Lynn Marecek; MaryAnne Anthony-Smith

9.8 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 Laws of Exponents to simply expressions with rational exponents

Be Prepared 9.8

Before you get started, take this readiness quiz.

Add: $\frac{7}{15} + \frac{5}{12}$ .
If you missed this problem, review Example 1.81.
Simplify: ${(4 x^{2} y^{5})}^{3}$ .
If you missed this problem, review Example 6.24.
Simplify: $5^{−3}$ .
If you missed this problem, review Example 6.89.

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.

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

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 $n \geq 2$ , $a^{\frac{1}{n}} = \sqrt[n]{a}$ .

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 9.103

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 $\sqrt[n]{a}$ .

ⓐ
$\begin{matrix} x^{\frac{1}{2}} \\ \begin{matrix} The denominator of the exponent is 2, so \\ the index of the radical is 2. We do not \\ show the index when it is 2. \end{matrix} & \sqrt{x} \end{matrix}$

ⓑ
$\begin{matrix} y^{\frac{1}{3}} \\ \begin{matrix} The denominator of the exponent is 3, so \\ the index is 3. \end{matrix} & \sqrt[3]{y} \end{matrix}$

ⓒ
$\begin{matrix} z^{\frac{1}{4}} \\ \begin{matrix} The denominator of the exponent is 4, so \\ the index is 4. \end{matrix} & \sqrt[4]{z} \end{matrix}$

Try It 9.205

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

Try It 9.206

Write as a radial expression: ⓐ $b^{\frac{1}{2}}$ ⓑ $z^{\frac{1}{3}}$ ⓒ $p^{\frac{1}{4}}$ .

Example 9.104

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

Solution

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

ⓐ
$\begin{matrix} \sqrt{x} \\ \begin{matrix} No index is shown, so it is 2. \\ The denominator of the exponent will be 2. \end{matrix} & x^{\frac{1}{2}} \end{matrix}$

ⓑ
$\begin{matrix} \sqrt[3]{y} \\ \begin{matrix} The index is 3, so the denominator of the \\ exponent is 3. \end{matrix} & y^{\frac{1}{3}} \end{matrix}$

ⓒ
$\begin{matrix} \sqrt[4]{z} \\ \begin{matrix} The index is 4, so the denominator of the \\ exponent is 4. \end{matrix} & z^{\frac{1}{4}} \end{matrix}$

Try It 9.207

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

Try It 9.208

Write with a rational exponent: ⓐ $\sqrt{v}$ ⓑ $\sqrt[3]{p}$ ⓒ $\sqrt[4]{p}$ .

Example 9.105

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 $a^{\frac{1}{n}}$ .

ⓐ
$\begin{matrix} \sqrt{5 y} \\ \begin{matrix} No index is shown, so it is 2. \\ The denominator of the exponent will be 2. \end{matrix} & {(5 y)}^{\frac{1}{2}} \end{matrix}$
ⓑ
$\begin{matrix} \sqrt[3]{4 x} \\ \begin{matrix} The index is 3, so the denominator of the \\ exponent is 3. \end{matrix} & {(4 x)}^{\frac{1}{3}} \end{matrix}$
ⓒ
$\begin{matrix} 3 \sqrt[4]{5 z} \\ \begin{matrix} The index is 4, so the denominator of the \\ exponent is 4. \end{matrix} & 3 {(5 z)}^{\frac{1}{4}} \end{matrix}$

Try It 9.209

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

Try It 9.210

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 9.106

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

Solution

ⓐ
$\begin{matrix} 25^{\frac{1}{2}} \\ Rewrite as a square root. & \sqrt{25} \\ Simplify. & 5 \end{matrix}$

ⓑ
$\begin{matrix} 64^{\frac{1}{3}} \\ Rewrite as a cube root. & \sqrt[3]{64} \\ Recognize 64 is a perfect cube. & \sqrt[3]{4^{3}} \\ Simplify. & 4 \end{matrix}$

ⓒ
$\begin{matrix} 256^{\frac{1}{4}} \\ Rewrite as a fourth root. & \sqrt[4]{256} \\ Recognize 256 is a perfect fourth power. & \sqrt[4]{4^{4}} \\ Simplify. & 4 \end{matrix}$

Try It 9.211

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

Try It 9.212

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 9.107

Simplify: ⓐ ${(−64)}^{\frac{1}{3}}$ ⓑ $− 64^{\frac{1}{3}}$ ⓒ ${(64)}^{- \frac{1}{3}}$ .

