Lynn Marecek; Andrea Honeycutt Mathis

8.3 Simplify Rational Exponents

Learning Objectives

By the end of this section, you will be able to:

Simplify expressions with $a^{\frac{1}{n}}$
Simplify expressions with $a^{\frac{m}{n}}$
Use the properties of exponents to simplify expressions with rational exponents

Be Prepared 8.7

Before you get started, take this readiness quiz.

Add: $\frac{7}{15} + \frac{5}{12} .$
If you missed this problem, review Example 1.28.

Be Prepared 8.8

Simplify: ${(4 x^{2} y^{5})}^{3} .$
If you missed this problem, review Example 5.18.

Be Prepared 8.9

Simplify: $5^{−3} .$
If you missed this problem, review Example 5.14.

Simplify Expressions with $a^{\frac{1}{n}}$

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 Power Property for Exponents says that ${(a^{m})}^{n} = a^{m \cdot n}$ when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number p such that ${(8^{p})}^{3} = 8 .$ We will use the Power Property of Exponents to find the value of p.

$\begin{matrix} {(8^{p})}^{3} & = & 8 \\ Multiply the exponents on the left. & 8^{3 p} & = & 8 \\ Write the exponent 1 on the right. & 8^{3 p} & = & 8^{1} \\ Since the bases are the same, the exponents must be equal. & 3 p & = & 1 \\ Solve for p . & p & = & \frac{1}{3} \end{matrix}$

So ${(8^{\frac{1}{3}})}^{3} = 8 .$ But we know also ${(\sqrt[3]{8})}^{3} = 8 .$ Then it must be that $8^{\frac{1}{3}} = \sqrt[3]{8} .$

This same logic can be used for any positive integer exponent n to show that $a^{\frac{1}{n}} = \sqrt[n]{a} .$

Rational Exponent $a^{\frac{1}{n}}$

If $\sqrt[n]{a}$ is a real number and $n \geq 2,$ then

a^{\frac{1}{n}} = \sqrt[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.

Example 8.26

Write as a radical expression: ⓐ $x^{\frac{1}{2}}$ ⓑ $y^{\frac{1}{3}}$ ⓒ $z^{\frac{1}{4}} .$

Solution

We want to write each expression in the form $\sqrt[n]{a} .$

ⓐ

x^{\frac{1}{2}}

The denominator of the rational exponent is 2, so
the index of the radical is 2. We do not show the
index when it is 2.

\sqrt{x}

ⓑ

y^{\frac{1}{3}}

The denominator of the exponent is 3, so the
index is 3.

\sqrt[3]{y}

ⓒ

z^{\frac{1}{4}}

The denominator of the exponent is 4, so the
index is 4.

\sqrt[4]{z}

Try It 8.51

Write as a radical expression: ⓐ $t^{\frac{1}{2}}$ ⓑ $m^{\frac{1}{3}}$ ⓒ $r^{\frac{1}{4}} .$

Try It 8.52

Write as a radial expression: ⓐ $b^{\frac{1}{6}}$ ⓑ $z^{\frac{1}{5}}$ ⓒ $p^{\frac{1}{4}} .$

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.

Example 8.27

Write with a rational exponent: ⓐ $\sqrt{5 y}$ ⓑ $\sqrt[3]{4 x}$ ⓒ $3 \sqrt[4]{5 z} .$

Solution

We want to write each radical in the form $a^{\frac{1}{n}} .$

ⓐ

\sqrt{5 y}

No index is shown, so it is 2.
The denominator of the exponent will be 2.

{(5 y)}^{\frac{1}{2}}

Put parentheses around the entire
expression

5 y .

ⓑ

\sqrt[3]{4 x}

The index is 3, so the denominator of the
exponent is 3. Include parentheses

(4 x) .

{(4 x)}^{\frac{1}{3}}

ⓒ

3 \sqrt[4]{5 z}

The index is 4, so the denominator of the
exponent is 4. Put parentheses only around
the

5 z

since 3 is not under the radical sign.

3 {(5 z)}^{\frac{1}{4}}

Try It 8.53

Write with a rational exponent: ⓐ $\sqrt{10 m}$ ⓑ $\sqrt[5]{3 n}$ ⓒ $3 \sqrt[4]{6 y} .$

Try It 8.54

Write with a rational exponent: ⓐ $\sqrt[7]{3 k}$ ⓑ $\sqrt[4]{5 j}$ ⓒ $8 \sqrt[3]{2 a} .$

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

Example 8.28

Simplify: ⓐ $25^{\frac{1}{2}}$ ⓑ $64^{\frac{1}{3}}$ ⓒ $256^{\frac{1}{4}} .$

Solution

ⓐ

25^{\frac{1}{2}}

Rewrite as a square root.

