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== 8.3#h(0.6em)Simplify 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 rational exponents, we can apply the properties of exponents to simplify expressions.

The #strong[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].

#math.equation(block: true, alt: "open parenthesis 8 to the power p close parenthesis cubed, equals, 8; Multiply the exponents on the left., 8 to the power 3 p, equals, 8; Write the exponent 1 on the right., 8 to the power 3 p, equals, 8 to the power 1; Since the bases are the same, the exponents must be equal., 3 p, equals, 1; Solve for p ., p, equals, the fraction 1 over 3")[$&  &  &  &  & attach(( 8^(p) ), t: 3) & = & 8 \ "Multiply the exponents on the left." &  &  &  &  & 8^(3 p) & = & 8 \ "Write the exponent 1 on the right." &  &  &  &  & 8^(3 p) & = & 8^(1) \ "Since the bases are the same, the exponents must be equal." &  &  &  &  & 3 p & = & 1 \ "Solve for" #h(0.2em) p . &  &  &  &  & p & = & frac(1, 3)$]

So #math.equation(block: false, alt: "open parenthesis 8 to the power the fraction 1 over 3 close parenthesis cubed equals 8 .")[$attach(( 8^(frac(1, 3)) ), t: 3) = 8 .$] 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 .")[$8^(frac(1, 3)) = root(3, 8) .$]

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 ,")[$n ≥ 2 ,$] then

#math.equation(block: true, alt: "a to the power the fraction 1 over n equals the nth root of a")[$a^(frac(1, n)) = root(n, a)$]
]

The denominator of the rational exponent is the index of the radical.

There will be times when working with expressions will be easier if you use 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")[$#h(5em) x^(frac(1, 2))$]]),
  [The denominator of the rational exponent is 2, so #linebreak() the index of the radical is 2. We do not show the #linebreak() index when it is 2.], [#math.equation(block: false, alt: "the square root of x")[$#h(5em) sqrt(x)$]],
))

ⓑ  #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")[$#h(7em) y^(frac(1, 3))$]]),
  [The denominator of the exponent is 3, so the #linebreak() index is 3.], [#math.equation(block: false, alt: "the cube root of y")[$#h(7em) root(3, y)$]],
))

ⓒ  #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")[$#h(7em) z^(frac(1, 4))$]]),
  [The denominator of the exponent is 4, so the #linebreak() index is 4.], [#math.equation(block: false, alt: "the 4th root of z")[$#h(7em) root(4, z)$]],
))
]
]

#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 6")[$b^(frac(1, 6))$] ⓑ #math.equation(block: false, alt: "z to the power the fraction 1 over 5")[$z^(frac(1, 5))$] ⓒ #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 6th root of b")[$root(6, b)$] ⓑ #math.equation(block: false, alt: "the 5th root of z")[$root(5, z)$] ⓒ #math.equation(block: false, alt: "the 4th root of p")[$root(4, p)$]
]
]

In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.

#examplebox("Example 2")[][
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")[$#h(4em) sqrt(5 y)$]]),
  [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")[$#h(4em) attach(( 5 y ), t: frac(1, 2))$]],
  [Put parentheses around the entire #linebreak() expression #math.equation(block: false, alt: "5 y .")[$5 y .$]], [],
))

ⓑ  #linebreak()

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

ⓒ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 the 4th root of 5 z")[$#h(4em) 3 #h(0.2em) root(4, 5 z)$]]),
  [The index is 4, so the denominator of the #linebreak() exponent is 4. Put parentheses only around #linebreak() the #math.equation(block: false, alt: "5 z")[$5 z$] since 3 is not under the radical sign.], [#math.equation(block: false, alt: "3 open parenthesis 5 z close parenthesis to the power the fraction 1 over 4")[$#h(4em) 3 attach(( 5 z ), t: frac(1, 4))$]],
))
]
]

#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")[$attach(( 3 n ), t: frac(1, 5))$] #linebreak() ⓒ #math.equation(block: false, alt: "3 open parenthesis 6 y close parenthesis to the power the fraction 1 over 4")[$3 attach(( 6 y ), t: frac(1, 4))$]
]
]

#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")[$attach(( 5 j ), t: frac(1, 4))$] #linebreak() ⓒ #math.equation(block: false, alt: "8 open parenthesis 2 a close parenthesis to the power the fraction 1 over 3")[$8 attach(( 2 a ), t: frac(1, 3))$]
]
]

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

#examplebox("Example 3")[][
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")[$#h(11em) 25^(frac(1, 2))$]]),
  [Rewrite as a square root.], [#math.equation(block: false, alt: "the square root of 25")[$#h(11em) sqrt(25)$]],
  [Simplify.], [#math.equation(block: false, alt: "5")[$#h(11em) 5$]],
))

ⓑ  #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")[$#h(8.5em) 64^(frac(1, 3))$]]),
  [Rewrite as a cube root.], [#math.equation(block: false, alt: "the cube root of 64")[$#h(8.5em) root(3, 64)$]],
  [Recognize 64 is a perfect cube.], [#math.equation(block: false, alt: "the cube root of 4 cubed")[$#h(8.5em) root(3, 4^(3))$]],
  [Simplify.], [#math.equation(block: false, alt: "4")[$#h(8.5em) 4$]],
))

ⓒ  #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")[$#h(4em) 256^(frac(1, 4))$]]),
  [Rewrite as a fourth root.], [#math.equation(block: false, alt: "the 4th root of 256")[$#h(4em) root(4, 256)$]],
  [Recognize 256 is a perfect fourth power.], [#math.equation(block: false, alt: "the 4th root of 4 to the power 4")[$#h(4em) root(4, 4^(4))$]],
  [Simplify.], [#math.equation(block: false, alt: "4")[$#h(4em) 4$]],
))
]
]

