, 4))$].

#solutionbox[
ⓐ 9 ⓑ #math.equation(block: false, alt: "y cubed")[$y^(3)$] ⓒ #emph[m]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "5 to the power the fraction 3 over 5 times 5 to the power the fraction 7 over 5")[$5^(frac(3, 5)) · 5^(frac(7, 5))$] ⓑ #math.equation(block: false, alt: "z to the power the fraction 1 over 8 times z to the power the fraction 7 over 8")[$z^(frac(1, 8)) · z^(frac(7, 8))$] ⓒ #math.equation(block: false, alt: "n to the power the fraction 2 over 7 times n to the power the fraction 5 over 7")[$n^(frac(2, 7)) · n^(frac(5, 7))$].

#solutionbox[
ⓐ 25 ⓑ #emph[z] ⓒ #emph[n]
]
]

We will use the Power Property in the next example.

#examplebox("Example 12")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis x to the power 4 close parenthesis to the power the fraction 1 over 2")[$attach(( x^(4) ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis y to the power 6 close parenthesis to the power the fraction 1 over 3")[$attach(( y^(6) ), t: frac(1, 3))$] ⓒ #math.equation(block: false, alt: "open parenthesis z to the power 9 close parenthesis to the power the fraction 2 over 3")[$attach(( z^(9) ), t: frac(2, 3))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis x to the power 4 close parenthesis to the power the fraction 1 over 2")[$attach(( x^(4) ), t: frac(1, 2))$]]),
  [To raise a power to a power, we multiply the exponents.], [#math.equation(block: false, alt: "x to the power 4 times the fraction 1 over 2")[$x^(4 · frac(1, 2))$]],
  [Simplify.], [#math.equation(block: false, alt: "x squared")[$x^(2)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis y to the power 6 close parenthesis to the power the fraction 1 over 3")[$attach(( y^(6) ), t: frac(1, 3))$]]),
  [To raise a power to a power, we multiply the exponents.], [#math.equation(block: false, alt: "y to the power 6 times the fraction 1 over 3")[$y^(6 · frac(1, 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "y squared")[$y^(2)$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis z to the power 9 close parenthesis to the power the fraction 2 over 3")[$attach(( z^(9) ), t: frac(2, 3))$]]),
  [To raise a power to a power, we multiply the exponents.], [#math.equation(block: false, alt: "z to the power 9 times the fraction 2 over 3")[$z^(9 · frac(2, 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "z to the power 6")[$z^(6)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis p to the power 10 close parenthesis to the power the fraction 1 over 5")[$attach(( p^(10) ), t: frac(1, 5))$] ⓑ #math.equation(block: false, alt: "open parenthesis q to the power 8 close parenthesis to the power the fraction 3 over 4")[$attach(( q^(8) ), t: frac(3, 4))$] ⓒ #math.equation(block: false, alt: "open parenthesis x to the power 6 close parenthesis to the power the fraction 4 over 3")[$attach(( x^(6) ), t: frac(4, 3))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "p squared")[$p^(2)$] ⓑ #math.equation(block: false, alt: "q to the power 6")[$q^(6)$] ⓒ #math.equation(block: false, alt: "x to the power 8")[$x^(8)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis r to the power 6 close parenthesis to the power the fraction 5 over 3")[$attach(( r^(6) ), t: frac(5, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis s to the power 12 close parenthesis to the power the fraction 3 over 4")[$attach(( s^(12) ), t: frac(3, 4))$] ⓒ #math.equation(block: false, alt: "open parenthesis m to the power 9 close parenthesis to the power the fraction 2 over 9")[$attach(( m^(9) ), t: frac(2, 9))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "r to the power 10")[$r^(10)$] ⓑ #math.equation(block: false, alt: "s to the power 9")[$s^(9)$] ⓒ #math.equation(block: false, alt: "m squared")[$m^(2)$]
]
]

The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

#examplebox("Example 13")[][
Simplify: ⓐ #math.equation(block: false, alt: "the fraction x to the power the fraction 4 over 3 over x to the power the fraction 1 over 3")[$frac(x^(frac(4, 3)), x^(frac(1, 3)))$] ⓑ #math.equation(block: false, alt: "the fraction y to the power the fraction 3 over 4 over y to the power the fraction 1 over 4")[$frac(y^(frac(3, 4)), y^(frac(1, 4)))$] ⓒ #math.equation(block: false, alt: "the fraction z to the power the fraction 2 over 3 over z to the power the fraction 5 over 3")[$frac(z^(frac(2, 3)), z^(frac(5, 3)))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction x to the power the fraction 4 over 3 over x to the power the fraction 1 over 3")[$frac(x^(frac(4, 3)), x^(frac(1, 3)))$]]),
  [To divide with the same base, we subtract the exponents.], [#math.equation(block: false, alt: "x to the power the fraction 4 over 3 minus the fraction 1 over 3")[$x^(frac(4, 3) − frac(1, 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "x")[$x$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction y to the power the fraction 3 over 4 over y to the power the fraction 1 over 4")[$frac(y^(frac(3, 4)), y^(frac(1, 4)))$]]),
  [To divide with the same base, we subtract the exponents.], [#math.equation(block: false, alt: "y to the power the fraction 3 over 4 minus the fraction 1 over 4")[$y^(frac(3, 4) − frac(1, 4))$]],
  [Simplify.], [#math.equation(block: false, alt: "y to the power the fraction 1 over 2")[$y^(frac(1, 2))$]],
))

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction z to the power the fraction 2 over 3 over z to the power the fraction 5 over 3")[$frac(z^(frac(2, 3)), z^(frac(5, 3)))$]]),
  [To divide with the same base, we subtract the exponents.], [#math.equation(block: false, alt: "z to the power the fraction 2 over 3 minus the fraction 5 over 3")[$z^(frac(2, 3) − frac(5, 3))$]],
  [Rewrite without a negative exponent.], [#math.equation(block: false, alt: "the fraction 1 over z")[$frac(1, z)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the fraction u to the power the fraction 5 over 4 over u to the power the fraction 1 over 4")[$frac(u^(frac(5, 4)), u^(frac(1, 4)))$] ⓑ #math.equation(block: false, alt: "the fraction v to the power the fraction 3 over 5 over v to the power the fraction 2 over 5")[$frac(v^(frac(3, 5)), v^(frac(2, 5)))$] ⓒ #math.equation(block: false, alt: "the fraction x to the power the fraction 2 over 3 over x to the power the fraction 5 over 3")[$frac(x^(frac(2, 3)), x^(frac(5, 3)))$].

#solutionbox[
ⓐ #emph[u] ⓑ #math.equation(block: false, alt: "v to the power the fraction 1 over 5")[$v^(frac(1, 5))$] ⓒ #math.equation(block: false, alt: "the fraction 1 over x")[$frac(1, x)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the fraction c to the power the fraction 12 over 5 over c to the power the fraction 2 over 5")[$frac(c^(frac(12, 5)), c^(frac(2, 5)))$] ⓑ #math.equation(block: false, alt: "the fraction m to the power the fraction 5 over 4 over m to the power the fraction 9 over 4")[$frac(m^(frac(5, 4)), m^(frac(9, 4)))$] ⓒ #math.equation(block: false, alt: "the fraction d to the power the fraction 1 over 5 over d to the power the fraction 6 over 5")[$frac(d^(frac(1, 5)), d^(frac(6, 5)))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "c squared")[$c^(2)$] ⓑ #math.equation(block: false, alt: "the fraction 1 over m")[$frac(1, m)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over d")[$frac(1, d)$]
]
]

Sometimes we need to use more than one property. In the next two examples, we will use both the Product to a Power Property and then the Power Property.

#examplebox("Example 14")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 27 u to the power the fraction 1 over 2 close parenthesis to the power the fraction 2 over 3")[$attach(( 27 u^(frac(1, 2)) ), t: frac(2, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 v to the power the fraction 1 over 4 close parenthesis to the power the fraction 2 over 3")[$attach(( 8 v^(frac(1, 4)) ), t: frac(2, 3))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 27 u to the power the fraction 1 over 2 close parenthesis to the power the fraction 2 over 3")[$attach(( 27 u^(frac(1, 2)) ), t: frac(2, 3))$]]),
  [First we use the Product to a Power Property.], [#math.equation(block: false, alt: "open parenthesis 27 close parenthesis to the power the fraction 2 over 3 open parenthesis u to the power the fraction 1 over 2 close parenthesis to the power the fraction 2 over 3")[$attach(( 27 ), t: frac(2, 3)) attach(( u^(frac(1, 2)) ), t: frac(2, 3))$]],
  [Rewrite 27 as a power of 3.], [#math.equation(block: false, alt: "open parenthesis 3 cubed close parenthesis to the power the fraction 2 over 3 open parenthesis u to the power the fraction 1 over 2 close parenthesis to the power the fraction 2 over 3")[$attach(( 3^(3) ), t: frac(2, 3)) attach(( u^(frac(1, 2)) ), t: frac(2, 3))$]],
  [To raise a power to a power, we multiply the exponents.], [#math.equation(block: false, alt: "open parenthesis 3 squared close parenthesis open parenthesis u to the power the fraction 1 over 3 close parenthesis")[$( 3^(2) ) ( u^(frac(1, 3)) )$]],
  [Simplify.], [#math.equation(block: false, alt: "9 u to the power the fraction 1 over 3")[$9 u^(frac(1, 3))$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 8 v to the power the fraction 1 over 4 close parenthesis to the power the fraction 2 over 3")[$attach(( 8 v^(frac(1, 4)) ), t: frac(2, 3))$]]),
  [First we use the Product to a Power Property.], [#math.equation(block: false, alt: "open parenthesis 8 close parenthesis to the power the fraction 2 over 3 open parenthesis v to the power the fraction 1 over 4 close parenthesis to the power the fraction 2 over 3")[$attach(( 8 ), t: frac(2, 3)) attach(( v^(frac(1, 4)) ), t: frac(2, 3))$]],
  [Rewrite 8 as a power of 2.], [#math.equation(block: false, alt: "open parenthesis 2 cubed close parenthesis to the power the fraction 2 over 3 open parenthesis v to the power the fraction 1 over 4 close parenthesis to the power the fraction 2 over 3")[$attach(( 2^(3) ), t: frac(2, 3)) attach(( v^(frac(1, 4)) ), t: frac(2, 3))$]],
  [To raise a power to a power, we multiply the exponents.], [#math.equation(block: false, alt: "open parenthesis 2 squared close parenthesis open parenthesis v to the power the fraction 1 over 6 close parenthesis")[$( 2^(2) ) ( v^(frac(1, 6)) )$]],
  [Simplify.], [#math.equation(block: false, alt: "4 v to the power the fraction 1 over 6")[$4 v^(frac(1, 6))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 32 x to the power the fraction 1 over 3 close parenthesis to the power the fraction 3 over 5")[$attach(( 32 x^(frac(1, 3)) ), t: frac(3, 5))$] ⓑ #math.equation(block: false, alt: "open parenthesis 64 y to the power the fraction 2 over 3 close parenthesis to the power the fraction 1 over 3")[$attach(( 64 y^(frac(2, 3)) ), t: frac(1, 3))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "8 x to the power the fraction 1 over 5")[$8 x^(frac(1, 5))$] ⓑ #math.equation(block: false, alt: "4 y to the power the fraction 2 over 9")[$4 y^(frac(2, 9))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 16 m to the power the fraction 1 over 3 close parenthesis to the power the fraction 3 over 2")[$attach(( 16 m^(frac(1, 3)) ), t: frac(3, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis 81 n to the power the fraction 2 over 5 close parenthesis to the power the fraction 3 over 2")[$attach(( 81 n^(frac(2, 5)) ), t: frac(3, 2))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "64 m to the power the fraction 1 over 2")[$64 m^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "729 n to the power the fraction 3 over 5")[$729 n^(frac(3, 5))$]
]
]

#examplebox("Example 15")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis m cubed n to the power 9 close parenthesis to the power the fraction 1 over 3")[$attach(( m^(3) n^(9) ), t: frac(1, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis p to the power 4 q to the power 8 close parenthesis to the power the fraction 1 over 4")[$attach(( p^(4) q^(8) ), t: frac(1, 4))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis m cubed n to the power 9 close parenthesis to the power the fraction 1 over 3")[$attach(( m^(3) n^(9) ), t: frac(1, 3))$]]),
  [First we use the Product to a Power Property.], [#math.equation(block: false, alt: "open parenthesis m cubed close parenthesis to the power the fraction 1 over 3 open parenthesis n to the power 9 close parenthesis to the power the fraction 1 over 3")[$attach(( m^(3) ), t: frac(1, 3)) attach(( n^(9) ), t: frac(1, 3))$]],
  [To raise a power to a power, we multiply the exponents.], [#math.equation(block: false, alt: "m n cubed")[$m n^(3)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis p to the power 4 q to the power 8 close parenthesis to the power the fraction 1 over 4")[$attach(( p^(4) q^(8) ), t: frac(1, 4))$]]),
  [First we use the Product to a Power Property.], [#math.equation(block: false, alt: "open parenthesis p to the power 4 close parenthesis to the power the fraction 1 over 4 open parenthesis q to the power 8 close parenthesis to the power the fraction 1 over 4")[$attach(( p^(4) ), t: frac(1, 4)) attach(( q^(8) ), t: frac(1, 4))$]],
  [To raise a power to a power, we multiply the exponents.], [#math.equation(block: false, alt: "p q squared")[$p q^(2)$]],
))
]
]

We will use both the Product and Quotient Properties in the next example.