\sqrt{25}

Simplify.

5

ⓑ

64^{\frac{1}{3}}

Rewrite as a cube root.

\sqrt[3]{64}

Recognize 64 is a perfect cube.

\sqrt[3]{4^{3}}

Simplify.

4

ⓒ

256^{\frac{1}{4}}

Rewrite as a fourth root.

\sqrt[4]{256}

Recognize 256 is a perfect fourth power.

\sqrt[4]{4^{4}}

Simplify.

4

Try It 8.55

Simplify: ⓐ $36^{\frac{1}{2}}$ ⓑ $8^{\frac{1}{3}}$ ⓒ $16^{\frac{1}{4}} .$

Try It 8.56

Simplify: ⓐ $100^{\frac{1}{2}}$ ⓑ $27^{\frac{1}{3}}$ ⓒ $81^{\frac{1}{4}} .$

Be careful of the placement of the negative signs in the next example. We will need to use the property $a^{− n} = \frac{1}{a^{n}}$ in one case.

Example 8.29

Simplify: ⓐ ${(−16)}^{\frac{1}{4}}$ ⓑ $− 16^{\frac{1}{4}}$ ⓒ ${(16)}^{- \frac{1}{4}} .$

Solution

ⓐ

{(−16)}^{\frac{1}{4}}

Rewrite as a fourth root.

\sqrt[4]{−16}

\sqrt[4]{{(−2)}^{4}}

Simplify.

No real solution.

ⓑ

− 16^{\frac{1}{4}}

The exponent only applies to the 16.
Rewrite as a fouth root.

− \sqrt[4]{16}

Rewrite 16 as

2^{4} .

− \sqrt[4]{2^{4}}

Simplify.

−2

ⓒ

{(16)}^{- \frac{1}{4}}

Rewrite using the property

a^{− n} = \frac{1}{a^{n}} .

\frac{1}{{(16)}^{\frac{1}{4}}}

Rewrite as a fourth root.

\frac{1}{\sqrt[4]{16}}

Rewrite 16 as

2^{4} .

\frac{1}{\sqrt[4]{2^{4}}}

Simplify.

\frac{1}{2}

Try It 8.57

Simplify: ⓐ ${(−64)}^{- \frac{1}{2}}$ ⓑ $− 64^{\frac{1}{2}}$ ⓒ ${(64)}^{- \frac{1}{2}} .$

Try It 8.58

Simplify: ⓐ ${(−256)}^{\frac{1}{4}}$ ⓑ $− 256^{\frac{1}{4}}$ ⓒ ${(256)}^{- \frac{1}{4}} .$

Simplify Expressions with $a^{\frac{m}{n}}$

We can look at $a^{\frac{m}{n}}$ in two ways. Remember the Power Property tells us to multiply the exponents and so ${(a^{\frac{1}{n}})}^{m}$ and ${(a^{m})}^{^{\frac{1}{n}}}$ both equal $a^{\frac{m}{n}} .$ If we write these expressions in radical form, we get

a^{\frac{m}{n}} = {(a^{\frac{1}{n}})}^{m} = {(\sqrt[n]{a})}^{m} and a^{\frac{m}{n}} = {(a^{m})}^{^{\frac{1}{n}}} = \sqrt[n]{a^{m}}

This leads us to the following definition.

Rational Exponent $a^{\frac{m}{n}}$

For any positive integers m and n,

a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m} and a^{\frac{m}{n}} = \sqrt[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.

Example 8.30

Write with a rational exponent: ⓐ $\sqrt{y^{3}}$ ⓑ ${(\sqrt[3]{2 x})}^{4}$ ⓒ $\sqrt{{(\frac{3 a}{4 b})}^{3}} .$

Solution

We want to use $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ to write each radical in the form $a^{\frac{m}{n}} .$

ⓐ

ⓑ

ⓒ

Try It 8.59

Write with a rational exponent: ⓐ $\sqrt{x^{5}}$ ⓑ ${(\sqrt[4]{3 y})}^{3}$ ⓒ $\sqrt{{(\frac{2 m}{3 n})}^{5}} .$

Try It 8.60

Write with a rational exponent: ⓐ $\sqrt[5]{a^{2}}$ ⓑ ${(\sqrt[3]{5 a b})}^{5}$ ⓒ $\sqrt{{(\frac{7 x y}{z})}^{3}} .$

Remember that $a^{− n} = \frac{1}{a^{n}} .$ The negative sign in the exponent does not change the sign of the expression.