Solution

ⓐ
$\begin{matrix} {(−64)}^{\frac{1}{3}} \\ Rewrite as a cube root. & \sqrt[3]{−64} \\ Rewrite −64 as a perfect cube. & \sqrt[3]{{(−4)}^{3}} \\ Simplify. & −4 \end{matrix}$

ⓑ
$\begin{matrix} − 64^{\frac{1}{3}} \\ The exponent applies only to the 64. & − (64^{\frac{1}{3}}) \\ Rewrite as a cube root. & − \sqrt[3]{64} \\ Rewrite 64 as 4^{3} . & − \sqrt[3]{4^{3}} \\ Simplify. & −4 \end{matrix}$

ⓒ
$\begin{matrix} {(64)}^{- \frac{1}{3}} \\ \begin{matrix} Rewrite as a fraction with \\ a positive exponent, using \\ the property, a^{− n} = \frac{1}{a^{n}} . \\ Write as a cube root. \end{matrix} & \frac{1}{\sqrt[3]{64}} \\ Rewrite 64 as 4^{3} . & \frac{1}{\sqrt[3]{4^{3}}} \\ Simplify. & \frac{1}{4} \end{matrix}$

Try It 9.213

Simplify: ⓐ ${(−125)}^{\frac{1}{3}}$ ⓑ $− 125^{\frac{1}{3}}$ ⓒ ${(125)}^{- \frac{1}{3}}$ .

Try It 9.214

Simplify: ⓐ ${(−32)}^{\frac{1}{5}}$ ⓑ $− 32^{\frac{1}{5}}$ ⓒ ${(32)}^{- \frac{1}{5}}$ .

Example 9.108

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

Solution

ⓐ
$\begin{matrix} {(−16)}^{\frac{1}{4}} \\ Rewrite as a fourth root. & \sqrt[4]{−16} \\ There is no real number whose fourth power is −16 . \end{matrix}$

ⓑ
$\begin{matrix} − 16^{\frac{1}{4}} \\ \begin{matrix} The exponent only applies to the 16. \\ Rewrite as a fourth root. \end{matrix} & − \sqrt[4]{16} \\ Rewrite 16 as 2^{4} . & − \sqrt[4]{2^{4}} \\ Simplify. & −2 \end{matrix}$

ⓒ
$\begin{matrix} {(16)}^{- \frac{1}{4}} \\ Rewrite using the property a^{− n} = \frac{1}{a^{n}} . & \frac{1}{{(16)}^{\frac{1}{4}}} \\ Rewrite as a fourth root. & \frac{1}{\sqrt[4]{16}} \\ Rewrite 16 as 2^{4} . & \frac{1}{\sqrt[4]{2^{4}}} \\ Simplify. & \frac{1}{2} \end{matrix}$

Try It 9.215

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

Try It 9.216

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

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

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

Suppose we raise $a^{\frac{1}{n}}$ to the power m.

$\begin{matrix} {(a^{\frac{1}{n}})}^{m} \\ Multiply the exponents. & a^{\frac{1}{n}}^{\cdot m} \\ Simplify. & a^{\frac{m}{n}} \\ So a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m} . \end{matrix}$

Now suppose we take $a^{m}$ to the $\frac{1}{n}$ power.

$\begin{matrix} {(a^{m})}^{\frac{1}{n}} \\ Multiply the exponents. & a^{m \cdot \frac{1}{n}} \\ Simplify. & a^{\frac{m}{n}} \\ So a^{\frac{m}{n}} = \sqrt[n]{a^{m}} also. \end{matrix}$

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.

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

For any positive integers m and n,

\begin{matrix} a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m} \\ a^{\frac{m}{n}} = \sqrt[n]{a^{m}} \end{matrix}

Example 9.109

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

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 9.217

Write with a rational exponent: ⓐ $\sqrt{x^{5}}$ ⓑ $\sqrt[4]{z^{3}}$ ⓒ $\sqrt[5]{y^{2}}$ .

