Rational Exponents - Elementary Algebra

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 9.21

Before you get started, take this readiness quiz.

Add: $\frac{7}{15} + \frac{5}{12}$ .
If you missed this problem, review Example 1.81.

Be Prepared 9.22

Simplify: ${(4 x^{2} y^{5})}^{3}$ .
If you missed this problem, review Example 6.24.

Be Prepared 9.23

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.

	$\begin{matrix} {(8^{p})}^{3} & = & 8 \end{matrix}$
Multiply the exponents on the left.	$\begin{matrix} 8^{3 p} & = & 8 \end{matrix}$
Write the exponent 1 on the right.	$\begin{matrix} 8^{3 p} & = & 8^{1} \end{matrix}$
The exponents must be equal.	$\begin{matrix} 3 p & = & 1 \end{matrix}$
Solve for $p .$	$\begin{matrix} p & = & \frac{1}{3} \end{matrix}$
	$\begin{matrix} So {(8^{\frac{1}{3}})}^{3} & = & 8 . \end{matrix}$
But we know also ${(\sqrt[3]{8})}^{3} = 8$ . Then it must be that $8^{\frac{1}{3}} = \sqrt[3]{8}$ .

Step-by-step solution for an exponential equation, illustrating the relationship between fractional exponents and roots.

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 $\sqrt[n]{a}$ .

ⓐ

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

This table illustrates the conversion of a fractional exponent (x^(1/2)) to its equivalent radical form (sqrt(x)), including a rule explanation.

ⓑ

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

This table demonstrates the relationship between fractional exponents and their radical equivalents, showing how the exponent's denominator determines the radical's index.

ⓒ

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

Conversion of fractional exponents to radical form, showing the exponent's denominator as the radical's index.

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

ⓐ

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

Illustrates the equivalence between square root notation and fractional exponent notation, with explanations for the transformation.

ⓑ

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

Conversion of cube roots to fractional exponents, illustrating how the root's index determines the exponent's denominator.

ⓒ

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

Illustrates the conversion of a fourth root expression to its equivalent exponential form, detailing the rule for the exponent's denominator.

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

ⓐ

	$\sqrt{5 y}$
No index is shown, so it is 2. The denominator of the exponent will be 2.	${(5 y)}^{\frac{1}{2}}$

Illustrates converting radical expressions to rational exponent form, showing how the root's index determines the fractional exponent.

ⓑ

	$\sqrt[3]{4 x}$
The index is 3, so the denominator of the exponent is 3.	${(4 x)}^{\frac{1}{3}}$

Illustrates converting the cube root of 4x to its equivalent exponential form with a fractional exponent, explaining the role of the index.

ⓒ

	$3 \sqrt[4]{5 z}$
The index is 4, so the denominator of the exponent is 4.	$3 {(5 z)}^{\frac{1}{4}}$

Conversion of a radical expression to an equivalent exponential form, showing an example with an explanation.

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

ⓐ

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

This table illustrates the step-by-step simplification of a numerical expression with a fractional exponent.

ⓑ

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

This table illustrates the step-by-step simplification of a numerical expression with a fractional exponent, converting it to a cube root for evaluation.

ⓒ

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

Step-by-step simplification of 256^(1/4), demonstrating rewriting as a root and final simplification.

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

ⓐ

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

Illustrates the step-by-step evaluation of a fractional exponent expression by conversion to a cube root and simplification.

ⓑ

	$− 64^{\frac{1}{3}}$
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.	$−4$

Detailed steps for simplifying the mathematical expression -64^(1/3) by rewriting it as a cube root and evaluating the final result.

ⓒ

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

This table demonstrates the step-by-step process for simplifying a mathematical expression with a negative fractional exponent.

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

ⓐ

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

Evaluation of (-16)^(1/4), illustrating it has no real number solution.

ⓑ

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

Step-by-step simplification of the mathematical expression -16^(1/4) to its final integer value.

ⓒ

	${(16)}^{- \frac{1}{4}}$
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.	$\frac{1}{2}$

This table shows the step-by-step simplification of the mathematical expression (16)^(-1/4), transforming it into its simplified fractional form of 1/2.