Example 8.31

Simplify: ⓐ $125^{\frac{2}{3}}$ ⓑ $16^{- \frac{3}{2}}$ ⓒ $32^{- \frac{2}{5}} .$

Solution

We will rewrite the expression as a radical first using the defintion, $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m} .$ 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.

ⓐ

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.

{(\sqrt[3]{125})}^{2}

Simplify.

{(5)}^{2}

25

ⓑ We will rewrite each expression first using $a^{− n} = \frac{1}{a^{n}}$ and then change to radical form.

16^{- \frac{3}{2}}

Rewrite using

a^{− n} = \frac{1}{a^{n}}

\frac{1}{16^{\frac{3}{2}}}

Change to radical form. The power of the radical is the
numerator of the exponent, 3. The index is the denominator
of the exponent, 2.

\frac{1}{{(\sqrt{16})}^{3}}

Simplify.

\frac{1}{4^{3}}

\frac{1}{64}

ⓒ

32^{- \frac{2}{5}}

Rewrite using

a^{− n} = \frac{1}{a^{n}} .

\frac{1}{32^{\frac{2}{5}}}

Change to radical form.

\frac{1}{{(\sqrt[5]{32})}^{2}}

Rewrite the radicand as a power.

\frac{1}{{(\sqrt[5]{2^{5}})}^{2}}

Simplify.

\frac{1}{2^{2}}

\frac{1}{4}

Try It 8.61

Simplify: ⓐ $27^{\frac{2}{3}}$ ⓑ $81^{- \frac{3}{2}}$ ⓒ $16^{- \frac{3}{4}} .$

Try It 8.62

Simplify: ⓐ $4^{\frac{3}{2}}$ ⓑ $27^{- \frac{2}{3}}$ ⓒ $625^{- \frac{3}{4}} .$

Example 8.32

Simplify: ⓐ $− 25^{\frac{3}{2}}$ ⓑ $− 25^{- \frac{3}{2}}$ ⓒ ${(−25)}^{\frac{3}{2}} .$

Solution

ⓐ

− 25^{\frac{3}{2}}

Rewrite in radical form.

− {(\sqrt{25})}^{3}

Simplify the radical.

− {(5)}^{3}

Simplify.

−125

ⓑ

− 25^{- \frac{3}{2}}

Rewrite using

a^{− n} = \frac{1}{a^{n}} .

− (\frac{1}{25^{\frac{3}{2}}})

Rewrite in radical form.

− (\frac{1}{{(\sqrt{25})}^{3}})

Simplify the radical.

− (\frac{1}{{(5)}^{3}})

Simplify.

- \frac{1}{125}

ⓒ

{(−25)}^{\frac{3}{2}}

Rewrite in radical form.

{(\sqrt{−25})}^{3}

There is no real number whose square root

is −25 .

Not a real number.

Try It 8.63

Simplify: ⓐ ${−16}^{\frac{3}{2}}$ ⓑ ${−16}^{- \frac{3}{2}}$ ⓒ ${(−16)}^{- \frac{3}{2}} .$

Try It 8.64

Simplify: ⓐ ${−81}^{\frac{3}{2}}$ ⓑ ${−81}^{- \frac{3}{2}}$ ⓒ ${(−81)}^{- \frac{3}{2}} .$

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.

Properties of Exponents

If a and b are real numbers and m and n are rational numbers, then

$\begin{matrix} Product Property & a^{m} \cdot a^{n} & = & a^{m + n} \\ Power Property & {(a^{m})}^{n} & = & a^{m \cdot n} \\ Product to a Power & {(a b)}^{m} & = & a^{m} b^{m} \\ Quotient Property & \frac{a^{m}}{a^{n}} & = & a^{m - n}, a \neq 0 \\ Zero Exponent Definition & a^{0} & = & 1, a \neq 0 \\ Quotient to a Power Property & {(\frac{a}{b})}^{m} & = & \frac{a^{m}}{b^{m}}, b \neq 0 \\ Negative Exponent Property & a^{− n} & = & \frac{1}{a^{n}}, a \neq 0 \end{matrix}$

We will apply these properties in the next example.