Try It 9.218

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

Example 9.110

Simplify: ⓐ $9^{\frac{3}{2}}$ ⓑ $125^{\frac{2}{3}}$ ⓒ $81^{\frac{3}{4}}$ .

Solution

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

ⓐ
$\begin{matrix} 9^{\frac{3}{2}} \\ \begin{matrix} The power of the radical is the numerator \\ of the exponent, 3. Since the denominator \\ of the exponent is 2, this is a square root. \end{matrix} & {(\sqrt{9})}^{3} \\ Simplify. & {(3)}^{3} \\ 27 \end{matrix}$
ⓑ
$\begin{matrix} 125^{\frac{2}{3}} \\ \begin{matrix} The power of the radical is the numerator \\ of the exponent, 2. The index of the radical \\ is the denominator of the exponent, 3. \end{matrix} & {(\sqrt[3]{125})}^{2} \\ Simplify. & {(5)}^{2} \\ 25 \end{matrix}$
ⓒ
$\begin{matrix} 81^{\frac{3}{4}} \\ \begin{matrix} The power of the radical is the numerator \\ of the exponent, 3. The index of the radical \\ is the denominator of the exponent, 4. \end{matrix} & {(\sqrt[4]{81})}^{3} \\ Simplify. & {(3)}^{3} \\ 27 \end{matrix}$

Try It 9.219

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

Try It 9.220

Simplify: ⓐ $8^{\frac{5}{3}}$ ⓑ $81^{\frac{3}{2}}$ ⓒ $16^{\frac{3}{4}}$ .

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

Example 9.111

Simplify: ⓐ $16^{- \frac{3}{2}}$ ⓑ $32^{- \frac{2}{5}}$ ⓒ $4^{- \frac{5}{2}}$ .

Solution

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

ⓐ
$\begin{matrix} 16^{- \frac{3}{2}} \\ Rewrite using b^{− p} = \frac{1}{b^{p}} . & \frac{1}{16^{\frac{3}{2}}} \\ \begin{matrix} Change to radical form. The power of the \\ radical is the numerator of the exponent, 3. \\ The index is the denominator of the \\ exponent, 2. \end{matrix} & \frac{1}{{(\sqrt{16})}^{3}} \\ Simplify. & \frac{1}{4^{3}} \\ \frac{1}{64} \end{matrix}$

ⓑ
$\begin{matrix} 32^{- \frac{2}{5}} \\ Rewrite using b^{− p} = \frac{1}{b^{p}} . & \frac{1}{32^{\frac{2}{5}}} \\ Change to radical form. & \frac{1}{{(\sqrt[5]{32})}^{2}} \\ Rewrite the radicand as a power. & \frac{1}{{(\sqrt[5]{2^{5}})}^{2}} \\ Simplify. & \frac{1}{2^{2}} \\ \frac{1}{4} \end{matrix}$

ⓒ
$\begin{matrix} 4^{- \frac{5}{2}} \\ Rewrite using b^{− p} = \frac{1}{b^{p}} . & \frac{1}{4^{\frac{5}{2}}} \\ Change to radical form. & \frac{1}{{(\sqrt{4})}^{5}} \\ Simplify. & \frac{1}{2^{5}} \\ \frac{1}{32} \end{matrix}$

Try It 9.221

Simplify: ⓐ $8^{- \frac{5}{3}}$ ⓑ $81^{- \frac{3}{2}}$ ⓒ $16^{- \frac{3}{4}}$ .

Try It 9.222

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

Example 9.112

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

Solution

ⓐ
$\begin{matrix} − 25^{\frac{3}{2}} \\ Rewrite in radical form. & − {(\sqrt{25})}^{3} \\ Simplify the radical. & − {(5)}^{3} \\ Simplify. & −125 \end{matrix}$

ⓑ
$\begin{matrix} − 25^{- \frac{3}{2}} \\ Rewrite using b^{− p} = \frac{1}{b^{p}} . & − (\frac{1}{25^{\frac{3}{2}}}) \\ Rewrite in radical form. & − (\frac{1}{{(\sqrt{25})}^{3}}) \\ Simplify the radical. & − (\frac{1}{{(5)}^{3}}) \\ Simplify. & - \frac{1}{125} \end{matrix}$