#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 4")[][
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")[$#h(9em) attach(( −16 ), t: frac(1, 4))$]]),
  [Rewrite as a fourth root.], [#math.equation(block: false, alt: "the 4th root of −16")[$#h(9em) root(4, −16)$]],
  [], [#math.equation(block: false, alt: "the 4th root of open parenthesis −2 close parenthesis to the power 4")[$#h(9em) root(4, attach(( −2 ), t: 4))$]],
  [Simplify.], [#math.equation(block: false, alt: "No real solution.")[$#h(9em) "No real solution."$]],
))

ⓑ  #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")[$#h(5.5em) "−" 16^(frac(1, 4))$]]),
  [The exponent only applies to the 16. #linebreak() Rewrite as a fouth root.], [#math.equation(block: false, alt: "− the 4th root of 16")[$#h(5.5em) "−" root(4, 16)$]],
  [Rewrite 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")[$#h(5.5em) "−" root(4, 2^(4))$]],
  [Simplify.], [#math.equation(block: false, alt: "−2")[$#h(5.5em) −2$]],
))

ⓒ  #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")[$#h(4.5em) attach(( 16 ), t: − frac(1, 4))$]]),
  [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")[$#h(4.5em) frac(1, attach(( 16 ), t: frac(1, 4)))$]],
  [Rewrite as a fourth root.], [#math.equation(block: false, alt: "the fraction 1 over the 4th root of 16")[$#h(4.5em) frac(1, root(4, 16))$]],
  [Rewrite 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")[$#h(4.5em) frac(1, root(4, 2^(4)))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 2")[$#h(4.5em) frac(1, 2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis −64 close parenthesis to the power minus 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[
ⓐ No real solution ⓑ #math.equation(block: false, alt: "−8")[$−8$] #linebreak() ⓒ #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[
ⓐ No real solution ⓑ #math.equation(block: false, alt: "−4")[$−4$] #linebreak() ⓒ #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")[$a^(frac(m, n))$]

We can look at #math.equation(block: false, alt: "a to the power the fraction m over n")[$a^(frac(m, n))$] in two ways. Remember the Power Property tells us to multiply the exponents and so #math.equation(block: false, alt: "open parenthesis a to the power the fraction 1 over n close parenthesis to the power m")[$attach(( a^(frac(1, n)) ), t: m)$] and #math.equation(block: false, alt: "open parenthesis a to the power m close parenthesis to the power to the power the fraction 1 over n")[$attach(( a^(m) ), t: frac(1, n))$] both equal #math.equation(block: false, alt: "a to the power the fraction m over n .")[$a^(frac(m, n)) .$] If we write these expressions in radical form, we get

#math.equation(block: true, alt: "a to the power the fraction m over n equals open parenthesis a to the power the fraction 1 over n close parenthesis to the power m equals open parenthesis the nth root of a close parenthesis to the power m and a to the power the fraction m over n equals open parenthesis a to the power m close parenthesis to the power to the power the fraction 1 over n equals the nth root of a to the power m")[$a^(frac(m, n)) = attach(( a^(frac(1, n)) ), t: m) = attach(( root(n, a) ), t: m) #h(1em) "and" #h(1em) a^(frac(m, n)) = attach(( a^(m) ), t: frac(1, n)) = root(n, a^(m))$]This leads us to the following definition.

#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 and a to the power the fraction m over n equals the nth root of a to the power m")[$a^(frac(m, n)) = attach(( root(n, a) ), t: m) #h(1em) "and" #h(1em) a^(frac(m, n)) = root(n, a^(m))$]
]

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, before raising it to the power indicated.

#examplebox("Example 5")[][
Write with a rational exponent: ⓐ #math.equation(block: false, alt: "the square root of y cubed")[$sqrt(y^(3))$] ⓑ #math.equation(block: false, alt: "open parenthesis the cube root of 2 x close parenthesis to the power 4")[$attach(( root(3, 2 x) ), t: 4)$] ⓒ #math.equation(block: false, alt: "the square root of open parenthesis the fraction 3 a over 4 b close parenthesis cubed .")[$sqrt(attach(( frac(3 a, 4 b) ), t: 3)) .$]

#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(table(
  columns: 1, align: left, inset: 6pt,
  [#figure(figph[Conversion of radical to exponential form: √y³ is equivalent to y^(3/2). The exponent's numerator (3) comes from the radicand's exponent, and the denominator (2) is the radical's index.], alt: "Conversion of radical to exponential form: √y³ is equivalent to y^(3/2). The exponent's numerator (3) comes from the radicand's exponent, and the denominator (2) is the radical's index.", caption: none)],
))

#linebreak() 
ⓑ  #linebreak()

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [#figure(figph[This image demonstrates how to convert radical expressions to rational exponents. The external exponent becomes the numerator and the radical index becomes the denominator, as seen with (³√2x)⁴ = (2x)^(4/3).], alt: "This image demonstrates how to convert radical expressions to rational exponents. The external exponent becomes the numerator and the radical index becomes the denominator, as seen with (³√2x)⁴ = (2x)^(4/3).", caption: none)],
))

#linebreak() 
ⓒ  #linebreak()