#examplebox("Example 16")[][
Simplify: ⓐ #math.equation(block: false, alt: "the fraction x to the power the fraction 3 over 4 times x to the power minus the fraction 1 over 4 over x to the power minus the fraction 6 over 4")[$frac(x^(frac(3, 4)) · x^(− frac(1, 4)), x^(− frac(6, 4)))$] ⓑ #math.equation(block: false, alt: "the fraction y to the power the fraction 4 over 3 times y over y to the power minus the fraction 2 over 3")[$frac(y^(frac(4, 3)) · y, y^(− frac(2, 3)))$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction x to the power the fraction 3 over 4 times x to the power minus the fraction 1 over 4 over x to the power minus the fraction 6 over 4")[$frac(x^(frac(3, 4)) · x^(− frac(1, 4)), x^(− frac(6, 4)))$]]),
  [Use the Product Property in the numerator, add the exponents.], [#math.equation(block: false, alt: "the fraction x to the power the fraction 2 over 4 over x to the power minus the fraction 6 over 4")[$frac(x^(frac(2, 4)), x^(− frac(6, 4)))$]],
  [Use the Quotient Property, subtract the exponents.], [#math.equation(block: false, alt: "x to the power the fraction 8 over 4")[$x^(frac(8, 4))$]],
  [Simplify.], [#math.equation(block: false, alt: "x squared")[$x^(2)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction y to the power the fraction 4 over 3 times y over y to the power minus the fraction 2 over 3")[$frac(y^(frac(4, 3)) · y, y^(− frac(2, 3)))$]]),
  [Use the Product Property in the numerator, add the exponents.], [#math.equation(block: false, alt: "the fraction y to the power the fraction 7 over 3 over y to the power minus the fraction 2 over 3")[$frac(y^(frac(7, 3)), y^(− frac(2, 3)))$]],
  [Use the Quotient Property, subtract the exponents.], [#math.equation(block: false, alt: "y to the power the fraction 9 over 3")[$y^(frac(9, 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "y cubed")[$y^(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the fraction m to the power the fraction 2 over 3 times m to the power minus the fraction 1 over 3 over m to the power minus the fraction 5 over 3")[$frac(m^(frac(2, 3)) · m^(− frac(1, 3)), m^(− frac(5, 3)))$] ⓑ #math.equation(block: false, alt: "the fraction n to the power the fraction 1 over 6 times n over n to the power minus the fraction 11 over 6")[$frac(n^(frac(1, 6)) · n, n^(− frac(11, 6)))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "m squared")[$m^(2)$] ⓑ #math.equation(block: false, alt: "n cubed")[$n^(3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the fraction u to the power the fraction 4 over 5 times u to the power minus the fraction 2 over 5 over u to the power minus the fraction 13 over 5")[$frac(u^(frac(4, 5)) · u^(− frac(2, 5)), u^(− frac(13, 5)))$] ⓑ #math.equation(block: false, alt: "the fraction v to the power the fraction 1 over 2 times v over v to the power minus the fraction 7 over 2")[$frac(v^(frac(1, 2)) · v, v^(− frac(7, 2)))$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "u cubed")[$u^(3)$] ⓑ #math.equation(block: false, alt: "v to the power 5")[$v^(5)$]
]
]


=== Key Concepts

- #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 rational numbers, then
- #strong[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")[$a^(m) · a^(n) = a^(m + n)$]
- #strong[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")[$attach(( a^(m) ), t: n) = a^(m · n)$]
- #strong[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")[$attach(( a b ), t: m) = a^(m) b^(m)$]
- #strong[Quotient Property]: #linebreak() 
#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 , a not equal to 0 , m greater than n")[$frac(a^(m), a^(n)) = a^(m − n) , #h(0.2em) #h(0.2em) a ≠ 0 , #h(0.2em) #h(0.2em) m > n$]#math.equation(block: true, 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 , a not equal to 0 , n greater than m")[$frac(a^(m), a^(n)) = frac(1, a^(n − m)) , #h(0.2em) #h(0.2em) a ≠ 0 , #h(0.2em) #h(0.2em) n > m$]
- #strong[Zero Exponent Definition] #math.equation(block: false, alt: "a to the power 0 equals 1")[$a^(0) = 1$], #math.equation(block: false, alt: "a not equal to 0")[$a ≠ 0$]
- #strong[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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m)) , #h(0.5em) b ≠ 0$]


=== Section Exercises


==== Practice Makes Perfect

#strong[Simplify Expressions with #math.equation(block: false, alt: "a to the power the fraction 1 over n")[$a^(frac(1, n))$]]

In the following exercises, write as a radical expression.

ⓐ #math.equation(block: false, alt: "x to the power the fraction 1 over 2")[$x^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "y to the power the fraction 1 over 3")[$y^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "z to the power the fraction 1 over 4")[$z^(frac(1, 4))$]

ⓐ #math.equation(block: false, alt: "r to the power the fraction 1 over 2")[$r^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "s to the power the fraction 1 over 3")[$s^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "t to the power the fraction 1 over 4")[$t^(frac(1, 4))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the square root of r")[$sqrt(r)$] ⓑ #math.equation(block: false, alt: "the cube root of s")[$root(3, s)$] ⓒ #math.equation(block: false, alt: "the 4th root of t")[$root(4, t)$]
]

ⓐ #math.equation(block: false, alt: "u to the power the fraction 1 over 5")[$u^(frac(1, 5))$] ⓑ #math.equation(block: false, alt: "v to the power the fraction 1 over 9")[$v^(frac(1, 9))$] ⓒ #math.equation(block: false, alt: "w to the power the fraction 1 over 20")[$w^(frac(1, 20))$]

ⓐ #math.equation(block: false, alt: "g to the power the fraction 1 over 7")[$g^(frac(1, 7))$] ⓑ #math.equation(block: false, alt: "h to the power the fraction 1 over 5")[$h^(frac(1, 5))$] ⓒ #math.equation(block: false, alt: "j to the power the fraction 1 over 25")[$j^(frac(1, 25))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the 7th root of g")[$root(7, g)$] ⓑ #math.equation(block: false, alt: "the 5th root of h")[$root(5, h)$] ⓒ #math.equation(block: false, alt: "the 25th root of j")[$root(25, j)$]
]

In the following exercises, write with a rational exponent.

ⓐ #math.equation(block: false, alt: "minus the 7th root of x")[$− root(7, x)$] ⓑ #math.equation(block: false, alt: "the 9th root of y")[$root(9, y)$] ⓒ #math.equation(block: false, alt: "the 5th root of f")[$root(5, f)$]

ⓐ #math.equation(block: false, alt: "the 8th root of r")[$root(8, r)$] ⓑ #figure(figph[A mathematical radical symbol is displayed, with '19' as the index above the left arm of the root and '5' as the radicand inside, indicating the 19th root of 5.], alt: "A mathematical radical symbol is displayed, with '19' as the index above the left arm of the root and '5' as the radicand inside, indicating the 19th root of 5.", caption: none)

 ⓒ #math.equation(block: false, alt: "the 4th root of t")[$root(4, t)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "r to the power the fraction 1 over 8")[$r^(frac(1, 8))$] ⓑ #math.equation(block: false, alt: "s to the power the fraction 1 over 10")[$s^(frac(1, 10))$] ⓒ #math.equation(block: false, alt: "t to the power the fraction 1 over 4")[$t^(frac(1, 4))$]
]

ⓐ #math.equation(block: false, alt: "the cube root of a")[$root(3, a)$] ⓑ #figure(figph[The mathematical expression for the twelfth root of 'b'.], alt: "The mathematical expression for the twelfth root of 'b'.", caption: none)

 ⓒ #math.equation(block: false, alt: "the square root of c")[$sqrt(c)$]

ⓐ #math.equation(block: false, alt: "the 5th root of u")[$root(5, u)$] ⓑ #math.equation(block: false, alt: "the square root of v")[$sqrt(v)$] ⓒ #figure(figph[A close-up view of the mathematical expression '16th root of W', stylized as a radical sign with '16' as the index and 'W' as the radicand.], alt: "A close-up view of the mathematical expression '16th root of W', stylized as a radical sign with '16' as the index and 'W' as the radicand.", caption: none)

#solutionbox[
ⓐ #math.equation(block: false, alt: "u to the power the fraction 1 over 5")[$u^(frac(1, 5))$] ⓑ #math.equation(block: false, alt: "v to the power the fraction 1 over 2")[$v^(frac(1, 2))$] ⓒ #math.equation(block: false, alt: "w to the power the fraction 1 over 16")[$w^(frac(1, 16))$]
]

ⓐ #math.equation(block: false, alt: "the cube root of 7 c")[$root(3, 7 c)$] ⓑ #math.equation(block: false, alt: "the 7th root of 12 d")[$root(7, 12 d)$] ⓒ #math.equation(block: false, alt: "3 the 4th root of 5 f")[$3 root(4, 5 f)$]

ⓐ #math.equation(block: false, alt: "the 4th root of 5 x")[$root(4, 5 x)$] ⓑ #math.equation(block: false, alt: "the 8th root of 9 y")[$root(8, 9 y)$] ⓒ #math.equation(block: false, alt: "7 the 5th root of 3 z")[$7 root(5, 3 z)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis 5 x close parenthesis to the power the fraction 1 over 4")[$attach(( 5 x ), t: frac(1, 4))$] ⓑ #math.equation(block: false, alt: "open parenthesis 9 y close parenthesis to the power the fraction 1 over 8")[$attach(( 9 y ), t: frac(1, 8))$] ⓒ #math.equation(block: false, alt: "7 open parenthesis 3 z close parenthesis to the power the fraction 1 over 5")[$7 attach(( 3 z ), t: frac(1, 5))$]
]

ⓐ #math.equation(block: false, alt: "the square root of 21 p")[$sqrt(21 p)$] ⓑ #math.equation(block: false, alt: "the 4th root of 8 q")[$root(4, 8 q)$] ⓒ #math.equation(block: false, alt: "4 the 6th root of 36 r")[$4 root(6, 36 r)$]