ⓒ
$\begin{matrix} {(−25)}^{\frac{3}{2}} \\ Rewrite in radical form. & {(\sqrt{−25})}^{3} \\ \begin{matrix} There is no real number whose \\ square root is −25 . \end{matrix} & Not a real number. \end{matrix}$

Try It 9.223

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

Try It 9.224

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

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

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

Summary of Exponent Properties

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

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

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

Example 9.113

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

Solution

ⓐ
$\begin{matrix} 2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}} \\ \begin{matrix} The bases are the same, so we add the \\ exponents. \end{matrix} & 2^{\frac{1}{2} + \frac{5}{2}} \\ Add the fractions. & 2^{\frac{6}{2}} \\ Simplify the exponent. & 2^{3} \\ Simplify. & 8 \end{matrix}$

ⓑ
$\begin{matrix} x^{\frac{2}{3}} \cdot x^{\frac{4}{3}} \\ \begin{matrix} The bases are the same, so we add the \\ exponents. \end{matrix} & x^{\frac{2}{3} + \frac{4}{3}} \\ Add the fractions. & x^{\frac{6}{3}} \\ Simplify. & x^{2} \end{matrix}$

ⓒ
$\begin{matrix} z^{\frac{3}{4}} \cdot z^{\frac{5}{4}} \\ \begin{matrix} The bases are the same, so we add the \\ exponents. \end{matrix} & z^{\frac{3}{4} + \frac{5}{4}} \\ Add the fractions. & z^{\frac{8}{4}} \\ Simplify. & z^{2} \end{matrix}$

Try It 9.225

Simplify: ⓐ $3^{\frac{2}{3}} \cdot 3^{\frac{4}{3}}$ ⓑ $y^{\frac{1}{3}} \cdot y^{\frac{8}{3}}$ ⓒ $m^{\frac{1}{4}} \cdot m^{\frac{3}{4}}$ .

Try It 9.226

Simplify: ⓐ $5^{\frac{3}{5}} \cdot 5^{\frac{7}{5}}$ ⓑ $z^{\frac{1}{8}} \cdot z^{\frac{7}{8}}$ ⓒ $n^{\frac{2}{7}} \cdot n^{\frac{5}{7}}$ .

We will use the Power Property in the next example.

Example 9.114

Simplify: ⓐ ${(x^{4})}^{\frac{1}{2}}$ ⓑ ${(y^{6})}^{\frac{1}{3}}$ ⓒ ${(z^{9})}^{\frac{2}{3}}$ .

Solution

ⓐ
$\begin{matrix} {(x^{4})}^{\frac{1}{2}} \\ \begin{matrix} To raise a power to a power, we multiply \\ the exponents. \end{matrix} & x^{4 \cdot \frac{1}{2}} \\ Simplify. & x^{2} \end{matrix}$

ⓑ
$\begin{matrix} {(y^{6})}^{\frac{1}{3}} \\ \begin{matrix} To raise a power to a power, we multiply \\ the exponents. \end{matrix} & y^{6 \cdot \frac{1}{3}} \\ Simplify. & y^{2} \end{matrix}$

ⓒ
$\begin{matrix} {(z^{9})}^{\frac{2}{3}} \\ \begin{matrix} To raise a power to a power, we multiply \\ the exponents. \end{matrix} & z^{9 \cdot \frac{2}{3}} \\ Simplify. & z^{6} \end{matrix}$

Try It 9.227

Simplify: ⓐ ${(p^{10})}^{\frac{1}{5}}$ ⓑ ${(q^{8})}^{\frac{3}{4}}$ ⓒ ${(x^{6})}^{\frac{4}{3}}$ .

Try It 9.228

Simplify: ⓐ ${(r^{6})}^{\frac{5}{3}}$ ⓑ ${(s^{12})}^{\frac{3}{4}}$ ⓒ ${(m^{9})}^{\frac{2}{9}}$ .

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

Example 9.115

Simplify: ⓐ $\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$ ⓑ $\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}$ ⓒ $\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}$ .