#figure(table(
  columns: 1, align: left, inset: 6pt,
  [#figure(figph[Illustration showing the relationship between radical form and exponential form, where the numerator of a fractional exponent is the power and the denominator is the radical index.], alt: "Illustration showing the relationship between radical form and exponential form, where the numerator of a fractional exponent is the power and the denominator is the radical index.", 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: "open parenthesis the 4th root of 3 y close parenthesis cubed")[$attach(( root(4, 3 y) ), t: 3)$] ⓒ #math.equation(block: false, alt: "the square root of open parenthesis the fraction 2 m over 3 n close parenthesis to the power 5 .")[$sqrt(attach(( frac(2 m, 3 n) ), t: 5)) .$]

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

#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: "open parenthesis the cube root of 5 a b close parenthesis to the power 5")[$attach(( root(3, 5 a b) ), t: 5)$] ⓒ #math.equation(block: false, alt: "the square root of open parenthesis the fraction 7 x y over z close parenthesis cubed .")[$sqrt(attach(( frac(7 x y, z) ), t: 3)) .$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "a to the power the fraction 2 over 5")[$a^(frac(2, 5))$] ⓑ #math.equation(block: false, alt: "open parenthesis 5 a b close parenthesis to the power the fraction 5 over 3")[$attach(( 5 a b ), t: frac(5, 3))$] #linebreak() ⓒ #math.equation(block: false, alt: "open parenthesis the fraction 7 x y over z close parenthesis to the power the fraction 3 over 2")[$attach(( frac(7 x y, z) ), t: frac(3, 2))$]
]
]

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

#examplebox("Example 6")[][
Simplify: ⓐ #math.equation(block: false, alt: "125 to the power the fraction 2 over 3")[$125^(frac(2, 3))$] ⓑ #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 .")[$32^(− frac(2, 5)) .$]

#solutionbox[
We will rewrite the expression as a radical first using the defintion, #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: "125 to the power the fraction 2 over 3")[$#h(4em) 125^(frac(2, 3))$]]),
  [The power of the radical is the numerator of the exponent, 2. #linebreak() The index of the radical is the denominator of the #linebreak() exponent, 3.], [#math.equation(block: false, alt: "open parenthesis the cube root of 125 close parenthesis squared")[$#h(4em) attach(( root(3, 125) ), t: 2)$]],
  [Simplify.], [#math.equation(block: false, alt: "open parenthesis 5 close parenthesis squared")[$#h(4em) attach(( 5 ), t: 2)$]],
  [], [#math.equation(block: false, alt: "25")[$#h(4em) 25$]],
))

ⓑ We will rewrite each expression first using #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n")[$a^("−" n) = frac(1, a^(n))$] 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")[$#h(4em) 16^(− frac(3, 2))$]]),
  [Rewrite using #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 16 to the power the fraction 3 over 2")[$#h(4em) frac(1, 16^(frac(3, 2)))$]],
  [Change to radical form. The power of the radical is the #linebreak() numerator of the exponent, 3. The index is the denominator #linebreak() of the exponent, 2.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis the square root of 16 close parenthesis cubed")[$#h(4em) frac(1, attach(( sqrt(16) ), t: 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 4 cubed")[$#h(4em) frac(1, 4^(3))$]],
  [], [#math.equation(block: false, alt: "the fraction 1 over 64")[$#h(4em) frac(1, 64)$]],
))

ⓒ  #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")[$#h(15em) 32^(− frac(2, 5))$]]),
  [Rewrite using #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 32 to the power the fraction 2 over 5")[$#h(15em) frac(1, 32^(frac(2, 5)))$]],
  [Change to radical form.], [#math.equation(block: false, alt: "the fraction 1 over open parenthesis the 5th root of 32 close parenthesis squared")[$#h(15em) frac(1, attach(( root(5, 32) ), t: 2))$]],
  [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")[$#h(15em) frac(1, attach(( root(5, 2^(5)) ), t: 2))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over 2 squared")[$#h(15em) frac(1, 2^(2))$]],
  [], [#math.equation(block: false, alt: "the fraction 1 over 4")[$#h(15em) frac(1, 4)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "27 to the power the fraction 2 over 3")[$27^(frac(2, 3))$] ⓑ #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[
ⓐ 9 ⓑ #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 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[
ⓐ 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 7")[][
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")[$#h(14em) "−" 25^(frac(3, 2))$]]),
  [Rewrite in radical form.], [#math.equation(block: false, alt: "− open parenthesis the square root of 25 close parenthesis cubed")[$#h(14em) "−" attach(( sqrt(25) ), t: 3)$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "− open parenthesis 5 close parenthesis cubed")[$#h(14em) "−" attach(( 5 ), t: 3)$]],
  [Simplify.], [#math.equation(block: false, alt: "−125")[$#h(14em) −125$]],
))

ⓑ  #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")[$#h(13em) "−" 25^(− frac(3, 2))$]]),
  [Rewrite using #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: "− open parenthesis the fraction 1 over 25 to the power the fraction 3 over 2 close parenthesis")[$#h(13em) "−" ( frac(1, 25^(frac(3, 2))) )$]],
  [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")[$#h(13em) "−" ( frac(1, attach(( sqrt(25) ), t: 3)) )$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "− open parenthesis the fraction 1 over open parenthesis 5 close parenthesis cubed close parenthesis")[$#h(13em) "−" ( frac(1, attach(( 5 ), t: 3)) )$]],
  [Simplify.], [#math.equation(block: false, alt: "minus the fraction 1 over 125")[$#h(13em) − frac(1, 125)$]],
))

ⓒ  #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")[$#h(4em) attach(( −25 ), t: frac(3, 2))$]]),
  [Rewrite in radical form.], [#math.equation(block: false, alt: "open parenthesis the square root of −25 close parenthesis cubed")[$#h(4em) attach(( sqrt(−25) ), t: 3)$]],
  [There is no real number whose square root #linebreak() #math.equation(block: false, alt: "is −25 .")[$"is" #h(0.2em) −25 .$]], [#math.equation(block: false, alt: "Not a real number.")[$#h(4em) "Not a real number."$]],
))
]
]

#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 Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponenets here to have them for reference as we simplify expressions.