ⓐ #math.equation(block: false, alt: "the cube root of 25 a")[$root(3, 25 a)$] ⓑ #math.equation(block: false, alt: "the square root of 3 b")[$sqrt(3 b)$] ⓒ #figure(figph[A mathematical expression showing the tenth root of 40c, represented as an n-th root symbol with 10 as the index and 40c as the radicand.], alt: "A mathematical expression showing the tenth root of 40c, represented as an n-th root symbol with 10 as the index and 40c as the radicand.", caption: none)

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis 25 a close parenthesis to the power the fraction 1 over 3")[$attach(( 25 a ), t: frac(1, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 b close parenthesis to the power the fraction 1 over 2")[$attach(( 3 b ), t: frac(1, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis 40 c close parenthesis to the power the fraction 1 over 10")[$attach(( 40 c ), t: frac(1, 10))$]
]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "81 to the power the fraction 1 over 2")[$81^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "125 to the power the fraction 1 over 3")[$125^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "64 to the power the fraction 1 over 2")[$64^(frac(1, 2))$]

ⓐ #math.equation(block: false, alt: "625 to the power the fraction 1 over 4")[$625^(frac(1, 4))$] ⓑ #math.equation(block: false, alt: "243 to the power the fraction 1 over 5")[$243^(frac(1, 5))$] ⓒ #math.equation(block: false, alt: "32 to the power the fraction 1 over 5")[$32^(frac(1, 5))$]

#solutionbox[
ⓐ 5 ⓑ 3 ⓒ 2
]

ⓐ #math.equation(block: false, alt: "16 to the power the fraction 1 over 4")[$16^(frac(1, 4))$] ⓑ #math.equation(block: false, alt: "16 to the power the fraction 1 over 2")[$16^(frac(1, 2))$] ⓒ #math.equation(block: false, alt: "3125 to the power the fraction 1 over 5")[$3125^(frac(1, 5))$]

ⓐ #math.equation(block: false, alt: "216 to the power the fraction 1 over 3")[$216^(frac(1, 3))$] ⓑ #math.equation(block: false, alt: "32 to the power the fraction 1 over 5")[$32^(frac(1, 5))$] ⓒ #math.equation(block: false, alt: "81 to the power the fraction 1 over 4")[$81^(frac(1, 4))$]

#solutionbox[
ⓐ 6 ⓑ 2 ⓒ 3
]

ⓐ #math.equation(block: false, alt: "open parenthesis −216 close parenthesis to the power the fraction 1 over 3")[$attach(( −216 ), t: frac(1, 3))$] ⓑ #math.equation(block: false, alt: "− 216 to the power the fraction 1 over 3")[$"−" 216^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "open parenthesis 216 close parenthesis to the power minus the fraction 1 over 3")[$attach(( 216 ), t: − frac(1, 3))$]

ⓐ #math.equation(block: false, alt: "open parenthesis −243 close parenthesis to the power the fraction 1 over 5")[$attach(( −243 ), t: frac(1, 5))$] ⓑ #math.equation(block: false, alt: "− 243 to the power the fraction 1 over 5")[$"−" 243^(frac(1, 5))$] ⓒ #math.equation(block: false, alt: "open parenthesis 243 close parenthesis to the power minus the fraction 1 over 5")[$attach(( 243 ), t: − frac(1, 5))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−3")[$−3$] ⓑ #math.equation(block: false, alt: "−3")[$−3$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis −1 close parenthesis to the power the fraction 1 over 3")[$attach(( −1 ), t: frac(1, 3))$] ⓑ #math.equation(block: false, alt: "−1 to the power the fraction 1 over 3")[$−1^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "open parenthesis 1 close parenthesis to the power minus the fraction 1 over 3")[$attach(( 1 ), t: − frac(1, 3))$]

ⓐ #math.equation(block: false, alt: "open parenthesis −1000 close parenthesis to the power the fraction 1 over 3")[$attach(( −1000 ), t: frac(1, 3))$] ⓑ #math.equation(block: false, alt: "− 1000 to the power the fraction 1 over 3")[$"−" 1000^(frac(1, 3))$] ⓒ #math.equation(block: false, alt: "open parenthesis 1000 close parenthesis to the power minus the fraction 1 over 3")[$attach(( 1000 ), t: − frac(1, 3))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−10")[$−10$] ⓑ #math.equation(block: false, alt: "−10")[$−10$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 10")[$frac(1, 10)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis −81 close parenthesis to the power the fraction 1 over 4")[$attach(( −81 ), t: frac(1, 4))$] ⓑ #math.equation(block: false, alt: "− 81 to the power the fraction 1 over 4")[$"−" 81^(frac(1, 4))$] ⓒ #math.equation(block: false, alt: "open parenthesis 81 close parenthesis to the power minus the fraction 1 over 4")[$attach(( 81 ), t: − frac(1, 4))$]

ⓐ #math.equation(block: false, alt: "open parenthesis −49 close parenthesis to the power the fraction 1 over 2")[$attach(( −49 ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "− 49 to the power the fraction 1 over 2")[$"−" 49^(frac(1, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis 49 close parenthesis to the power minus the fraction 1 over 2")[$attach(( 49 ), t: − frac(1, 2))$]

#solutionbox[
ⓐ not a real number ⓑ #math.equation(block: false, alt: "−7")[$−7$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 7")[$frac(1, 7)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis −36 close parenthesis to the power the fraction 1 over 2")[$attach(( −36 ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "minus 36 to the power the fraction 1 over 2")[$− 36^(frac(1, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis 36 close parenthesis to the power minus the fraction 1 over 2")[$attach(( 36 ), t: − frac(1, 2))$]

ⓐ #math.equation(block: false, alt: "open parenthesis −1 close parenthesis to the power the fraction 1 over 4")[$attach(( −1 ), t: frac(1, 4))$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 close parenthesis to the power minus the fraction 1 over 4")[$attach(( 1 ), t: − frac(1, 4))$] ⓒ #math.equation(block: false, alt: "− 1 to the power the fraction 1 over 4")[$"−" 1^(frac(1, 4))$]

#solutionbox[
ⓐ not a real number ⓑ #math.equation(block: false, alt: "1")[$1$] ⓒ #math.equation(block: false, alt: "−1")[$−1$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis −100 close parenthesis to the power the fraction 1 over 2")[$attach(( −100 ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "− 100 to the power the fraction 1 over 2")[$"−" 100^(frac(1, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis 100 close parenthesis to the power minus the fraction 1 over 2")[$attach(( 100 ), t: − frac(1, 2))$]

ⓐ #math.equation(block: false, alt: "open parenthesis −32 close parenthesis to the power the fraction 1 over 5")[$attach(( −32 ), t: frac(1, 5))$] ⓑ #math.equation(block: false, alt: "open parenthesis 243 close parenthesis to the power minus the fraction 1 over 5")[$attach(( 243 ), t: − frac(1, 5))$] ⓒ #math.equation(block: false, alt: "− 125 to the power the fraction 1 over 3")[$"−" 125^(frac(1, 3))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−2")[$−2$] ⓑ #math.equation(block: false, alt: "the fraction 1 over 3")[$frac(1, 3)$] #linebreak() ⓒ #math.equation(block: false, alt: "−5")[$−5$]
]

#strong[Simplify Expressions with #math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$]]

In the following exercises, write with a rational exponent.

ⓐ #math.equation(block: false, alt: "the square root of m to the power 5")[$sqrt(m^(5))$] ⓑ #math.equation(block: false, alt: "the cube root of n squared")[$root(3, n^(2))$] ⓒ #math.equation(block: false, alt: "the 4th root of p cubed")[$root(4, p^(3))$]

ⓐ #math.equation(block: false, alt: "the 4th root of r to the power 7")[$root(4, r^(7))$] ⓑ #math.equation(block: false, alt: "the 5th root of s cubed")[$root(5, s^(3))$] ⓒ #math.equation(block: false, alt: "the cube root of t to the power 7")[$root(3, t^(7))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "r to the power the fraction 7 over 4")[$r^(frac(7, 4))$] ⓑ #math.equation(block: false, alt: "s to the power the fraction 3 over 5")[$s^(frac(3, 5))$] ⓒ #math.equation(block: false, alt: "t to the power the fraction 7 over 3")[$t^(frac(7, 3))$]
]

ⓐ #math.equation(block: false, alt: "the 5th root of u squared")[$root(5, u^(2))$] ⓑ #math.equation(block: false, alt: "the 5th root of v to the power 8")[$root(5, v^(8))$] ⓒ #math.equation(block: false, alt: "the 9th root of w to the power 4")[$root(9, w^(4))$]

ⓐ #math.equation(block: false, alt: "the cube root of a")[$root(3, a)$] ⓑ #math.equation(block: false, alt: "the square root of b to the power 5")[$sqrt(b^(5))$] ⓒ #math.equation(block: false, alt: "the cube root of c to the power 5")[$root(3, c^(5))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "a to the power the fraction 1 over 3")[$a^(frac(1, 3))$] ⓑ #math.equation(block: false, alt: "b to the power the fraction 5 over 2")[$b^(frac(5, 2))$] ⓒ #math.equation(block: false, alt: "c to the power the fraction 5 over 3")[$c^(frac(5, 3))$]
]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "16 to the power the fraction 3 over 2")[$16^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "8 to the power the fraction 2 over 3")[$8^(frac(2, 3))$] ⓒ #math.equation(block: false, alt: "10,000 to the power the fraction 3 over 4")[$10,000^(frac(3, 4))$]

ⓐ #math.equation(block: false, alt: "1000 to the power the fraction 2 over 3")[$1000^(frac(2, 3))$] ⓑ #math.equation(block: false, alt: "25 to the power the fraction 3 over 2")[$25^(frac(3, 2))$] ⓒ #math.equation(block: false, alt: "32 to the power the fraction 3 over 5")[$32^(frac(3, 5))$]

#solutionbox[
ⓐ 100 ⓑ 125 ⓒ 8
]

ⓐ #math.equation(block: false, alt: "27 to the power the fraction 5 over 3")[$27^(frac(5, 3))$] ⓑ #math.equation(block: false, alt: "16 to the power the fraction 5 over 4")[$16^(frac(5, 4))$] ⓒ #math.equation(block: false, alt: "32 to the power the fraction 2 over 5")[$32^(frac(2, 5))$]

ⓐ #math.equation(block: false, alt: "16 to the power the fraction 3 over 2")[$16^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "125 to the power the fraction 5 over 3")[$125^(frac(5, 3))$] ⓒ #math.equation(block: false, alt: "64 to the power the fraction 4 over 3")[$64^(frac(4, 3))$]

#solutionbox[
ⓐ 64 ⓑ 3125 ⓒ 256
]

ⓐ #math.equation(block: false, alt: "32 to the power the fraction 2 over 5")[$32^(frac(2, 5))$] ⓑ #math.equation(block: false, alt: "27 to the power minus the fraction 2 over 3")[$27^(− frac(2, 3))$] ⓒ #math.equation(block: false, alt: "25 to the power minus the fraction 3 over 2")[$25^(− frac(3, 2))$]

ⓐ #math.equation(block: false, alt: "64 to the power the fraction 5 over 2")[$64^(frac(5, 2))$] ⓑ #math.equation(block: false, alt: "81 to the power minus the fraction 3 over 2")[$81^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "27 to the power minus the fraction 4 over 3")[$27^(− frac(4, 3))$]

#solutionbox[
ⓐ 32,768 ⓑ #math.equation(block: false, alt: "the fraction 1 over 729")[$frac(1, 729)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 81")[$frac(1, 81)$]
]

ⓐ #math.equation(block: false, alt: "25 to the power the fraction 3 over 2")[$25^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "9 to the power minus the fraction 3 over 2")[$9^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −64 close parenthesis to the power the fraction 2 over 3")[$attach(( −64 ), t: frac(2, 3))$]