Solution

ⓐ
$\begin{matrix} \frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}} \\ \begin{matrix} To divide with the same base, we subtract \\ the exponents. \end{matrix} & x^{\frac{4}{3} - \frac{1}{3}} \\ Simplify. & x \end{matrix}$
ⓑ
$\begin{matrix} \frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}} \\ \begin{matrix} To divide with the same base, we subtract \\ the exponents. \end{matrix} & y^{\frac{3}{4} - \frac{1}{4}} \\ Simplify. & y^{\frac{1}{2}} \end{matrix}$
ⓒ
$\begin{matrix} \frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}} \\ \begin{matrix} To divide with the same base, we subtract \\ the exponents. \end{matrix} & z^{\frac{2}{3} - \frac{5}{3}} \\ Rewrite without a negative exponent. & \frac{1}{z} \end{matrix}$

Try It 9.229

Simplify: ⓐ $\frac{u^{\frac{5}{4}}}{u^{\frac{1}{4}}}$ ⓑ $\frac{v^{\frac{3}{5}}}{v^{\frac{2}{5}}}$ ⓒ $\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}$ .

Try It 9.230

Simplify: ⓐ $\frac{c^{\frac{12}{5}}}{c^{\frac{2}{5}}}$ ⓑ $\frac{m^{\frac{5}{4}}}{m^{\frac{9}{4}}}$ ⓒ $\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}$ .

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

Example 9.116

Simplify: ⓐ ${(27 u^{\frac{1}{2}})}^{\frac{2}{3}}$ ⓑ ${(8 v^{\frac{1}{4}})}^{\frac{2}{3}}$ .

Solution

ⓐ
$\begin{matrix} {(27 u^{\frac{1}{2}})}^{\frac{2}{3}} \\ \begin{matrix} First we use the Product to a Power \\ Property. \end{matrix} & {(27)}^{\frac{2}{3}} {(u^{\frac{1}{2}})}^{\frac{2}{3}} \\ Rewrite 27 as a power of 3. & {(3^{3})}^{\frac{2}{3}} {(u^{\frac{1}{2}})}^{\frac{2}{3}} \\ \begin{matrix} To raise a power to a power, we multiply \\ the exponents. \end{matrix} & (3^{2}) (u^{\frac{1}{3}}) \\ Simplify. & 9 u^{\frac{1}{3}} \end{matrix}$
ⓑ
$\begin{matrix} {(8 v^{\frac{1}{4}})}^{\frac{2}{3}} \\ \begin{matrix} First we use the Product to a Power \\ Property. \end{matrix} & {(8)}^{\frac{2}{3}} {(v^{\frac{1}{4}})}^{\frac{2}{3}} \\ Rewrite 8 as a power of 2. & {(2^{3})}^{\frac{2}{3}} {(v^{\frac{1}{4}})}^{\frac{2}{3}} \\ \begin{matrix} To raise a power to a power, we multiply \\ the exponents. \end{matrix} & (2^{2}) (v^{\frac{1}{6}}) \\ Simplify. & 4 v^{\frac{1}{6}} \end{matrix}$

Try It 9.231

Simplify: ⓐ ${(32 x^{\frac{1}{3}})}^{\frac{3}{5}}$ ⓑ ${(64 y^{\frac{2}{3}})}^{\frac{1}{3}}$ .

Try It 9.232

Simplify: ⓐ ${(16 m^{\frac{1}{3}})}^{\frac{3}{2}}$ ⓑ ${(81 n^{\frac{2}{5}})}^{\frac{3}{2}}$ .

Example 9.117

Simplify: ⓐ ${(m^{3} n^{9})}^{\frac{1}{3}}$ ⓑ ${(p^{4} q^{8})}^{\frac{1}{4}}$ .

Solution

ⓐ
$\begin{matrix} {(m^{3} n^{9})}^{\frac{1}{3}} \\ \begin{matrix} First we use the Product to a Power \\ Property. \end{matrix} & {(m^{3})}^{\frac{1}{3}} {(n^{9})}^{\frac{1}{3}} \\ \begin{matrix} To raise a power to a power, we multiply \\ the exponents. \end{matrix} & m n^{3} \end{matrix}$
ⓑ
$\begin{matrix} {(p^{4} q^{8})}^{\frac{1}{4}} \\ \begin{matrix} First we use the Product to a Power \\ Property. \end{matrix} & {(p^{4})}^{\frac{1}{4}} {(q^{8})}^{\frac{1}{4}} \\ \begin{matrix} To raise a power to a power, we multiply \\ the exponents. \end{matrix} & p q^{2} \end{matrix}$