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

If #emph[a] and #emph[b] are real numbers and #emph[m] and #emph[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; 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 a to the power m over b to the power m , b not equal to 0; Negative Exponent Property, a to the power − n, equals, the fraction 1 over a to the power")[$"Product Property" &  &  & #h(4em) a^(m) · a^(n) & = & a^(m + n) \ "Power Property" &  &  & #h(4em) attach(( a^(m) ), t: n) & = & a^(m · n) \ "Product to a Power" &  &  & #h(4em) attach(( a b ), t: m) & = & a^(m) b^(m) \ "Quotient Property" &  &  & #h(4em) frac(a^(m), a^(n)) & = & a^(m − n) , #h(0.2em) a ≠ 0 \ "Zero Exponent Definition" &  &  & #h(4em) a^(0) & = & 1 , #h(0.2em) a ≠ 0 \ "Quotient to a Power Property" &  &  & #h(4em) attach(( frac(a, b) ), t: m) & = & frac(a^(m), b^(m)) , #h(0.2em) b ≠ 0 \ "Negative Exponent Property" &  &  & #h(4em) a^("−" n) & = & frac(1, a^(n)) , #h(0.2em) a ≠ 0$]
]

We will apply these properties in the next example.

#examplebox("Example 8")[][
Simplify: ⓐ #math.equation(block: false, alt: "x to the power the fraction 1 over 2 times x to the power the fraction 5 over 6")[$x^(frac(1, 2)) · x^(frac(5, 6))$] ⓑ #math.equation(block: false, alt: "open parenthesis z to the power 9 close parenthesis to the power the fraction 2 over 3")[$attach(( z^(9) ), t: frac(2, 3))$] ⓒ #math.equation(block: false, alt: "the fraction x to the power the fraction 1 over 3 over x to the power the fraction 5 over 3 .")[$frac(x^(frac(1, 3)), x^(frac(5, 3))) .$]

#solutionbox[
ⓐ The Product Property tells us that when we multiply the same base, we add the exponents.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "x to the power the fraction 1 over 2 times x to the power the fraction 5 over 6")[$#h(5em) x^(frac(1, 2)) · x^(frac(5, 6))$]]),
  [The bases are the same, so we add the #linebreak() exponents.], [#math.equation(block: false, alt: "x to the power the fraction 1 over 2 plus the fraction 5 over 6")[$#h(5em) x^(frac(1, 2) + frac(5, 6))$]],
  [Add the fractions.], [#math.equation(block: false, alt: "x to the power the fraction 8 over 6")[$#h(5em) x^(frac(8, 6))$]],
  [Simplify the exponent.], [#math.equation(block: false, alt: "x to the power the fraction 4 over 3")[$#h(5em) x^(frac(4, 3))$]],
))

ⓑ The Power Property tells us that when we raise a power to a power, we multiply the exponents.

#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")[$#h(4.5em) attach(( z^(9) ), t: frac(2, 3))$]]),
  [To raise a power to a power, we multiply #linebreak() the exponents.], [#math.equation(block: false, alt: "z to the power 9 times the fraction 2 over 3")[$#h(4.5em) z^(9 · frac(2, 3))$]],
  [Simplify.], [#math.equation(block: false, alt: "z to the power 6")[$#h(4.5em) z^(6)$]],
))

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

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction x to the power the fraction 1 over 3 over x to the power the fraction 5 over 3")[$#h(4em) frac(x^(frac(1, 3)), x^(frac(5, 3)))$]]),
  [], [#math.equation(block: false, alt: "the fraction x to the power the fraction 1 over 3 over x to the power the fraction 5 over 3")[$#h(4em) frac(x^(frac(1, 3)), x^(frac(5, 3)))$]],
  [To divide with the same base, we subtract #linebreak() the exponents.], [#math.equation(block: false, alt: "the fraction 1 over x to the power the fraction 5 over 3 minus the fraction 1 over 3")[$#h(4em) frac(1, x^(frac(5, 3) − frac(1, 3)))$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 1 over x to the power the fraction 4 over 3")[$#h(4em) frac(1, x^(frac(4, 3)))$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "x to the power the fraction 1 over 6 times x to the power the fraction 4 over 3")[$x^(frac(1, 6)) · x^(frac(4, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis x to the power 6 close parenthesis to the power the fraction 4 over 3")[$attach(( x^(6) ), t: frac(4, 3))$] ⓒ #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[
ⓐ #math.equation(block: false, alt: "x to the power the fraction 3 over 2")[$x^(frac(3, 2))$] ⓑ #math.equation(block: false, alt: "x to the power 8")[$x^(8)$] ⓒ #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: "y to the power the fraction 3 over 4 times y to the power the fraction 5 over 8")[$y^(frac(3, 4)) · y^(frac(5, 8))$] ⓑ #math.equation(block: false, alt: "open parenthesis m to the power 9 close parenthesis to the power the fraction 2 over 9")[$attach(( m^(9) ), t: frac(2, 9))$] ⓒ #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: "y to the power the fraction 11 over 8")[$y^(frac(11, 8))$] ⓑ #math.equation(block: false, alt: "m squared")[$m^(2)$] ⓒ #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 example, we will use both the #strong[Product to a Power Property] and then the #strong[Power Property].