ⓐ #math.equation(block: false, alt: "100 to the power the fraction 3 over 2")[$100^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "49 to the power minus the fraction 5 over 2")[$49^(− frac(5, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −100 close parenthesis to the power the fraction 3 over 2")[$attach(( −100 ), t: frac(3, 2))$]

#solutionbox[
ⓐ 1000 ⓑ #math.equation(block: false, alt: "the fraction 1 over 16,807")[$frac(1, 16","807)$] ⓒ not a real number
]

ⓐ #math.equation(block: false, alt: "− 9 to the power the fraction 3 over 2")[$"−" 9^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "− 9 to the power minus the fraction 3 over 2")[$"−" 9^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −9 close parenthesis to the power the fraction 3 over 2")[$attach(( −9 ), t: frac(3, 2))$]

ⓐ #math.equation(block: false, alt: "− 64 to the power the fraction 3 over 2")[$"−" 64^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "− 64 to the power minus the fraction 3 over 2")[$"−" 64^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −64 close parenthesis to the power the fraction 3 over 2")[$attach(( −64 ), t: frac(3, 2))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−512")[$−512$] #linebreak() ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 512")[$− frac(1, 512)$] ⓒ not a real number
]

ⓐ #math.equation(block: false, alt: "− 100 to the power the fraction 3 over 2")[$"−" 100^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "− 100 to the power minus the fraction 3 over 2")[$"−" 100^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −100 close parenthesis to the power the fraction 3 over 2")[$attach(( −100 ), t: frac(3, 2))$]

ⓐ #math.equation(block: false, alt: "− 49 to the power the fraction 3 over 2")[$"−" 49^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "− 49 to the power minus the fraction 3 over 2")[$"−" 49^(− frac(3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −49 close parenthesis to the power the fraction 3 over 2")[$attach(( −49 ), t: frac(3, 2))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−343")[$−343$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 343")[$− frac(1, 343)$] ⓒ not a real number
]

#strong[Use the Laws of Exponents to Simplify Expressions with Rational Exponents]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "4 to the power the fraction 5 over 8 times 4 to the power the fraction 11 over 8")[$4^(frac(5, 8)) · 4^(frac(11, 8))$] ⓑ #math.equation(block: false, alt: "m to the power the fraction 7 over 12 times m to the power the fraction 17 over 12")[$m^(frac(7, 12)) · m^(frac(17, 12))$] ⓒ #math.equation(block: false, alt: "p to the power the fraction 3 over 7 times p to the power the fraction 18 over 7")[$p^(frac(3, 7)) · p^(frac(18, 7))$]

ⓐ #math.equation(block: false, alt: "6 to the power the fraction 5 over 2 times 6 to the power the fraction 1 over 2")[$6^(frac(5, 2)) · 6^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "n to the power the fraction 2 over 10 times n to the power the fraction 8 over 10")[$n^(frac(2, 10)) · n^(frac(8, 10))$] ⓒ #math.equation(block: false, alt: "q to the power the fraction 2 over 5 times q to the power the fraction 13 over 5")[$q^(frac(2, 5)) · q^(frac(13, 5))$]

#solutionbox[
ⓐ 216 ⓑ #math.equation(block: false, alt: "n")[$n$] ⓒ #math.equation(block: false, alt: "q cubed")[$q^(3)$]
]

ⓐ #math.equation(block: false, alt: "5 to the power the fraction 1 over 2 times 5 to the power the fraction 7 over 2")[$5^(frac(1, 2)) · 5^(frac(7, 2))$] ⓑ #math.equation(block: false, alt: "c to the power the fraction 3 over 4 times c to the power the fraction 9 over 4")[$c^(frac(3, 4)) · c^(frac(9, 4))$] ⓒ #math.equation(block: false, alt: "d to the power the fraction 3 over 5 times d to the power the fraction 2 over 5")[$d^(frac(3, 5)) · d^(frac(2, 5))$]

ⓐ #math.equation(block: false, alt: "10 to the power the fraction 1 over 3 times 10 to the power the fraction 5 over 3")[$10^(frac(1, 3)) · 10^(frac(5, 3))$] ⓑ #math.equation(block: false, alt: "x to the power the fraction 5 over 6 times x to the power the fraction 7 over 6")[$x^(frac(5, 6)) · x^(frac(7, 6))$] ⓒ #math.equation(block: false, alt: "y to the power the fraction 11 over 8 times y to the power the fraction 21 over 8")[$y^(frac(11, 8)) · y^(frac(21, 8))$]

#solutionbox[
ⓐ 100 ⓑ #math.equation(block: false, alt: "x squared")[$x^(2)$] ⓒ #math.equation(block: false, alt: "y to the power 4")[$y^(4)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis m to the power 6 close parenthesis to the power the fraction 5 over 2")[$attach(( m^(6) ), t: frac(5, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis n to the power 9 close parenthesis to the power the fraction 4 over 3")[$attach(( n^(9) ), t: frac(4, 3))$] ⓒ #math.equation(block: false, alt: "open parenthesis p to the power 12 close parenthesis to the power the fraction 3 over 4")[$attach(( p^(12) ), t: frac(3, 4))$]

ⓐ #math.equation(block: false, alt: "open parenthesis a to the power 12 close parenthesis to the power the fraction 1 over 6")[$attach(( a^(12) ), t: frac(1, 6))$] ⓑ #math.equation(block: false, alt: "open parenthesis b to the power 15 close parenthesis to the power the fraction 3 over 5")[$attach(( b^(15) ), t: frac(3, 5))$] ⓒ #math.equation(block: false, alt: "open parenthesis c to the power 11 close parenthesis to the power the fraction 1 over 11")[$attach(( c^(11) ), t: frac(1, 11))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "a squared")[$a^(2)$] ⓑ #math.equation(block: false, alt: "b to the power 9")[$b^(9)$] #linebreak() ⓒ #math.equation(block: false, alt: "c")[$c$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis x to the power 12 close parenthesis to the power the fraction 2 over 3")[$attach(( x^(12) ), t: frac(2, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis y to the power 20 close parenthesis to the power the fraction 2 over 5")[$attach(( y^(20) ), t: frac(2, 5))$] ⓒ #math.equation(block: false, alt: "open parenthesis z to the power 16 close parenthesis to the power the fraction 1 over 16")[$attach(( z^(16) ), t: frac(1, 16))$]

ⓐ #math.equation(block: false, alt: "open parenthesis h to the power 6 close parenthesis to the power the fraction 4 over 3")[$attach(( h^(6) ), t: frac(4, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis k to the power 12 close parenthesis to the power the fraction 3 over 4")[$attach(( k^(12) ), t: frac(3, 4))$] ⓒ #math.equation(block: false, alt: "open parenthesis j to the power 10 close parenthesis to the power the fraction 7 over 5")[$attach(( j^(10) ), t: frac(7, 5))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "h to the power 8")[$h^(8)$] ⓑ #math.equation(block: false, alt: "k to the power 9")[$k^(9)$] ⓒ #math.equation(block: false, alt: "j to the power 14")[$j^(14)$]
]

ⓐ #math.equation(block: false, alt: "the fraction x to the power the fraction 7 over 2 over x to the power the fraction 5 over 2")[$frac(x^(frac(7, 2)), x^(frac(5, 2)))$] ⓑ #math.equation(block: false, alt: "the fraction y to the power the fraction 5 over 2 over y to the power the fraction 1 over 2")[$frac(y^(frac(5, 2)), y^(frac(1, 2)))$] ⓒ #math.equation(block: false, alt: "the fraction r to the power the fraction 4 over 5 over r to the power the fraction 9 over 5")[$frac(r^(frac(4, 5)), r^(frac(9, 5)))$]

ⓐ #math.equation(block: false, alt: "the fraction s to the power the fraction 11 over 5 over s to the power the fraction 6 over 5")[$frac(s^(frac(11, 5)), s^(frac(6, 5)))$] ⓑ #math.equation(block: false, alt: "the fraction z to the power the fraction 7 over 3 over z to the power the fraction 1 over 3")[$frac(z^(frac(7, 3)), z^(frac(1, 3)))$] ⓒ #math.equation(block: false, alt: "the fraction w to the power the fraction 2 over 7 over w to the power the fraction 9 over 7")[$frac(w^(frac(2, 7)), w^(frac(9, 7)))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "s")[$s$] ⓑ #math.equation(block: false, alt: "z squared")[$z^(2)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over w")[$frac(1, w)$]
]

ⓐ #math.equation(block: false, alt: "the fraction t to the power the fraction 12 over 5 over t to the power the fraction 7 over 5")[$frac(t^(frac(12, 5)), t^(frac(7, 5)))$] ⓑ #math.equation(block: false, alt: "the fraction x to the power the fraction 3 over 2 over x to the power the fraction 1 over 2")[$frac(x^(frac(3, 2)), x^(frac(1, 2)))$]ⓒ #math.equation(block: false, alt: "the fraction m to the power the fraction 13 over 8 over m to the power the fraction 5 over 8")[$frac(m^(frac(13, 8)), m^(frac(5, 8)))$]

ⓐ #math.equation(block: false, alt: "the fraction u to the power the fraction 13 over 9 over u to the power the fraction 4 over 9")[$frac(u^(frac(13, 9)), u^(frac(4, 9)))$] ⓑ #math.equation(block: false, alt: "the fraction r to the power the fraction 15 over 7 over r to the power the fraction 8 over 7")[$frac(r^(frac(15, 7)), r^(frac(8, 7)))$] ⓒ #math.equation(block: false, alt: "the fraction n to the power the fraction 3 over 5 over n to the power the fraction 8 over 5")[$frac(n^(frac(3, 5)), n^(frac(8, 5)))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "u")[$u$] ⓑ #math.equation(block: false, alt: "r")[$r$] ⓒ #math.equation(block: false, alt: "the fraction 1 over n")[$frac(1, n)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis 9 p to the power the fraction 2 over 3 close parenthesis to the power the fraction 5 over 2")[$attach(( 9 p^(frac(2, 3)) ), t: frac(5, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis 27 q to the power the fraction 3 over 2 close parenthesis to the power the fraction 4 over 3")[$attach(( 27 q^(frac(3, 2)) ), t: frac(4, 3))$]

ⓐ #math.equation(block: false, alt: "open parenthesis 81 r to the power the fraction 4 over 5 close parenthesis to the power the fraction 1 over 4")[$attach(( 81 r^(frac(4, 5)) ), t: frac(1, 4))$] ⓑ #math.equation(block: false, alt: "open parenthesis 64 s to the power the fraction 3 over 7 close parenthesis to the power the fraction 1 over 6")[$attach(( 64 s^(frac(3, 7)) ), t: frac(1, 6))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "3 r to the power the fraction 1 over 5")[$3 r^(frac(1, 5))$] ⓑ #math.equation(block: false, alt: "2 s to the power the fraction 1 over 14")[$2 s^(frac(1, 14))$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis 16 u to the power the fraction 1 over 3 close parenthesis to the power the fraction 3 over 4")[$attach(( 16 u^(frac(1, 3)) ), t: frac(3, 4))$] ⓑ #math.equation(block: false, alt: "open parenthesis 100 v to the power the fraction 2 over 5 close parenthesis to the power the fraction 3 over 2")[$attach(( 100 v^(frac(2, 5)) ), t: frac(3, 2))$]

ⓐ #math.equation(block: false, alt: "open parenthesis 27 m to the power the fraction 3 over 4 close parenthesis to the power the fraction 2 over 3")[$attach(( 27 m^(frac(3, 4)) ), t: frac(2, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis 625 n to the power the fraction 8 over 3 close parenthesis to the power the fraction 3 over 4")[$attach(( 625 n^(frac(8, 3)) ), t: frac(3, 4))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "9 m to the power the fraction 1 over 2")[$9 m^(frac(1, 2))$] ⓑ #math.equation(block: false, alt: "125 n squared")[$125 n^(2)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis x to the power 8 y to the power 10 close parenthesis to the power the fraction 1 over 2")[$attach(( x^(8) y^(10) ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis a to the power 9 b to the power 12 close parenthesis to the power the fraction 1 over 3")[$attach(( a^(9) b^(12) ), t: frac(1, 3))$]