Example 8.33

Simplify: ⓐ $x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}$ ⓑ ${(z^{9})}^{\frac{2}{3}}$ ⓒ $\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}} .$

Solution

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

x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}

The bases are the same, so we add the
exponents.

x^{\frac{1}{2} + \frac{5}{6}}

Add the fractions.

x^{\frac{8}{6}}

Simplify the exponent.

x^{\frac{4}{3}}

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

{(z^{9})}^{\frac{2}{3}}

To raise a power to a power, we multiply
the exponents.

z^{9 \cdot \frac{2}{3}}

Simplify.

z^{6}

\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}

\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}

To divide with the same base, we subtract
the exponents.

\frac{1}{x^{\frac{5}{3} - \frac{1}{3}}}

Simplify.

\frac{1}{x^{\frac{4}{3}}}

Try It 8.65

Simplify: ⓐ $x^{\frac{1}{6}} \cdot x^{\frac{4}{3}}$ ⓑ ${(x^{6})}^{\frac{4}{3}}$ ⓒ $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}} .$

Try It 8.66

Simplify: ⓐ $y^{\frac{3}{4}} \cdot y^{\frac{5}{8}}$ ⓑ ${(m^{9})}^{\frac{2}{9}}$ ⓒ $\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}} .$

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

Example 8.34

Simplify: ⓐ ${(27 u^{\frac{1}{2}})}^{\frac{2}{3}}$ ⓑ ${(m^{\frac{2}{3}} n^{\frac{1}{2}})}^{\frac{3}{2}} .$

Solution

ⓐ

{(27 u^{\frac{1}{2}})}^{\frac{2}{3}}

First we use the Product to a Power
Property.

{(27)}^{\frac{2}{3}} {(u^{\frac{1}{2}})}^{\frac{2}{3}}

Rewrite 27 as a power of 3.

{(3^{3})}^{\frac{2}{3}} {(u^{\frac{1}{2}})}^{\frac{2}{3}}

To raise a power to a power, we multiply
the exponents.

(3^{2}) (u^{\frac{1}{3}})

Simplify.

9 u^{\frac{1}{3}}

ⓑ

{(m^{\frac{2}{3}} n^{\frac{1}{2}})}^{\frac{3}{2}}

First we use the Product to a Power
Property.

{(m^{\frac{2}{3}})}^{\frac{3}{2}} {(n^{\frac{1}{2}})}^{\frac{3}{2}}

To raise a power to a power, we multiply
the exponents.

m n^{\frac{3}{4}}

Try It 8.67

Simplify: ⓐ ${(32 x^{\frac{1}{3}})}^{\frac{3}{5}}$ ⓑ ${(x^{\frac{3}{4}} y^{\frac{1}{2}})}^{\frac{2}{3}} .$

Try It 8.68

Simplify: ⓐ ${(81 n^{\frac{2}{5}})}^{\frac{3}{2}}$ ⓑ ${(a^{\frac{3}{2}} b^{\frac{1}{2}})}^{\frac{4}{3}} .$

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

Example 8.35

Simplify: ⓐ $\frac{x^{\frac{3}{4}} \cdot x^{- \frac{1}{4}}}{x^{- \frac{6}{4}}}$ ⓑ ${(\frac{16 x^{\frac{4}{3}} y^{- \frac{5}{6}}}{x^{- \frac{2}{3}} y^{\frac{1}{6}}})}^{\frac{1}{2}} .$

Solution

ⓐ

\frac{x^{\frac{3}{4}} \cdot x^{- \frac{1}{4}}}{x^{- \frac{6}{4}}}

Use the Product Property in the numerator,
add the exponents.

\frac{x^{\frac{2}{4}}}{x^{- \frac{6}{4}}}

Use the Quotient Property, subtract the
exponents.

x^{\frac{8}{4}}

Simplify.

x^{2}

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

{(\frac{16 x^{\frac{4}{3}} y^{- \frac{5}{6}}}{x^{- \frac{2}{3}} y^{\frac{1}{6}}})}^{\frac{1}{2}}

Use the Quotient Property, subtract the
exponents.

{(\frac{16 x^{\frac{6}{3}}}{y^{\frac{6}{6}}})}^{\frac{1}{2}}

Simplify.