#examplebox("Example 9")[][
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 m to the power the fraction 2 over 3 n to the power the fraction 1 over 2 close parenthesis to the power the fraction 3 over 2 .")[$attach(( m^(frac(2, 3)) n^(frac(1, 2)) ), t: frac(3, 2)) .$]

#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")[$#h(4em) attach(( 27 u^(frac(1, 2)) ), t: frac(2, 3))$]]),
  [First we use the Product to a Power #linebreak() 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")[$#h(4em) attach(( 27 ), t: frac(2, 3)) attach(( u^(frac(1, 2)) ), t: frac(2, 3))$]],
  [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")[$#h(4em) attach(( 3^(3) ), t: frac(2, 3)) attach(( u^(frac(1, 2)) ), t: frac(2, 3))$]],
  [To raise a power to a power, we multiply #linebreak() 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")[$#h(4em) ( 3^(2) ) ( u^(frac(1, 3)) )$]],
  [Simplify.], [#math.equation(block: false, alt: "9 u to the power the fraction 1 over 3")[$#h(4em) 9 u^(frac(1, 3))$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis m to the power the fraction 2 over 3 n to the power the fraction 1 over 2 close parenthesis to the power the fraction 3 over 2")[$#h(4em) attach(( m^(frac(2, 3)) n^(frac(1, 2)) ), t: frac(3, 2))$]]),
  [First we use the Product to a Power #linebreak() Property.], [#math.equation(block: false, alt: "open parenthesis m to the power the fraction 2 over 3 close parenthesis to the power the fraction 3 over 2 open parenthesis n to the power the fraction 1 over 2 close parenthesis to the power the fraction 3 over 2")[$#h(4em) attach(( m^(frac(2, 3)) ), t: frac(3, 2)) attach(( n^(frac(1, 2)) ), t: frac(3, 2))$]],
  [To raise a power to a power, we multiply #linebreak() the exponents.], [#math.equation(block: false, alt: "m n to the power the fraction 3 over 4")[$#h(4em) m n^(frac(3, 4))$]],
))
]
]

#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 x to the power the fraction 3 over 4 y to the power the fraction 1 over 2 close parenthesis to the power the fraction 2 over 3 .")[$attach(( x^(frac(3, 4)) y^(frac(1, 2)) ), t: frac(2, 3)) .$]

#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: "x to the power the fraction 1 over 2 y to the power the fraction 1 over 3")[$x^(frac(1, 2)) y^(frac(1, 3))$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #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")[$attach(( 81 n^(frac(2, 5)) ), t: frac(3, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis a to the power the fraction 3 over 2 b to the power the fraction 1 over 2 close parenthesis to the power the fraction 4 over 3 .")[$attach(( a^(frac(3, 2)) b^(frac(1, 2)) ), t: frac(4, 3)) .$]

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

We will use both the #strong[Product Property] and the #strong[Quotient Property] in the next example.

#examplebox("Example 10")[][
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: "open parenthesis the fraction 16 x to the power the fraction 4 over 3 y to the power minus the fraction 5 over 6 over x to the power minus the fraction 2 over 3 y to the power the fraction 1 over 6 close parenthesis to the power the fraction 1 over 2 .")[$attach(( frac(16 #h(0.2em) x^(frac(4, 3)) y^(− frac(5, 6)), x^(− frac(2, 3)) y^(frac(1, 6))) ), t: frac(1, 2)) .$]

#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")[$#h(4em) frac(x^(frac(3, 4)) · x^(− frac(1, 4)), x^(− frac(6, 4)))$]]),
  [Use the Product Property in the numerator, #linebreak() 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")[$#h(4em) frac(x^(frac(2, 4)), x^(− frac(6, 4)))$]],
  [Use the Quotient Property, subtract the #linebreak() exponents.], [#math.equation(block: false, alt: "x to the power the fraction 8 over 4")[$#h(4em) x^(frac(8, 4))$]],
  [Simplify.], [#math.equation(block: false, alt: "x squared")[$#h(4em) x^(2)$]],
))

ⓑ Follow the order of operations to simplify inside the parenthese first.

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis the fraction 16 x to the power the fraction 4 over 3 y to the power minus the fraction 5 over 6 over x to the power minus the fraction 2 over 3 y to the power the fraction 1 over 6 close parenthesis to the power the fraction 1 over 2")[$#h(4em) attach(( frac(16 #h(0.2em) x^(frac(4, 3)) y^(− frac(5, 6)), x^(− frac(2, 3)) y^(frac(1, 6))) ), t: frac(1, 2))$]]),
  [Use the Quotient Property, subtract the #linebreak() exponents.], [#math.equation(block: false, alt: "open parenthesis the fraction 16 x to the power the fraction 6 over 3 over y to the power the fraction 6 over 6 close parenthesis to the power the fraction 1 over 2")[$#h(4em) attach(( frac(16 x^(frac(6, 3)), y^(frac(6, 6))) ), t: frac(1, 2))$]],
  [Simplify.], [#math.equation(block: false, alt: "open parenthesis the fraction 16 x squared over y close parenthesis to the power the fraction 1 over 2")[$#h(4em) attach(( frac(16 x^(2), y) ), t: frac(1, 2))$]],
  [Use the Product to a Power Property, #linebreak() multiply the exponents.], [#math.equation(block: false, alt: "the fraction 4 x over y to the power the fraction 1 over 2")[$#h(4em) frac(4 x, y^(frac(1, 2)))$]],
))
]
]