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

Example 9.118

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

Solution

ⓐ
$\begin{matrix} \frac{x^{\frac{3}{4}} \cdot x^{- \frac{1}{4}}}{x^{- \frac{6}{4}}} \\ \begin{matrix} Use the Product Property in the numerator, \\ add the exponents. \end{matrix} & \frac{x^{\frac{2}{4}}}{x^{- \frac{6}{4}}} \\ \begin{matrix} Use the Quotient Property, subtract the \\ exponents. \end{matrix} & x^{\frac{8}{4}} \\ Simplify. & x^{2} \end{matrix}$
ⓑ
$\begin{matrix} \frac{y^{\frac{4}{3}} \cdot y}{y^{- \frac{2}{3}}} \\ \begin{matrix} Use the Product Property in the numerator, \\ add the exponents. \end{matrix} & \frac{y^{\frac{7}{3}}}{y^{- \frac{2}{3}}} \\ \begin{matrix} Use the Quotient Property, subtract the \\ exponents. \end{matrix} & y^{\frac{9}{3}} \\ Simplify. & y^{3} \end{matrix}$

Try It 9.233

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

Try It 9.234

Simplify: ⓐ $\frac{u^{\frac{4}{5}} \cdot u^{- \frac{2}{5}}}{u^{- \frac{13}{5}}}$ ⓑ $\frac{v^{\frac{1}{2}} \cdot v}{v^{- \frac{7}{2}}}$ .

Section 9.8 Exercises

Practice Makes Perfect

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

In the following exercises, write as a radical expression.

524.

ⓐ $x^{\frac{1}{2}}$
ⓑ $y^{\frac{1}{3}}$
ⓒ $z^{\frac{1}{4}}$

525.

ⓐ $r^{\frac{1}{2}}$
ⓑ $s^{\frac{1}{3}}$
ⓒ $t^{\frac{1}{4}}$

526.

ⓐ $u^{\frac{1}{5}}$
ⓑ $v^{\frac{1}{9}}$
ⓒ $w^{\frac{1}{20}}$

527.

ⓐ $g^{\frac{1}{7}}$
ⓑ $h^{\frac{1}{5}}$
ⓒ $j^{\frac{1}{25}}$

In the following exercises, write with a rational exponent.

528.

ⓐ $- \sqrt[7]{x}$
ⓑ $\sqrt[9]{y}$
ⓒ $\sqrt[5]{f}$

529.

ⓐ $\sqrt[8]{r}$
ⓑ

530.

ⓐ $\sqrt[3]{a}$
ⓑ

531.

ⓐ $\sqrt[5]{u}$
ⓑ $\sqrt{v}$
ⓒ

532.

ⓐ $\sqrt[3]{7 c}$
ⓑ $\sqrt[7]{12 d}$
ⓒ $3 \sqrt[4]{5 f}$

533.

ⓐ $\sqrt[4]{5 x}$
ⓑ $\sqrt[8]{9 y}$
ⓒ $7 \sqrt[5]{3 z}$

534.

ⓐ $\sqrt{21 p}$
ⓑ $\sqrt[4]{8 q}$
ⓒ $4 \sqrt[6]{36 r}$

535.

ⓐ $\sqrt[3]{25 a}$
ⓑ $\sqrt{3 b}$
ⓒ

In the following exercises, simplify.

536.

ⓐ $81^{\frac{1}{2}}$
ⓑ $125^{\frac{1}{3}}$
ⓒ $64^{\frac{1}{2}}$

537.

ⓐ $625^{\frac{1}{4}}$
ⓑ $243^{\frac{1}{5}}$
ⓒ $32^{\frac{1}{5}}$

538.

ⓐ $16^{\frac{1}{4}}$
ⓑ $16^{\frac{1}{2}}$
ⓒ $3125^{\frac{1}{5}}$

539.

ⓐ $216^{\frac{1}{3}}$
ⓑ $32^{\frac{1}{5}}$
ⓒ $81^{\frac{1}{4}}$

540.

ⓐ ${(−216)}^{\frac{1}{3}}$
ⓑ $− 216^{\frac{1}{3}}$
ⓒ ${(216)}^{- \frac{1}{3}}$

541.