ⓐ #math.equation(block: false, alt: "open parenthesis r to the power 8 s to the power 4 close parenthesis to the power the fraction 1 over 4")[$attach(( r^(8) s^(4) ), t: frac(1, 4))$] ⓑ #math.equation(block: false, alt: "open parenthesis u to the power 15 v to the power 20 close parenthesis to the power the fraction 1 over 5")[$attach(( u^(15) v^(20) ), t: frac(1, 5))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "r squared s")[$r^(2) s$] ⓑ #math.equation(block: false, alt: "u cubed v to the power 4")[$u^(3) v^(4)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis a to the power 6 b to the power 16 close parenthesis to the power the fraction 1 over 2")[$attach(( a^(6) b^(16) ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis j to the power 9 k to the power 6 close parenthesis to the power the fraction 2 over 3")[$attach(( j^(9) k^(6) ), t: frac(2, 3))$]

ⓐ #math.equation(block: false, alt: "open parenthesis r to the power 16 s to the power 10 close parenthesis to the power the fraction 1 over 2")[$attach(( r^(16) s^(10) ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis u to the power 10 v to the power 5 close parenthesis to the power the fraction 4 over 5")[$attach(( u^(10) v^(5) ), t: frac(4, 5))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "r to the power 8 s to the power 5")[$r^(8) s^(5)$] ⓑ #math.equation(block: false, alt: "u to the power 8 v to the power 4")[$u^(8) v^(4)$]
]

ⓐ #math.equation(block: false, alt: "the fraction r to the power the fraction 5 over 2 times r to the power minus the fraction 1 over 2 over r to the power minus the fraction 3 over 2")[$frac(r^(frac(5, 2)) · r^(− frac(1, 2)), r^(− frac(3, 2)))$] ⓑ #math.equation(block: false, alt: "the fraction s to the power the fraction 1 over 5 times s over s to the power minus the fraction 9 over 5")[$frac(s^(frac(1, 5)) · s, s^(− frac(9, 5)))$]

ⓐ #math.equation(block: false, alt: "the fraction a to the power the fraction 3 over 4 times a to the power minus the fraction 1 over 4 over a to the power minus the fraction 10 over 4")[$frac(a^(frac(3, 4)) · a^(− frac(1, 4)), a^(− frac(10, 4)))$] ⓑ #math.equation(block: false, alt: "the fraction b to the power the fraction 2 over 3 times b over b to the power minus the fraction 7 over 3")[$frac(b^(frac(2, 3)) · b, b^(− frac(7, 3)))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "a cubed")[$a^(3)$] ⓑ #math.equation(block: false, alt: "b to the power 4")[$b^(4)$]
]

ⓐ #math.equation(block: false, alt: "the fraction c to the power the fraction 5 over 3 times c to the power minus the fraction 1 over 3 over c to the power minus the fraction 2 over 3")[$frac(c^(frac(5, 3)) · c^(− frac(1, 3)), c^(− frac(2, 3)))$] ⓑ #math.equation(block: false, alt: "the fraction d to the power the fraction 3 over 5 times d over d to the power minus the fraction 2 over 5")[$frac(d^(frac(3, 5)) · d, d^(− frac(2, 5)))$]

ⓐ #math.equation(block: false, alt: "the fraction m to the power the fraction 7 over 4 times m to the power minus the fraction 5 over 4 over m to the power minus the fraction 2 over 4")[$frac(m^(frac(7, 4)) · m^(− frac(5, 4)), m^(− frac(2, 4)))$] ⓑ #math.equation(block: false, alt: "the fraction n to the power the fraction 3 over 7 times n over n to the power minus the fraction 4 over 7")[$frac(n^(frac(3, 7)) · n, n^(− frac(4, 7)))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "m")[$m$] ⓑ #math.equation(block: false, alt: "n squared")[$n^(2)$]
]

#math.equation(block: true, alt: "4 to the power the fraction 5 over 2 times 4 to the power the fraction 1 over 2")[$4^(frac(5, 2)) · 4^(frac(1, 2))$]

#math.equation(block: true, alt: "n to the power the fraction 2 over 6 times n to the power the fraction 4 over 6")[$n^(frac(2, 6)) · n^(frac(4, 6))$]

#solutionbox[
#math.equation(block: true, alt: "n")[$n$]
]

#math.equation(block: true, alt: "open parenthesis a to the power 24 close parenthesis to the power the fraction 1 over 6")[$attach(( a^(24) ), t: frac(1, 6))$]

#math.equation(block: true, alt: "open parenthesis b to the power 10 close parenthesis to the power the fraction 3 over 5")[$attach(( b^(10) ), t: frac(3, 5))$]

#solutionbox[
#math.equation(block: true, alt: "b to the power 6")[$b^(6)$]
]

#math.equation(block: true, alt: "the fraction w to the power the fraction 2 over 5 over w to the power the fraction 7 over 5")[$frac(w^(frac(2, 5)), w^(frac(7, 5)))$]

#math.equation(block: true, alt: "the fraction z to the power the fraction 2 over 3 over z to the power the fraction 8 over 3")[$frac(z^(frac(2, 3)), z^(frac(8, 3)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over z squared")[$frac(1, z^(2))$]
]

#math.equation(block: true, alt: "open parenthesis 27 r to the power the fraction 3 over 5 close parenthesis to the power the fraction 1 over 3")[$attach(( 27 r^(frac(3, 5)) ), t: frac(1, 3))$]

#math.equation(block: true, alt: "open parenthesis 64 s to the power the fraction 3 over 5 close parenthesis to the power the fraction 1 over 6")[$attach(( 64 s^(frac(3, 5)) ), t: frac(1, 6))$]

#solutionbox[
#math.equation(block: true, alt: "2 s to the power the fraction 1 over 10")[$2 s^(frac(1, 10))$]
]

#math.equation(block: true, alt: "open parenthesis r to the power 9 s to the power 12 close parenthesis to the power the fraction 1 over 3")[$attach(( r^(9) s^(12) ), t: frac(1, 3))$]

#math.equation(block: true, alt: "open parenthesis u to the power 12 v to the power 18 close parenthesis to the power the fraction 1 over 6")[$attach(( u^(12) v^(18) ), t: frac(1, 6))$]

#solutionbox[
#math.equation(block: true, alt: "u squared v cubed")[$u^(2) v^(3)$]
]


==== Everyday Math

#strong[Landscaping] Joe wants to have a square garden plot in his backyard. He has enough compost to cover an area of 144 square feet. Simplify #math.equation(block: false, alt: "144 to the power the fraction 1 over 2")[$144^(frac(1, 2))$] to find the length of each side of his garden.

#strong[Landscaping] Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify #math.equation(block: false, alt: "242 to the power the fraction 1 over 2")[$242^(frac(1, 2))$] to find the length of each side of his patio.Round to the nearest tenth of a foot.

#solutionbox[
15.6 feet
]

#strong[Gravity] While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify #math.equation(block: false, alt: "the fraction 12 to the power the fraction 1 over 2 over 16 to the power the fraction 1 over 2")[$frac(12^(frac(1, 2)), 16^(frac(1, 2)))$] to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

#strong[Gravity] An airplane dropped a flare from a height of 1024 feet above a lake. Simplify #math.equation(block: false, alt: "the fraction 1024 to the power the fraction 1 over 2 over 16 to the power the fraction 1 over 2")[$frac(1024^(frac(1, 2)), 16^(frac(1, 2)))$] to find how many seconds it took for the flare to reach the water.

#solutionbox[
8 seconds
]


==== Writing Exercises

Show two different algebraic methods to simplify #math.equation(block: false, alt: "4 to the power the fraction 3 over 2 .")[$4^(frac(3, 2)) .$] Explain all your steps.

Explain why the expression #math.equation(block: false, alt: "open parenthesis −16 close parenthesis to the power the fraction 3 over 2")[$attach(( −16 ), t: frac(3, 2))$] cannot be evaluated.

#solutionbox[
Answers will vary.
]


=== Chapter 9 Review Exercises


==== Simplify and Use Square Roots

#strong[Simplify Expressions with Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 64")[$sqrt(64)$]

#math.equation(block: true, alt: "the square root of 144")[$sqrt(144)$]

#solutionbox[
12
]

#math.equation(block: true, alt: "minus the square root of 25")[$− sqrt(25)$]

#math.equation(block: true, alt: "minus the square root of 81")[$− sqrt(81)$]

#solutionbox[
#math.equation(block: true, alt: "−9")[$−9$]
]

#math.equation(block: true, alt: "the square root of −9")[$sqrt(−9)$]

#math.equation(block: true, alt: "the square root of −36")[$sqrt(−36)$]

#solutionbox[
not a real number
]

#math.equation(block: true, alt: "the square root of 64 plus the square root of 225")[$sqrt(64) + sqrt(225)$]

#math.equation(block: true, alt: "the square root of 64 plus 225")[$sqrt(64 + 225)$]

#solutionbox[
17
]

#strong[Estimate Square Roots]

In the following exercises, estimate each square root between two consecutive whole numbers.

#math.equation(block: true, alt: "the square root of 28")[$sqrt(28)$]

#math.equation(block: true, alt: "the square root of 155")[$sqrt(155)$]

#solutionbox[
#math.equation(block: true, alt: "12 less than the square root of 155 less than 13")[$12 < sqrt(155) < 13$]
]

#strong[Approximate Square Roots]

In the following exercises, approximate each square root and round to two decimal places.

#math.equation(block: true, alt: "the square root of 15")[$sqrt(15)$]

#math.equation(block: true, alt: "the square root of 57")[$sqrt(57)$]

#solutionbox[
7.55
]

#strong[Simplify Variable Expressions with Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of q squared")[$sqrt(q^(2))$]

#math.equation(block: true, alt: "the square root of 64 b squared")[$sqrt(64 b^(2))$]

#solutionbox[
#math.equation(block: true, alt: "8 b")[$8 b$]
]

#math.equation(block: true, alt: "− the square root of 121 a squared")[$"−" sqrt(121 a^(2))$]

#math.equation(block: true, alt: "the square root of 225 m squared n squared")[$sqrt(225 m^(2) n^(2))$]

#solutionbox[
#math.equation(block: true, alt: "15 m n")[$15 m n$]
]

#math.equation(block: true, alt: "− the square root of 100 q squared")[$"−" sqrt(100 q^(2))$]

#math.equation(block: true, alt: "the square root of 49 y squared")[$sqrt(49 y^(2))$]

#solutionbox[
#math.equation(block: true, alt: "7 y")[$7 y$]
]

#math.equation(block: true, alt: "the square root of 4 a squared b squared")[$sqrt(4 a^(2) b^(2))$]

#math.equation(block: true, alt: "the square root of 121 c squared d squared")[$sqrt(121 c^(2) d^(2))$]

#solutionbox[
#math.equation(block: true, alt: "11 c d")[$11 c d$]
]


==== Simplify Square Roots

#strong[Use the Product Property to Simplify Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 300")[$sqrt(300)$]

#math.equation(block: true, alt: "the square root of 98")[$sqrt(98)$]

#solutionbox[
#math.equation(block: true, alt: "7 the square root of 2")[$7 sqrt(2)$]
]

#math.equation(block: true, alt: "the square root of x to the power 13")[$sqrt(x^(13))$]

#math.equation(block: true, alt: "the square root of y to the power 19")[$sqrt(y^(19))$]

#solutionbox[
#math.equation(block: true, alt: "y to the power 9 the square root of y")[$y^(9) sqrt(y)$]
]

#math.equation(block: true, alt: "the square root of 16 m to the power 4")[$sqrt(16 m^(4))$]