#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: "open parenthesis the fraction 25 m to the power the fraction 1 over 6 n to the power the fraction 11 over 6 over m to the power the fraction 2 over 3 n to the power minus the fraction 1 over 6 close parenthesis to the power the fraction 1 over 2 .")[$attach(( frac(25 m^(frac(1, 6)) n^(frac(11, 6)), m^(frac(2, 3)) n^(− frac(1, 6))) ), t: frac(1, 2)) .$]

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

#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: "open parenthesis the fraction 27 x to the power the fraction 4 over 5 y to the power the fraction 1 over 6 over x to the power the fraction 1 over 5 y to the power minus the fraction 5 over 6 close parenthesis to the power the fraction 1 over 3 .")[$attach(( frac(27 x^(frac(4, 5)) y^(frac(1, 6)), x^(frac(1, 5)) y^(− frac(5, 6))) ), t: frac(1, 3)) .$]

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with simplifying rational exponents.

- #link("https://openstax.org/l/37RatExpont1")[Review-Rational Exponents]
- #link("https://openstax.org/l/37RatExpont2")[Using Laws of Exponents on Radicals: Properties of Rational Exponents]
]


=== Key Concepts

- #strong[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 ,")[$n ≥ 2 ,$] then #math.equation(block: false, alt: "a to the power the fraction 1 over n equals the nth root of a .")[$a^(frac(1, n)) = root(n, a) .$]
- #strong[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], #linebreak() 
#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")[$a^(frac(m, n)) = attach(( root(n, a) ), t: m)$] and #math.equation(block: false, alt: "a to the power the fraction m over n equals the nth root of a to the power m")[$a^(frac(m, n)) = root(n, a^(m))$]
- #strong[Properties of Exponents]
- If #emph[a, b] are real numbers and #emph[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] #math.equation(block: false, alt: "the fraction a to the power m over a to the power n equals a to the power m minus n , a not equal to 0")[$frac(a^(m), a^(n)) = a^(m − n) , #h(0.2em) a ≠ 0$]
- #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")[$attach(( frac(a, b) ), t: m) = frac(a^(m), b^(m)) , #h(0.2em) b ≠ 0$]
- #strong[Negative Exponent Property] #math.equation(block: false, alt: "a to the power − n equals the fraction 1 over a to the power n , a not equal to 0")[$a^("−" n) = frac(1, a^(n)) , a ≠ 0$]


==== 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")[$z^(frac(1, 4))$]

#solutionbox[
ⓐ #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")[$root(4, z)$]
]

ⓐ #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")[$t^(frac(1, 4))$]

ⓐ #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")[$w^(frac(1, 20))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "the 5th root of u")[$root(5, u)$] ⓑ #math.equation(block: false, alt: "the 9th root of v")[$root(9, v)$] ⓒ #math.equation(block: false, alt: "the 20th root of w")[$root(20, w)$]
]

ⓐ #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")[$j^(frac(1, 25))$]

In the following exercises, write with a rational exponent.

ⓐ #math.equation(block: false, alt: "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")[$root(5, f)$]

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

ⓐ #math.equation(block: false, alt: "the 8th root of r")[$root(8, r)$] ⓑ #math.equation(block: false, alt: "the 10th root of s")[$root(10, s)$] ⓒ #math.equation(block: false, alt: "the 4th root of t")[$root(4, t)$]

ⓐ #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: "2 the 4th root of 6 b")[$2 root(4, 6 b)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis 7 c close parenthesis to the power the fraction 1 over 3")[$attach(( 7 c ), t: frac(1, 3))$] ⓑ #math.equation(block: false, alt: "open parenthesis 12 d close parenthesis to the power the fraction 1 over 7")[$attach(( 12 d ), t: frac(1, 7))$] #linebreak() ⓒ #math.equation(block: false, alt: "2 open parenthesis 6 b close parenthesis to the power the fraction 1 over 4")[$2 attach(( 6 b ), t: frac(1, 4))$]
]

ⓐ #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")[$7 root(5, 3 z)$]

ⓐ #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")[$4 root(6, 36 r)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "open parenthesis 21 p close parenthesis to the power the fraction 1 over 2")[$attach(( 21 p ), t: frac(1, 2))$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 q close parenthesis to the power the fraction 1 over 4")[$attach(( 8 q ), t: frac(1, 4))$] #linebreak() ⓒ #math.equation(block: false, alt: "4 open parenthesis 36 r close parenthesis to the power the fraction 1 over 6")[$4 attach(( 36 r ), t: frac(1, 6))$]
]

ⓐ #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")[$sqrt(3 b)$] ⓒ #math.equation(block: false, alt: "the 8th root of 40 c")[$root(8, 40 c)$]

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")[$64^(frac(1, 2))$]

#solutionbox[
ⓐ 9 ⓑ 5 ⓒ 8
]

ⓐ #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")[$32^(frac(1, 5))$]

ⓐ #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: "625 to the power the fraction 1 over 4")[$625^(frac(1, 4))$]

#solutionbox[
ⓐ 2 ⓑ 4 ⓒ 5
]

ⓐ #math.equation(block: false, alt: "64 to the power the fraction 1 over 3")[$64^(frac(1, 3))$] ⓑ #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")[$81^(frac(1, 4))$]