{(\frac{16 x^{2}}{y})}^{\frac{1}{2}}

Use the Product to a Power Property,
multiply the exponents.

\frac{4 x}{y^{\frac{1}{2}}}

Try It 8.69

Simplify: ⓐ $\frac{m^{\frac{2}{3}} \cdot m^{- \frac{1}{3}}}{m^{- \frac{5}{3}}}$ ⓑ ${(\frac{25 m^{\frac{1}{6}} n^{\frac{11}{6}}}{m^{\frac{2}{3}} n^{- \frac{1}{6}}})}^{\frac{1}{2}} .$

Try It 8.70

Simplify: ⓐ $\frac{u^{\frac{4}{5}} \cdot u^{- \frac{2}{5}}}{u^{- \frac{13}{5}}}$ ⓑ ${(\frac{27 x^{\frac{4}{5}} y^{\frac{1}{6}}}{x^{\frac{1}{5}} y^{- \frac{5}{6}}})}^{\frac{1}{3}} .$

Media

Access these online resources for additional instruction and practice with simplifying rational exponents.

Section 8.3 Exercises

Practice Makes Perfect

Simplify expressions with $a^{\frac{1}{n}}$

In the following exercises, write as a radical expression.

119.

120.

121.

122.

In the following exercises, write with a rational exponent.

123.

124.

125.

126.

127.

128.

In the following exercises, simplify.

129.

130.

131.

132.

133.

134.

135.

136.

137.

138.

139.

140.

Simplify Expressions with $a^{\frac{m}{n}}$

In the following exercises, write with a rational exponent.

141.

142.

143.

144.

In the following exercises, simplify.

145.

146.

147.

148.

149.

150.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

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

151.

152.

153.

154.

155.

ⓐ ${(27 q^{\frac{3}{2}})}^{\frac{4}{3}}$ ⓑ ${(a^{\frac{1}{3}} b^{\frac{2}{3}})}^{\frac{3}{2}}$

156.

ⓐ ${(64 s^{\frac{3}{7}})}^{\frac{1}{6}}$ ⓑ ${(m^{\frac{4}{3}} n^{\frac{1}{2}})}^{\frac{3}{4}}$

157.

ⓐ ${(16 u^{\frac{1}{3}})}^{\frac{3}{4}}$ ⓑ ${(4 p^{\frac{1}{3}} q^{\frac{1}{2}})}^{\frac{3}{2}}$

158.

ⓐ ${(625 n^{\frac{8}{3}})}^{\frac{3}{4}}$ ⓑ ${(9 x^{\frac{2}{5}} y^{\frac{3}{5}})}^{\frac{5}{2}}$

159.

ⓐ $\frac{r^{\frac{5}{2}} \cdot r^{- \frac{1}{2}}}{r^{- \frac{3}{2}}}$ ⓑ ${(\frac{36 s^{\frac{1}{5}} t^{- \frac{3}{2}}}{s^{- \frac{9}{5}} t^{\frac{1}{2}}})}^{\frac{1}{2}}$

160.

ⓐ $\frac{a^{\frac{3}{4}} \cdot a^{- \frac{1}{4}}}{a^{- \frac{10}{4}}}$ ⓑ ${(\frac{27 b^{\frac{2}{3}} c^{- \frac{5}{2}}}{b^{- \frac{7}{3}} c^{\frac{1}{2}}})}^{\frac{1}{3}}$

161.

ⓐ $\frac{c^{\frac{5}{3}} \cdot c^{- \frac{1}{3}}}{c^{- \frac{2}{3}}}$ ⓑ ${(\frac{8 x^{\frac{5}{3}} y^{- \frac{1}{2}}}{27 x^{- \frac{4}{3}} y^{\frac{5}{2}}})}^{\frac{1}{3}}$

162.

ⓐ $\frac{m^{\frac{7}{4}} \cdot m^{- \frac{5}{4}}}{m^{- \frac{2}{4}}}$ ⓑ ${(\frac{16 m^{\frac{1}{5}} n^{\frac{3}{2}}}{81 m^{\frac{9}{5}} n^{- \frac{1}{2}}})}^{\frac{1}{4}}$

Writing Exercises

163.

Show two different algebraic methods to simplify $4^{\frac{3}{2}} .$ Explain all your steps.

164.

Explain why the expression ${(- 16)}^{\frac{3}{2}}$ cannot be evaluated.

Self Check

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

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

8.3 Simplify Rational Exponents