#math.equation(block: true, alt: "the square root of 36 n to the power 13")[$sqrt(36 n^(13))$]

#solutionbox[
#math.equation(block: true, alt: "6 n to the power 6 the square root of n")[$6 n^(6) sqrt(n)$]
]

#math.equation(block: true, alt: "the square root of 288 m to the power 21")[$sqrt(288 m^(21))$]

#math.equation(block: true, alt: "the square root of 150 n to the power 7")[$sqrt(150 n^(7))$]

#solutionbox[
#math.equation(block: true, alt: "5 n cubed the square root of 6 n")[$5 n^(3) sqrt(6 n)$]
]

#math.equation(block: true, alt: "the square root of 48 r to the power 5 s to the power 4")[$sqrt(48 r^(5) s^(4))$]

#math.equation(block: true, alt: "the square root of 108 r to the power 5 s cubed")[$sqrt(108 r^(5) s^(3))$]

#solutionbox[
#math.equation(block: true, alt: "6 r squared s the square root of 3 r s")[$6 r^(2) s sqrt(3 r s)$]
]

#math.equation(block: true, alt: "the fraction 10 minus the square root of 50 over 5")[$frac(10 − sqrt(50), 5)$]

#math.equation(block: true, alt: "the fraction 6 plus the square root of 72 over 6")[$frac(6 + sqrt(72), 6)$]

#solutionbox[
#math.equation(block: true, alt: "1 plus the square root of 2")[$1 + sqrt(2)$]
]

#strong[Use the Quotient Property to Simplify Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of the fraction 16 over 25")[$sqrt(frac(16, 25))$]

#math.equation(block: true, alt: "the square root of the fraction 81 over 36")[$sqrt(frac(81, 36))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 over 2")[$frac(3, 2)$]
]

#math.equation(block: true, alt: "the square root of the fraction x to the power 8 over x to the power 4")[$sqrt(frac(x^(8), x^(4)))$]

#math.equation(block: true, alt: "the square root of the fraction y to the power 6 over y squared")[$sqrt(frac(y^(6), y^(2)))$]

#solutionbox[
#math.equation(block: true, alt: "y squared")[$y^(2)$]
]

#math.equation(block: true, alt: "the square root of the fraction 98 p to the power 6 over 2 p squared")[$sqrt(frac(98 p^(6), 2 p^(2)))$]

#math.equation(block: true, alt: "the square root of the fraction 72 q to the power 8 over 2 q to the power 4")[$sqrt(frac(72 q^(8), 2 q^(4)))$]

#solutionbox[
#math.equation(block: true, alt: "6 q squared")[$6 q^(2)$]
]

#math.equation(block: true, alt: "the square root of the fraction 65 over 121")[$sqrt(frac(65, 121))$]

#math.equation(block: true, alt: "the square root of the fraction 26 over 169")[$sqrt(frac(26, 169))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 26 over 13")[$frac(sqrt(26), 13)$]
]

#math.equation(block: true, alt: "the square root of the fraction 64 x to the power 4 over 25 x squared")[$sqrt(frac(64 x^(4), 25 x^(2)))$]

#math.equation(block: true, alt: "the square root of the fraction 36 r to the power 10 over 16 r to the power 5")[$sqrt(frac(36 r^(10), 16 r^(5)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 3 r squared the square root of r over 2")[$frac(3 r^(2) sqrt(r), 2)$]
]

#math.equation(block: true, alt: "the square root of the fraction 48 p cubed q to the power 5 over 27 p q")[$sqrt(frac(48 p^(3) q^(5), 27 p q))$]

#math.equation(block: true, alt: "the square root of the fraction 12 r to the power 5 s to the power 7 over 75 r squared s")[$sqrt(frac(12 r^(5) s^(7), 75 r^(2) s))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 r s cubed the square root of r over 5")[$frac(2 r s^(3) sqrt(r), 5)$]
]


==== Add and Subtract Square Roots

#strong[Add and Subtract Like Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "3 the square root of 2 plus the square root of 2")[$3 sqrt(2) + sqrt(2)$]

#math.equation(block: true, alt: "5 the square root of 5 plus 7 the square root of 5")[$5 sqrt(5) + 7 sqrt(5)$]

#solutionbox[
#math.equation(block: true, alt: "12 the square root of 5")[$12 sqrt(5)$]
]

#math.equation(block: true, alt: "4 the square root of y plus 4 the square root of y")[$4 sqrt(y) + 4 sqrt(y)$]

#math.equation(block: true, alt: "6 the square root of m minus 2 the square root of m")[$6 sqrt(m) − 2 sqrt(m)$]

#solutionbox[
#math.equation(block: true, alt: "4 the square root of m")[$4 sqrt(m)$]
]

#math.equation(block: true, alt: "−3 the square root of 7 plus 2 the square root of 7 minus the square root of 7")[$−3 sqrt(7) + 2 sqrt(7) − sqrt(7)$]

#math.equation(block: true, alt: "8 the square root of 13 plus 2 the square root of 3 plus 3 the square root of 13")[$8 sqrt(13) + 2 sqrt(3) + 3 sqrt(13)$]

#solutionbox[
#math.equation(block: true, alt: "11 the square root of 13 plus 2 the square root of 3")[$11 sqrt(13) + 2 sqrt(3)$]
]

#math.equation(block: true, alt: "3 the square root of 5 x y minus the square root of 5 x y plus 3 the square root of 5 x y")[$3 sqrt(5 x y) − sqrt(5 x y) + 3 sqrt(5 x y)$]

#math.equation(block: true, alt: "2 the square root of 3 r s plus the square root of 3 r s minus 5 the square root of r s")[$2 sqrt(3 r s) + sqrt(3 r s) − 5 sqrt(r s)$]

#solutionbox[
#math.equation(block: true, alt: "3 the square root of 3 r s minus 5 the square root of r s")[$3 sqrt(3 r s) − 5 sqrt(r s)$]
]

#strong[Add and Subtract Square Roots that Need Simplification]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 32 plus 3 the square root of 2")[$sqrt(32) + 3 sqrt(2)$]

#math.equation(block: true, alt: "the square root of 8 plus 3 the square root of 2")[$sqrt(8) + 3 sqrt(2)$]

#solutionbox[
#math.equation(block: true, alt: "5 the square root of 2")[$5 sqrt(2)$]
]

#math.equation(block: true, alt: "the square root of 72 plus the square root of 50")[$sqrt(72) + sqrt(50)$]

#math.equation(block: true, alt: "the square root of 48 plus the square root of 75")[$sqrt(48) + sqrt(75)$]

#solutionbox[
#math.equation(block: true, alt: "9 the square root of 3")[$9 sqrt(3)$]
]

#math.equation(block: true, alt: "3 the square root of 32 plus the square root of 98")[$3 sqrt(32) + sqrt(98)$]

#math.equation(block: true, alt: "the fraction 1 over 3 the square root of 27 minus the fraction 1 over 8 the square root of 192")[$frac(1, 3) sqrt(27) − frac(1, 8) sqrt(192)$]

#solutionbox[
#math.equation(block: true, alt: "0")[$0$]
]

#math.equation(block: true, alt: "the square root of 50 y to the power 5 minus the square root of 72 y to the power 5")[$sqrt(50 y^(5)) − sqrt(72 y^(5))$]

#math.equation(block: true, alt: "6 the square root of 18 n to the power 4 minus 3 the square root of 8 n to the power 4 plus n squared the square root of 50")[$6 sqrt(18 n^(4)) − 3 sqrt(8 n^(4)) + n^(2) sqrt(50)$]

#solutionbox[
#math.equation(block: true, alt: "17 n squared the square root of 2")[$17 n^(2) sqrt(2)$]
]


==== Multiply Square Roots

#strong[Multiply Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 2 times the square root of 20")[$sqrt(2) · sqrt(20)$]

#math.equation(block: true, alt: "2 the square root of 2 times 6 the square root of 14")[$2 sqrt(2) · 6 sqrt(14)$]

#solutionbox[
#math.equation(block: true, alt: "24 the square root of 7")[$24 sqrt(7)$]
]

#math.equation(block: true, alt: "the square root of 2 m squared times the square root of 20 m to the power 4")[$sqrt(2 m^(2)) · sqrt(20 m^(4))$]

#math.equation(block: true, alt: "open parenthesis 6 the square root of 2 y close parenthesis open parenthesis 3 the square root of 50 y cubed close parenthesis")[$( 6 sqrt(2 y) ) ( 3 sqrt(50 y^(3)) )$]

#solutionbox[
#math.equation(block: true, alt: "180 y squared")[$180 y^(2)$]
]

#math.equation(block: true, alt: "open parenthesis 6 the square root of 3 v to the power 4 close parenthesis open parenthesis 5 the square root of 30 v close parenthesis")[$( 6 sqrt(3 v^(4)) ) ( 5 sqrt(30 v) )$]

#math.equation(block: true, alt: "open parenthesis the square root of 8 close parenthesis squared")[$attach(( sqrt(8) ), t: 2)$]

#solutionbox[
8
]

#math.equation(block: true, alt: "open parenthesis − the square root of 10 close parenthesis squared")[$attach(( "−" sqrt(10) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 2 the square root of 5 close parenthesis open parenthesis 5 the square root of 5 close parenthesis")[$( 2 sqrt(5) ) ( 5 sqrt(5) )$]

#solutionbox[
50
]

#math.equation(block: true, alt: "open parenthesis −3 the square root of 3 close parenthesis open parenthesis 5 the square root of 18 close parenthesis")[$( −3 sqrt(3) ) ( 5 sqrt(18) )$]

#strong[Use Polynomial Multiplication to Multiply Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "10 open parenthesis 2 minus the square root of 7 close parenthesis")[$10 ( 2 − sqrt(7) )$]

#solutionbox[
#math.equation(block: true, alt: "20 minus 10 the square root of 7")[$20 − 10 sqrt(7)$]
]

#math.equation(block: true, alt: "the square root of 3 open parenthesis 4 plus the square root of 12 close parenthesis")[$sqrt(3) ( 4 + sqrt(12) )$]

#math.equation(block: true, alt: "open parenthesis 5 plus the square root of 2 close parenthesis open parenthesis 3 minus the square root of 2 close parenthesis")[$( 5 + sqrt(2) ) ( 3 − sqrt(2) )$]

#solutionbox[
#math.equation(block: true, alt: "13 minus 2 the square root of 2")[$13 − 2 sqrt(2)$]
]

#math.equation(block: true, alt: "open parenthesis 5 minus 3 the square root of 7 close parenthesis open parenthesis 1 minus 2 the square root of 7 close parenthesis")[$( 5 − 3 sqrt(7) ) ( 1 − 2 sqrt(7) )$]

#math.equation(block: true, alt: "open parenthesis 1 minus 3 the square root of x close parenthesis open parenthesis 5 plus 2 the square root of x close parenthesis")[$( 1 − 3 sqrt(x) ) ( 5 + 2 sqrt(x) )$]

#solutionbox[
#math.equation(block: true, alt: "5 minus 13 the square root of x minus 6 x")[$5 − 13 sqrt(x) − 6 x$]
]

#math.equation(block: true, alt: "open parenthesis 3 plus 4 the square root of y close parenthesis open parenthesis 10 minus the square root of y close parenthesis")[$( 3 + 4 sqrt(y) ) ( 10 − sqrt(y) )$]

#math.equation(block: true, alt: "open parenthesis 1 plus 6 the square root of p close parenthesis squared")[$attach(( 1 + 6 sqrt(p) ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "1 plus 12 the square root of p plus 36 p")[$1 + 12 sqrt(p) + 36 p$]
]

#math.equation(block: true, alt: "open parenthesis 2 minus 6 the square root of 5 close parenthesis squared")[$attach(( 2 − 6 sqrt(5) ), t: 2)$]