ⓐ #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")[$attach(( 216 ), t: − frac(1, 3))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−6")[$−6$] ⓑ #math.equation(block: false, alt: "−6")[$−6$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$]
]

ⓐ #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")[$attach(( 1000 ), t: − frac(1, 3))$]

ⓐ #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")[$attach(( 81 ), t: − frac(1, 4))$]

#solutionbox[
ⓐ not real ⓑ #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 −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")[$attach(( 49 ), t: − frac(1, 2))$]

ⓐ #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: "− 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")[$attach(( 36 ), t: − frac(1, 2))$]

#solutionbox[
ⓐ not real ⓑ #math.equation(block: false, alt: "−6")[$−6$] ⓒ #math.equation(block: false, alt: "the fraction 1 over 6")[$frac(1, 6)$]
]

ⓐ #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: "16 to the power minus the fraction 1 over 4")[$16^(− frac(1, 4))$]

ⓐ #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")[$attach(( 100 ), t: − frac(1, 2))$]

#solutionbox[
ⓐ not real ⓑ #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 −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")[$"−" 125^(frac(1, 3))$]

#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: "open parenthesis the cube root of 3 y close parenthesis to the power 7")[$attach(( root(3, 3 y) ), t: 7)$] ⓒ #math.equation(block: false, alt: "the 5th root of open parenthesis the fraction 4 x over 5 y close parenthesis cubed")[$root(5, attach(( frac(4 x, 5 y) ), t: 3))$]

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

ⓐ #math.equation(block: false, alt: "the 4th root of r to the power 7")[$root(4, r^(7))$] ⓑ #math.equation(block: false, alt: "open parenthesis the 5th root of 2 p q close parenthesis cubed")[$attach(( root(5, 2 p q) ), t: 3)$] ⓒ #math.equation(block: false, alt: "the 4th root of open parenthesis the fraction 12 m over 7 n close parenthesis cubed")[$root(4, attach(( frac(12 m, 7 n) ), t: 3))$]

ⓐ #math.equation(block: false, alt: "the 5th root of u squared")[$root(5, u^(2))$] ⓑ #math.equation(block: false, alt: "open parenthesis the cube root of 6 x close parenthesis to the power 5")[$attach(( root(3, 6 x) ), t: 5)$] ⓒ #math.equation(block: false, alt: "the 4th root of open parenthesis the fraction 18 a over 5 b close parenthesis to the power 7")[$root(4, attach(( frac(18 a, 5 b) ), t: 7))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "u to the power the fraction 2 over 5")[$u^(frac(2, 5))$] ⓑ #math.equation(block: false, alt: "open parenthesis 6 x close parenthesis to the power the fraction 5 over 3")[$attach(( 6 x ), t: frac(5, 3))$] ⓒ #math.equation(block: false, alt: "open parenthesis the fraction 18 a over 5 b close parenthesis to the power the fraction 7 over 4")[$attach(( frac(18 a, 5 b) ), t: frac(7, 4))$]
]

ⓐ #math.equation(block: false, alt: "the cube root of a")[$root(3, a)$] ⓑ #math.equation(block: false, alt: "open parenthesis the 4th root of 21 v close parenthesis cubed")[$attach(( root(4, 21 v) ), t: 3)$] ⓒ #math.equation(block: false, alt: "the 4th root of open parenthesis the fraction 2 x y over 5 z close parenthesis squared")[$root(4, attach(( frac(2 x y, 5 z) ), t: 2))$]

In the following exercises, simplify.

ⓐ #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 the fraction −3 over 2")[$81^(frac(−3, 2))$] ⓒ #math.equation(block: false, alt: "open parenthesis −27 close parenthesis to the power the fraction 2 over 3")[$attach(( −27 ), t: frac(2, 3))$]

#solutionbox[
ⓐ 32,768 ⓑ #math.equation(block: false, alt: "the fraction 1 over 729")[$frac(1, 729)$] ⓒ 9
]

ⓐ #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: "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: "open parenthesis −25 close parenthesis to the power the fraction 1 over 2")[$attach(( −25 ), t: frac(1, 2))$]

#solutionbox[
ⓐ 4 ⓑ #math.equation(block: false, alt: "the fraction 1 over 9")[$frac(1, 9)$] ⓒ not real
]

ⓐ #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")[$attach(( −100 ), t: frac(3, 2))$]

ⓐ #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")[$attach(( −9 ), t: frac(3, 2))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−27")[$−27$] ⓑ #math.equation(block: false, alt: "minus the fraction 1 over 27")[$− frac(1, 27)$] ⓒ not real
]

ⓐ #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")[$attach(( −64 ), t: frac(3, 2))$]

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

In the following exercises, simplify. Assume all variables are positive.

ⓐ #math.equation(block: false, alt: "c to the power the fraction 1 over 4 times c to the power the fraction 5 over 8")[$c^(frac(1, 4)) · c^(frac(5, 8))$] ⓑ #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: "the fraction r to the power the fraction 4 over 5 over r to the power the fraction 9 over 5")[$frac(r^(frac(4, 5)), r^(frac(9, 5)))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "c to the power the fraction 7 over 8")[$c^(frac(7, 8))$] ⓑ #math.equation(block: false, alt: "p to the power 9")[$p^(9)$] ⓒ #math.equation(block: false, alt: "the fraction 1 over r")[$frac(1, r)$]
]

ⓐ #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: "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: "the fraction w to the power the fraction 2 over 7 over w to the power the fraction 9 over 7")[$frac(w^(frac(2, 7)), w^(frac(9, 7)))$]