ⓐ ${(−243)}^{\frac{1}{5}}$
ⓑ $− 243^{\frac{1}{5}}$
ⓒ ${(243)}^{- \frac{1}{5}}$

542.

ⓐ ${(−1)}^{\frac{1}{3}}$
ⓑ ${−1}^{\frac{1}{3}}$
ⓒ ${(1)}^{- \frac{1}{3}}$

543.

ⓐ ${(−1000)}^{\frac{1}{3}}$
ⓑ $− 1000^{\frac{1}{3}}$
ⓒ ${(1000)}^{- \frac{1}{3}}$

544.

ⓐ ${(−81)}^{\frac{1}{4}}$
ⓑ $− 81^{\frac{1}{4}}$
ⓒ ${(81)}^{- \frac{1}{4}}$

545.

ⓐ ${(−49)}^{\frac{1}{2}}$
ⓑ $− 49^{\frac{1}{2}}$
ⓒ ${(49)}^{- \frac{1}{2}}$

546.

ⓐ ${(−36)}^{\frac{1}{2}}$
ⓑ $- 36^{\frac{1}{2}}$
ⓒ ${(36)}^{- \frac{1}{2}}$

547.

ⓐ ${(−1)}^{\frac{1}{4}}$
ⓑ ${(1)}^{- \frac{1}{4}}$
ⓒ $− 1^{\frac{1}{4}}$

548.

ⓐ ${(−100)}^{\frac{1}{2}}$
ⓑ $− 100^{\frac{1}{2}}$
ⓒ ${(100)}^{- \frac{1}{2}}$

549.

ⓐ ${(−32)}^{\frac{1}{5}}$
ⓑ ${(243)}^{- \frac{1}{5}}$
ⓒ $− 125^{\frac{1}{3}}$

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

In the following exercises, write with a rational exponent.

550.

ⓐ $\sqrt{m^{5}}$
ⓑ $\sqrt[3]{n^{2}}$
ⓒ $\sqrt[4]{p^{3}}$

551.

ⓐ $\sqrt[4]{r^{7}}$
ⓑ $\sqrt[5]{s^{3}}$
ⓒ $\sqrt[3]{t^{7}}$

552.

ⓐ $\sqrt[5]{u^{2}}$
ⓑ $\sqrt[5]{v^{8}}$
ⓒ $\sqrt[9]{w^{4}}$

553.

ⓐ $\sqrt[3]{a}$
ⓑ $\sqrt{b^{5}}$
ⓒ $\sqrt[3]{c^{5}}$

In the following exercises, simplify.

554.

ⓐ $16^{\frac{3}{2}}$
ⓑ $8^{\frac{2}{3}}$
ⓒ ${10,000}^{\frac{3}{4}}$

555.

556.

557.

558.

559.

560.

561.

562.

563.

564.

565.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

566.

ⓐ $4^{\frac{5}{8}} \cdot 4^{\frac{11}{8}}$
ⓑ $m^{\frac{7}{12}} \cdot m^{\frac{17}{12}}$
ⓒ $p^{\frac{3}{7}} \cdot p^{\frac{18}{7}}$

567.

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$
ⓑ $n^{\frac{2}{10}} \cdot n^{\frac{8}{10}}$
ⓒ $q^{\frac{2}{5}} \cdot q^{\frac{13}{5}}$

568.

ⓐ $5^{\frac{1}{2}} \cdot 5^{\frac{7}{2}}$
ⓑ $c^{\frac{3}{4}} \cdot c^{\frac{9}{4}}$
ⓒ $d^{\frac{3}{5}} \cdot d^{\frac{2}{5}}$

569.

ⓐ $10^{\frac{1}{3}} \cdot 10^{\frac{5}{3}}$
ⓑ $x^{\frac{5}{6}} \cdot x^{\frac{7}{6}}$
ⓒ $y^{\frac{11}{8}} \cdot y^{\frac{21}{8}}$

570.

571.

572.

573.

574.

ⓐ $\frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}$
ⓑ $\frac{y^{\frac{5}{2}}}{y^{\frac{1}{2}}}$
ⓒ $\frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}$

575.

ⓐ $\frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}}$
ⓑ $\frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}}$
ⓒ $\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

576.