#math.equation(block: true, alt: "open parenthesis 3 plus 2 the square root of 7 close parenthesis open parenthesis 3 minus 2 the square root of 7 close parenthesis")[$( 3 + 2 sqrt(7) ) ( 3 − 2 sqrt(7) )$]

#solutionbox[
#math.equation(block: true, alt: "−19")[$−19$]
]

#math.equation(block: true, alt: "open parenthesis 6 minus the square root of 11 close parenthesis open parenthesis 6 plus the square root of 11 close parenthesis")[$( 6 − sqrt(11) ) ( 6 + sqrt(11) )$]


==== Divide Square Roots

#strong[Divide Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the fraction the square root of 75 over 10")[$frac(sqrt(75), 10)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 3 over 2")[$frac(sqrt(3), 2)$]
]

#math.equation(block: true, alt: "the fraction 2 minus the square root of 12 over 6")[$frac(2 − sqrt(12), 6)$]

#math.equation(block: true, alt: "the fraction the square root of 48 over the square root of 27")[$frac(sqrt(48), sqrt(27))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 4 over 3")[$frac(4, 3)$]
]

#math.equation(block: true, alt: "the fraction the square root of 75 x to the power 7 over the square root of 3 x cubed")[$frac(sqrt(75 x^(7)), sqrt(3 x^(3)))$]

#math.equation(block: true, alt: "the fraction the square root of 20 y to the power 5 over the square root of 2 y")[$frac(sqrt(20 y^(5)), sqrt(2 y))$]

#solutionbox[
#math.equation(block: true, alt: "y squared the square root of 10")[$y^(2) sqrt(10)$]
]

#math.equation(block: true, alt: "the fraction the square root of 98 p to the power 6 q to the power 4 over the square root of 2 p to the power 4 q to the power 8")[$frac(sqrt(98 p^(6) q^(4)), sqrt(2 p^(4) q^(8)))$]

#strong[Rationalize a One Term Denominator]

In the following exercises, rationalize the denominator.

#math.equation(block: true, alt: "the fraction 10 over the square root of 15")[$frac(10, sqrt(15))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 2 the square root of 15 over 3")[$frac(2 sqrt(15), 3)$]
]

#math.equation(block: true, alt: "the fraction 6 over the square root of 6")[$frac(6, sqrt(6))$]

#math.equation(block: true, alt: "the fraction 5 over 3 the square root of 5")[$frac(5, 3 sqrt(5))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 5 over 3")[$frac(sqrt(5), 3)$]
]

#math.equation(block: true, alt: "the fraction 10 over 2 the square root of 6")[$frac(10, 2 sqrt(6))$]

#math.equation(block: true, alt: "the square root of the fraction 3 over 28")[$sqrt(frac(3, 28))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 21 over 14")[$frac(sqrt(21), 14)$]
]

#math.equation(block: true, alt: "the square root of the fraction 9 over 75")[$sqrt(frac(9, 75))$]

#strong[Rationalize a Two Term Denominator]

In the following exercises, rationalize the denominator.

#math.equation(block: true, alt: "the fraction 4 over 4 plus the square root of 27")[$frac(4, 4 + sqrt(27))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 16 minus 12 the square root of 3 over −11")[$frac(16 − 12 sqrt(3), −11)$]
]

#math.equation(block: true, alt: "the fraction 5 over 2 minus the square root of 10")[$frac(5, 2 − sqrt(10))$]

#math.equation(block: true, alt: "the fraction 4 over 2 minus the square root of 5")[$frac(4, 2 − sqrt(5))$]

#solutionbox[
#math.equation(block: true, alt: "−8 minus 4 the square root of 5")[$−8 − 4 sqrt(5)$]
]

#math.equation(block: true, alt: "the fraction 5 over 4 minus the square root of 8")[$frac(5, 4 − sqrt(8))$]

#math.equation(block: true, alt: "the fraction the square root of 2 over the square root of p plus the square root of 3")[$frac(sqrt(2), sqrt(p) + sqrt(3))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 2 p minus the square root of 6 over p minus 3")[$frac(sqrt(2 p) − sqrt(6), p − 3)$]
]

#math.equation(block: true, alt: "the fraction the square root of x minus the square root of 2 over the square root of x plus the square root of 2")[$frac(sqrt(x) − sqrt(2), sqrt(x) + sqrt(2))$]


==== Solve Equations with Square Roots

#strong[Solve Radical Equations]

In the following exercises, solve the equation.

#math.equation(block: true, alt: "the square root of 7 z plus 1 equals 6")[$sqrt(7 z + 1) = 6$]

#solutionbox[
5
]

#math.equation(block: true, alt: "the square root of 4 u minus 2 minus 4 equals 0")[$sqrt(4 u − 2) − 4 = 0$]

#math.equation(block: true, alt: "the square root of 6 m plus 4 minus 5 equals 0")[$sqrt(6 m + 4) − 5 = 0$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 7 over 2")[$frac(7, 2)$]
]

#math.equation(block: true, alt: "the square root of 2 u minus 3 plus 2 equals 0")[$sqrt(2 u − 3) + 2 = 0$]

#math.equation(block: true, alt: "the square root of u minus 4 plus 4 equals u")[$sqrt(u − 4) + 4 = u$]

#solutionbox[
4 and 5
]

#math.equation(block: true, alt: "the square root of v minus 9 plus 9 equals 0")[$sqrt(v − 9) + 9 = 0$]

#math.equation(block: true, alt: "the square root of r minus 4 minus r equals −10")[$sqrt(r − 4) − r = −10$]

#solutionbox[
13
]

#math.equation(block: true, alt: "the square root of s minus 9 minus s equals −9")[$sqrt(s − 9) − s = −9$]

#math.equation(block: true, alt: "2 the square root of 2 x minus 7 minus 4 equals 8")[$2 sqrt(2 x − 7) − 4 = 8$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 43 over 2")[$frac(43, 2)$]
]

#math.equation(block: true, alt: "the square root of 2 minus x equals the square root of 2 x minus 7")[$sqrt(2 − x) = sqrt(2 x − 7)$]

#math.equation(block: true, alt: "the square root of a plus 3 equals the square root of a plus 9")[$sqrt(a) + 3 = sqrt(a + 9)$]

#solutionbox[
0
]

#math.equation(block: true, alt: "the square root of r plus 3 equals the square root of r plus 4")[$sqrt(r) + 3 = sqrt(r + 4)$]

#math.equation(block: true, alt: "the square root of u plus 2 equals the square root of u plus 5")[$sqrt(u) + 2 = sqrt(u + 5)$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 16")[$frac(1, 16)$]
]

#math.equation(block: true, alt: "the square root of n plus 11 minus 1 equals the square root of n plus 4")[$sqrt(n + 11) − 1 = sqrt(n + 4)$]

#math.equation(block: true, alt: "the square root of y plus 5 plus 1 equals the square root of 2 y plus 3")[$sqrt(y + 5) + 1 = sqrt(2 y + 3)$]

#solutionbox[
11
]

#strong[Use Square Roots in Applications]

In the following exercises, solve. Round approximations to one decimal place.

A pallet of sod will cover an area of about 600 square feet. Trinh wants to order a pallet of sod to make a square lawn in his backyard. Use the formula #math.equation(block: false, alt: "s equals the square root of A")[$s = sqrt(A)$] to find the length of each side of his lawn.

A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the package to reach the hiker.

#solutionbox[
7.5 seconds
]

Officer Morales measured the skid marks of one of the cars involved in an accident. The length of the skid marks was 245 feet. Use the formula #math.equation(block: false, alt: "s equals the square root of 24 d")[$s = sqrt(24 d)$] to find the speed of the car before the brakes were applied.


==== Higher Roots

#strong[Simplify Expressions with Higher Roots]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "the 6th root of 64")[$root(6, 64)$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of 64")[$root(3, 64)$]

#solutionbox[
ⓐ 2 ⓑ 4
]

ⓐ #math.equation(block: false, alt: "the cube root of −27")[$root(3, −27)$] #linebreak() ⓑ #math.equation(block: false, alt: "the 4th root of −64")[$root(4, −64)$]

ⓐ #math.equation(block: false, alt: "the 9th root of d to the power 9")[$root(9, d^(9))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 8th root of v to the power 8")[$root(8, v^(8))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "d")[$d$] ⓑ #math.equation(block: false, alt: "| v |")[$| v |$]
]

ⓐ #math.equation(block: false, alt: "the 5th root of a to the power 10")[$root(5, a^(10))$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of b to the power 27")[$root(3, b^(27))$]

ⓐ #math.equation(block: false, alt: "the 4th root of 16 x to the power 8")[$root(4, 16 x^(8))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 6th root of 64 y to the power 12")[$root(6, 64 y^(12))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "2 x squared")[$2 x^(2)$] ⓑ #math.equation(block: false, alt: "2 y squared")[$2 y^(2)$]
]

ⓐ #math.equation(block: false, alt: "the 7th root of 128 r to the power 14")[$root(7, 128 r^(14))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 4th root of 81 s to the power 24")[$root(4, 81 s^(24))$]

#strong[Use the Product Property to Simplify Expressions with Higher Roots]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "the 9th root of d to the power 9")[$root(9, d^(9))$] #linebreak() ⓑ #figure(figph[A mathematical expression showing the 11th root of m raised to the power of 17, written as '11th root of m^17'.], alt: "A mathematical expression showing the 11th root of m raised to the power of 17, written as '11th root of m^17'.", caption: none)

#solutionbox[
ⓐ #math.equation(block: false, alt: "d")[$d$] ⓑ #math.equation(block: false, alt: "m the 11th root of m to the power 6")[$m root(11, m^(6))$]
]

ⓐ #math.equation(block: false, alt: "the cube root of 54")[$root(3, 54)$] #linebreak() ⓑ #math.equation(block: false, alt: "the 4th root of 128")[$root(4, 128)$]

ⓐ #math.equation(block: false, alt: "the 5th root of 64 c to the power 8")[$root(5, 64 c^(8))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 4th root of 48 d to the power 7")[$root(4, 48 d^(7))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "2 c the 5th root of 2 c cubed")[$2 c root(5, 2 c^(3))$] ⓑ #math.equation(block: false, alt: "2 d the 4th root of 3 d cubed")[$2 d root(4, 3 d^(3))$]
]

ⓐ #math.equation(block: false, alt: "the cube root of 343 q to the power 7")[$root(3, 343 q^(7))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 6th root of 192 r to the power 9")[$root(6, 192 r^(9))$]

ⓐ #math.equation(block: false, alt: "the cube root of −500")[$root(3, −500)$] #linebreak() ⓑ #math.equation(block: false, alt: "the 4th root of −16")[$root(4, −16)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−5 the cube root of 4")[$−5 root(3, 4)$] ⓑ not a real number
]

#strong[Use the Quotient Property to Simplify Expressions with Higher Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "the 5th root of the fraction r to the power 10 over r to the power 5")[$root(5, frac(r^(10), r^(5)))$]

#math.equation(block: true, alt: "the cube root of the fraction w to the power 12 over w squared")[$root(3, frac(w^(12), w^(2)))$]

#solutionbox[
#math.equation(block: true, alt: "w cubed the cube root of w")[$w^(3) root(3, w)$]
]

#math.equation(block: true, alt: "the 4th root of the fraction 64 y to the power 8 over 4 y to the power 5")[$root(4, frac(64 y^(8), 4 y^(5)))$]

#math.equation(block: true, alt: "the cube root of the fraction 54 z to the power 9 over 2 z cubed")[$root(3, frac(54 z^(9), 2 z^(3)))$]

#solutionbox[
#math.equation(block: true, alt: "3 z squared")[$3 z^(2)$]
]

#math.equation(block: true, alt: "the 6th root of the fraction 64 a to the power 7 over b squared")[$root(6, frac(64 a^(7), b^(2)))$]

#strong[Add and Subtract Higher Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "4 the 5th root of 20 minus 2 the 5th root of 20")[$4 root(5, 20) − 2 root(5, 20)$]

#solutionbox[
#math.equation(block: true, alt: "2 the 5th root of 20")[$2 root(5, 20)$]
]

#math.equation(block: true, alt: "4 the cube root of 18 plus 3 the cube root of 18")[$4 root(3, 18) + 3 root(3, 18)$]

#math.equation(block: true, alt: "the 4th root of 1250 minus the 4th root of 162")[$root(4, 1250) − root(4, 162)$]

#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 640 c to the power 5 minus the cube root of −80 c cubed")[$root(3, 640 c^(5)) − root(3, −80 c^(3))$]

#math.equation(block: true, alt: "the 5th root of 96 t to the power 8 plus the 5th root of 486 t to the power 4")[$root(5, 96 t^(8)) + root(5, 486 t^(4))$]

#solutionbox[
#math.equation(block: true, alt: "2 t the 5th root of 3 t cubed plus 3 the 5th root of 2 t to the power 4")[$2 t root(5, 3 t^(3)) + 3 root(5, 2 t^(4))$]
]


==== Rational Exponents

#strong[Simplify Expressions with #math.equation(block: false, alt: "a to the power the fraction 1 over n")[$a^(frac(1, n))$]]

In the following exercises, write as a radical expression.