ⓐ #math.equation(block: false, alt: "y to the power the fraction 1 over 2 times y to the power the fraction 3 over 4")[$y^(frac(1, 2)) · y^(frac(3, 4))$] ⓑ #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: "the fraction m to the power the fraction 5 over 8 over m to the power the fraction 13 over 8")[$frac(m^(frac(5, 8)), m^(frac(13, 8)))$]

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

ⓐ #math.equation(block: false, alt: "q to the power the fraction 2 over 3 times q to the power the fraction 5 over 6")[$q^(frac(2, 3)) · q^(frac(5, 6))$] ⓑ #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: "the fraction n to the power the fraction 3 over 5 over n to the power the fraction 8 over 5")[$frac(n^(frac(3, 5)), n^(frac(8, 5)))$]

ⓐ #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 a to the power the fraction 1 over 3 b to the power the fraction 2 over 3 close parenthesis to the power the fraction 3 over 2")[$attach(( a^(frac(1, 3)) b^(frac(2, 3)) ), t: frac(3, 2))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "81 q squared")[$81 q^(2)$] ⓑ #math.equation(block: false, alt: "a to the power the fraction 1 over 2 b")[$a^(frac(1, 2)) b$]
]

ⓐ #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")[$attach(( 64 s^(frac(3, 7)) ), t: frac(1, 6))$] ⓑ #math.equation(block: false, alt: "open parenthesis m to the power the fraction 4 over 3 n to the power the fraction 1 over 2 close parenthesis to the power the fraction 3 over 4")[$attach(( m^(frac(4, 3)) n^(frac(1, 2)) ), t: frac(3, 4))$]

ⓐ #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")[$attach(( 16 #h(0.2em) u^(frac(1, 3)) ), t: frac(3, 4))$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 p to the power the fraction 1 over 3 q to the power the fraction 1 over 2 close parenthesis to the power the fraction 3 over 2")[$attach(( 4 #h(0.2em) p^(frac(1, 3)) q^(frac(1, 2)) ), t: frac(3, 2))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "8 u to the power the fraction 1 over 4")[$8 u^(frac(1, 4))$] ⓑ #math.equation(block: false, alt: "8 p to the power the fraction 1 over 2 q to the power the fraction 3 over 4")[$8 p^(frac(1, 2)) q^(frac(3, 4))$]
]

ⓐ #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")[$attach(( 625 #h(0.2em) n^(frac(8, 3)) ), t: frac(3, 4))$] ⓑ #math.equation(block: false, alt: "open parenthesis 9 x to the power the fraction 2 over 5 y to the power the fraction 3 over 5 close parenthesis to the power the fraction 5 over 2")[$attach(( 9 #h(0.2em) x^(frac(2, 5)) y^(frac(3, 5)) ), t: frac(5, 2))$]

ⓐ #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: "open parenthesis the fraction 36 s to the power the fraction 1 over 5 t to the power minus the fraction 3 over 2 over s to the power minus the fraction 9 over 5 t to the power the fraction 1 over 2 close parenthesis to the power the fraction 1 over 2")[$attach(( frac(36 #h(0.2em) s^(frac(1, 5)) t^(− frac(3, 2)), s^(− frac(9, 5)) t^(frac(1, 2))) ), t: frac(1, 2))$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "r to the power the fraction 7 over 2")[$r^(frac(7, 2))$] ⓑ #math.equation(block: false, alt: "the fraction 6 s over t")[$frac(6 s, t)$]
]

ⓐ #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: "open parenthesis the fraction 27 b to the power the fraction 2 over 3 c to the power minus the fraction 5 over 2 over b to the power minus the fraction 7 over 3 c to the power the fraction 1 over 2 close parenthesis to the power the fraction 1 over 3")[$attach(( frac(27 #h(0.2em) b^(frac(2, 3)) c^(− frac(5, 2)), b^(− frac(7, 3)) c^(frac(1, 2))) ), t: frac(1, 3))$]

ⓐ #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: "open parenthesis the fraction 8 x to the power the fraction 5 over 3 y to the power minus the fraction 1 over 2 over 27 x to the power minus the fraction 4 over 3 y to the power the fraction 5 over 2 close parenthesis to the power the fraction 1 over 3")[$attach(( frac(8 #h(0.2em) x^(frac(5, 3)) #h(0.2em) y^(− frac(1, 2)), 27 #h(0.2em) x^(− frac(4, 3)) #h(0.2em) y^(frac(5, 2))) ), t: frac(1, 3))$]

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

ⓐ #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: "open parenthesis the fraction 16 m to the power the fraction 1 over 5 n to the power the fraction 3 over 2 over 81 m to the power the fraction 9 over 5 n to the power minus the fraction 1 over 2 close parenthesis to the power the fraction 1 over 4")[$attach(( frac(16 #h(0.2em) m^(frac(1, 5)) #h(0.2em) n^(frac(3, 2)), 81 #h(0.2em) m^(frac(9, 5)) #h(0.2em) n^(− frac(1, 2))) ), t: frac(1, 4))$]


==== 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.

#solutionbox[
Answers will vary.
]

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


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions with a to the power of 1 divided by n.”, “simplify expression with a to the power of m divided by n”, and “use the laws of exponents to simplify expression with rational exponents”. The other columns are left blank so that the learner may indicate their mastery level for each topic.], alt: "This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions with a to the power of 1 divided by n.”, “simplify expression with a to the power of m divided by n”, and “use the laws of exponents to simplify expression with rational exponents”. The other columns are left blank so that the learner may indicate their mastery level for each topic.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?