ⓐ $\frac{t^{\frac{12}{5}}}{t^{\frac{7}{5}}}$
ⓑ $\frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}}$
ⓒ $\frac{m^{\frac{13}{8}}}{m^{\frac{5}{8}}}$

577.

ⓐ $\frac{u^{\frac{13}{9}}}{u^{\frac{4}{9}}}$
ⓑ $\frac{r^{\frac{15}{7}}}{r^{\frac{8}{7}}}$
ⓒ $\frac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}$

578.

ⓐ ${(9 p^{\frac{2}{3}})}^{\frac{5}{2}}$
ⓑ ${(27 q^{\frac{3}{2}})}^{\frac{4}{3}}$

579.

ⓐ ${(81 r^{\frac{4}{5}})}^{\frac{1}{4}}$
ⓑ ${(64 s^{\frac{3}{7}})}^{\frac{1}{6}}$

580.

ⓐ ${(16 u^{\frac{1}{3}})}^{\frac{3}{4}}$
ⓑ ${(100 v^{\frac{2}{5}})}^{\frac{3}{2}}$

581.

ⓐ ${(27 m^{\frac{3}{4}})}^{\frac{2}{3}}$
ⓑ ${(625 n^{\frac{8}{3}})}^{\frac{3}{4}}$

582.

ⓐ ${(x^{8} y^{10})}^{\frac{1}{2}}$
ⓑ ${(a^{9} b^{12})}^{\frac{1}{3}}$

583.

ⓐ ${(r^{8} s^{4})}^{\frac{1}{4}}$
ⓑ ${(u^{15} v^{20})}^{\frac{1}{5}}$

584.

ⓐ ${(a^{6} b^{16})}^{\frac{1}{2}}$
ⓑ ${(j^{9} k^{6})}^{\frac{2}{3}}$

585.

ⓐ ${(r^{16} s^{10})}^{\frac{1}{2}}$
ⓑ ${(u^{10} v^{5})}^{\frac{4}{5}}$

586.

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

587.

ⓐ $\frac{a^{\frac{3}{4}} \cdot a^{- \frac{1}{4}}}{a^{- \frac{10}{4}}}$
ⓑ $\frac{b^{\frac{2}{3}} \cdot b}{b^{- \frac{7}{3}}}$

588.

ⓐ $\frac{c^{\frac{5}{3}} \cdot c^{- \frac{1}{3}}}{c^{- \frac{2}{3}}}$
ⓑ $\frac{d^{\frac{3}{5}} \cdot d}{d^{- \frac{2}{5}}}$

589.

ⓐ $\frac{m^{\frac{7}{4}} \cdot m^{- \frac{5}{4}}}{m^{- \frac{2}{4}}}$
ⓑ $\frac{n^{\frac{3}{7}} \cdot n}{n^{- \frac{4}{7}}}$

590.

$4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}}$

591.

$n^{\frac{2}{6}} \cdot n^{\frac{4}{6}}$

592.

${(a^{24})}^{\frac{1}{6}}$

593.

${(b^{10})}^{\frac{3}{5}}$

594.

$\frac{w^{\frac{2}{5}}}{w^{\frac{7}{5}}}$

595.

$\frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}}$

596.

${(27 r^{\frac{3}{5}})}^{\frac{1}{3}}$

597.

${(64 s^{\frac{3}{5}})}^{\frac{1}{6}}$

598.

${(r^{9} s^{12})}^{\frac{1}{3}}$

599.

${(u^{12} v^{18})}^{\frac{1}{6}}$

Everyday Math

600.

Landscaping Joe wants to have a square garden plot in his backyard. He has enough compost to cover an area of 144 square feet. Simplify $144^{\frac{1}{2}}$ to find the length of each side of his garden.

601.

Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify $242^{\frac{1}{2}}$ to find the length of each side of his patio.Round to the nearest tenth of a foot.

602.

Gravity While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify $\frac{12^{\frac{1}{2}}}{16^{\frac{1}{2}}}$ to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

603.

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify $\frac{1024^{\frac{1}{2}}}{16^{\frac{1}{2}}}$ to find how many seconds it took for the flare to reach the water.

Writing Exercises

604.

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

605.

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

9.8 Rational Exponents