#math.equation(block: true, alt: "r to the power the fraction 1 over 8")[$r^(frac(1, 8))$]

#math.equation(block: true, alt: "s to the power the fraction 1 over 10")[$s^(frac(1, 10))$]

#solutionbox[
#figure(figph[A close-up image showing a mathematical expression: the square root of nineteen divided by S. It appears to be a variable or a value S under the radical symbol, with 19 also inside.], alt: "A close-up image showing a mathematical expression: the square root of nineteen divided by S. It appears to be a variable or a value S under the radical symbol, with 19 also inside.", caption: none)
]

In the following exercises, write with a rational exponent.

#math.equation(block: true, alt: "the 5th root of u")[$root(5, u)$]

#math.equation(block: true, alt: "the 6th root of v")[$root(6, v)$]

#solutionbox[
#math.equation(block: true, alt: "v to the power the fraction 1 over 6")[$v^(frac(1, 6))$]
]

#math.equation(block: true, alt: "the cube root of 9 m")[$root(3, 9 m)$]

#math.equation(block: true, alt: "the 6th root of 10 z")[$root(6, 10 z)$]

#solutionbox[
#math.equation(block: true, alt: "open parenthesis 10 z close parenthesis to the power the fraction 1 over 6")[$attach(( 10 z ), t: frac(1, 6))$]
]

In the following exercises, simplify.

#math.equation(block: true, alt: "16 to the power the fraction 1 over 4")[$16^(frac(1, 4))$]

#math.equation(block: true, alt: "32 to the power the fraction 1 over 5")[$32^(frac(1, 5))$]

#solutionbox[
2
]

#math.equation(block: true, alt: "open parenthesis −125 close parenthesis to the power the fraction 1 over 3")[$attach(( −125 ), t: frac(1, 3))$]

#math.equation(block: true, alt: "open parenthesis 125 close parenthesis to the power minus the fraction 1 over 3")[$attach(( 125 ), t: − frac(1, 3))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 5")[$frac(1, 5)$]
]

#math.equation(block: true, alt: "open parenthesis −9 close parenthesis to the power the fraction 1 over 2")[$attach(( −9 ), t: frac(1, 2))$]

#math.equation(block: true, alt: "open parenthesis 36 close parenthesis to the power minus the fraction 1 over 2")[$attach(( 36 ), t: − frac(1, 2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 6")[$frac(1, 6)$]
]

#strong[Simplify Expressions with #math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$]]

In the following exercises, write with a rational exponent.

#math.equation(block: true, alt: "the cube root of q to the power 5")[$root(3, q^(5))$]

#math.equation(block: true, alt: "the 5th root of n to the power 8")[$root(5, n^(8))$]

#solutionbox[
#math.equation(block: true, alt: "n to the power the fraction 8 over 5")[$n^(frac(8, 5))$]
]

In the following exercises, simplify.

#math.equation(block: true, alt: "27 to the power minus the fraction 2 over 3")[$27^(− frac(2, 3))$]

#math.equation(block: true, alt: "64 to the power the fraction 5 over 2")[$64^(frac(5, 2))$]

#solutionbox[
32,768
]

#math.equation(block: true, alt: "36 to the power the fraction 3 over 2")[$36^(frac(3, 2))$]

#math.equation(block: true, alt: "81 to the power minus the fraction 5 over 2")[$81^(− frac(5, 2))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 59,049")[$frac(1, 59","049)$]
]

#strong[Use the Laws of Exponents to Simplify Expressions with Rational Exponents]

In the following exercises, simplify.

#math.equation(block: true, alt: "3 to the power the fraction 4 over 5 times 3 to the power the fraction 6 over 5")[$3^(frac(4, 5)) · 3^(frac(6, 5))$]

#math.equation(block: true, alt: "open parenthesis x to the power 6 close parenthesis to the power the fraction 4 over 3")[$attach(( x^(6) ), t: frac(4, 3))$]

#solutionbox[
#math.equation(block: true, alt: "x to the power 8")[$x^(8)$]
]

#math.equation(block: true, alt: "the fraction z to the power the fraction 5 over 2 over z to the power the fraction 7 over 5")[$frac(z^(frac(5, 2)), z^(frac(7, 5)))$]

#math.equation(block: true, alt: "open parenthesis 16 s to the power the fraction 9 over 4 close parenthesis to the power the fraction 1 over 4")[$attach(( 16 s^(frac(9, 4)) ), t: frac(1, 4))$]

#solutionbox[
#math.equation(block: true, alt: "2 s to the power the fraction 9 over 16")[$2 s^(frac(9, 16))$]
]

#math.equation(block: true, alt: "open parenthesis m to the power 8 n to the power 12 close parenthesis to the power the fraction 1 over 4")[$attach(( m^(8) n^(12) ), t: frac(1, 4))$]

#math.equation(block: true, alt: "the fraction z to the power the fraction 2 over 3 times z to the power minus the fraction 1 over 3 over z to the power minus the fraction 5 over 3")[$frac(z^(frac(2, 3)) · z^(− frac(1, 3)), z^(− frac(5, 3)))$]

#solutionbox[
#math.equation(block: true, alt: "z squared")[$z^(2)$]
]


=== Practice Test

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 81 plus 144")[$sqrt(81 + 144)$]

#math.equation(block: true, alt: "the square root of 169 m to the power 4 n squared")[$sqrt(169 m^(4) n^(2))$]

#solutionbox[
#math.equation(block: true, alt: "13 m squared | n |")[$13 m^(2) | n |$]
]

#math.equation(block: true, alt: "the square root of 36 n to the power 13")[$sqrt(36 n^(13))$]

#math.equation(block: true, alt: "3 the square root of 13 plus 5 the square root of 2 plus the square root of 13")[$3 sqrt(13) + 5 sqrt(2) + sqrt(13)$]

#solutionbox[
#math.equation(block: true, alt: "4 the square root of 13 plus 5 the square root of 2")[$4 sqrt(13) + 5 sqrt(2)$]
]

#math.equation(block: true, alt: "5 the square root of 20 plus 2 the square root of 125")[$5 sqrt(20) + 2 sqrt(125)$]

#math.equation(block: true, alt: "open parenthesis 3 the square root of 6 y close parenthesis open parenthesis 2 the square root of 50 y cubed close parenthesis")[$( 3 sqrt(6 y) ) ( 2 sqrt(50 y^(3)) )$]

#solutionbox[
#math.equation(block: true, alt: "60 y squared the square root of 3")[$60 y^(2) sqrt(3)$]
]

#math.equation(block: true, alt: "open parenthesis 2 minus 5 the square root of x close parenthesis open parenthesis 3 plus the square root of x close parenthesis")[$( 2 − 5 sqrt(x) ) ( 3 + sqrt(x) )$]

#math.equation(block: true, alt: "open parenthesis 1 minus 2 the square root of q close parenthesis squared")[$attach(( 1 − 2 sqrt(q) ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "1 minus 4 the square root of q plus 4 q")[$1 − 4 sqrt(q) + 4 q$]
]

ⓐ #math.equation(block: false, alt: "the 4th root of a to the power 12")[$root(4, a^(12))$] #linebreak() ⓑ #math.equation(block: false, alt: "the cube root of b to the power 21")[$root(3, b^(21))$]

ⓐ #math.equation(block: false, alt: "the 4th root of 81 x to the power 12")[$root(4, 81 x^(12))$] #linebreak() ⓑ #math.equation(block: false, alt: "the 6th root of 64 y to the power 18")[$root(6, 64 y^(18))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "3 x cubed")[$3 x^(3)$] ⓑ #math.equation(block: false, alt: "2 y cubed")[$2 y^(3)$]
]

#math.equation(block: true, alt: "the square root of the fraction 64 r to the power 12 over 25 r to the power 6")[$sqrt(frac(64 r^(12), 25 r^(6)))$]

#math.equation(block: true, alt: "the square root of the fraction 14 y cubed over 7 y")[$sqrt(frac(14 y^(3), 7 y))$]

#solutionbox[
#math.equation(block: true, alt: "y the square root of 2")[$y sqrt(2)$]
]

#math.equation(block: true, alt: "the fraction the 5th root of 256 x to the power 7 over the 5th root of 4 x squared")[$frac(root(5, 256 x^(7)), root(5, 4 x^(2)))$]

#math.equation(block: true, alt: "the 4th root of 512 minus 2 the 4th root of 32")[$root(4, 512) − 2 root(4, 32)$]

#solutionbox[
#math.equation(block: true, alt: "0")[$0$]
]

ⓐ #math.equation(block: false, alt: "256 to the power the fraction 1 over 4")[$256^(frac(1, 4))$] #linebreak() ⓑ #math.equation(block: false, alt: "243 to the power the fraction 1 over 5")[$243^(frac(1, 5))$]

#math.equation(block: true, alt: "49 to the power the fraction 3 over 2")[$49^(frac(3, 2))$]

#solutionbox[
343
]

#math.equation(block: true, alt: "25 to the power minus the fraction 5 over 2")[$25^(− frac(5, 2))$]

#math.equation(block: true, alt: "the fraction w to the power the fraction 3 over 4 over w to the power the fraction 7 over 4")[$frac(w^(frac(3, 4)), w^(frac(7, 4)))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over w")[$frac(1, w)$]
]

#math.equation(block: true, alt: "open parenthesis 27 s to the power the fraction 3 over 5 close parenthesis to the power the fraction 1 over 3")[$attach(( 27 s^(frac(3, 5)) ), t: frac(1, 3))$]

In the following exercises, rationalize the denominator.

#math.equation(block: true, alt: "the fraction 3 over 2 the square root of 6")[$frac(3, 2 sqrt(6))$]

#solutionbox[
#math.equation(block: true, alt: "the fraction the square root of 6 over 4")[$frac(sqrt(6), 4)$]
]

#math.equation(block: true, alt: "the fraction the square root of 3 over the square root of x plus the square root of 5")[$frac(sqrt(3), sqrt(x) + sqrt(5))$]

In the following exercises, solve.

#math.equation(block: true, alt: "3 the square root of 2 x minus 3 minus 20 equals 7")[$3 sqrt(2 x − 3) − 20 = 7$]

#solutionbox[
42
]

#math.equation(block: true, alt: "the square root of 3 u minus 2 equals the square root of 5 u plus 1")[$sqrt(3 u − 2) = sqrt(5 u + 1)$]

In the following exercise, solve.

A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula #math.equation(block: false, alt: "t equals the fraction the square root of h over 4")[$t = frac(sqrt(h), 4)$] to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.

#solutionbox[
6.1 seconds
]