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.

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

Steps to simplify an exponential expression with a fractional power, demonstrating its equivalence to a radical form.

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

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

This table demonstrates the steps to simplify (a^m)^(1/n) to a^(m/n), illustrating its equivalence with the n-th root of (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.

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.

ⓐ

	$9^{\frac{3}{2}}$
The power of the radical is the numerator of the exponent, 3. Since the denominator of the exponent is 2, this is a square root.	${(\sqrt{9})}^{3}$
Simplify.	${(3)}^{3}$
	$27$

This table illustrates the step-by-step evaluation of the exponential expression 9^(3/2) by converting it to radical 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.	${(5)}^{2}$
	$25$

Steps to simplify an expression with a rational exponent by converting it to radical form.

ⓒ

	$81^{\frac{3}{4}}$
The power of the radical is the numerator of the exponent, 3. The index of the radical is the denominator of the exponent, 4.	${(\sqrt[4]{81})}^{3}$
Simplify.	${(3)}^{3}$
	$27$

Steps to simplify an expression with a rational exponent by converting it to radical form and calculating the result.

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.

ⓐ

	$16^{- \frac{3}{2}}$
Rewrite using $b^{− p} = \frac{1}{b^{p}} .$	$\frac{1}{16^{\frac{3}{2}}}$
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.	$\frac{1}{4^{3}}$
	$\frac{1}{64}$

Step-by-step simplification of an expression with a negative fractional exponent, showing the application of exponent and radical rules.

ⓑ

	$32^{- \frac{2}{5}}$
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.	$\frac{1}{2^{2}}$
	$\frac{1}{4}$

Detailed steps to simplify 32^(-2/5) by converting to a fraction, radical form, and evaluating to 1/4.

ⓒ

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

Step-by-step simplification of an expression with a negative fractional exponent, demonstrating conversions to positive exponents and radical form.

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

ⓐ

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

Step-by-step simplification of the expression -25^(3/2) by rewriting it in radical form and evaluating it to -125.

ⓑ

	$− 25^{- \frac{3}{2}}$
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.	$- \frac{1}{125}$

This table illustrates the step-by-step simplification of the mathematical expression -25^(-3/2) by applying exponent rules and converting to radical form.

ⓒ

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

This table evaluates the expression (-25)^(3/2), demonstrating its transformation into radical form and concluding it is not a real number.

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

ⓐ

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

Step-by-step simplification of the exponential expression 2^(1/2) * 2^(5/2) to 8.

ⓑ

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

Steps demonstrating the simplification of an algebraic expression involving the multiplication of exponential terms with the same base.

ⓒ

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

Step-by-step simplification of an algebraic expression with fractional exponents using exponent rules.

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

ⓐ

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

Simplifying an expression with a power raised to a power by multiplying exponents, with steps and examples.

ⓑ

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

This table illustrates the step-by-step simplification of an algebraic expression involving a power raised to a fractional power, applying the rule for multiplying exponents.

ⓒ

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

Demonstration of simplifying the exponential expression (z^9)^(2/3) by applying the power to a power rule.

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

ⓐ

	$\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$
To divide with the same base, we subtract the exponents.	$x^{\frac{4}{3} - \frac{1}{3}}$
Simplify.	$x$

Step-by-step simplification of an exponential expression involving division with the same base by subtracting fractional exponents.

ⓑ

	$\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}$
To divide with the same base, we subtract the exponents.	$y^{\frac{3}{4} - \frac{1}{4}}$
Simplify.	$y^{\frac{1}{2}}$

Steps to simplify an exponential expression by dividing terms with the same base and subtracting their fractional exponents.

ⓒ

	$\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}$
To divide with the same base, we subtract the exponents.	$z^{\frac{2}{3} - \frac{5}{3}}$
Rewrite without a negative exponent.	$\frac{1}{z}$

This table demonstrates the simplification of an exponential expression by applying rules for dividing with the same base and rewriting negative exponents.

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

ⓐ

	${(27 u^{\frac{1}{2}})}^{\frac{2}{3}}$
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.	$9 u^{\frac{1}{3}}$

Demonstrates the step-by-step simplification of the expression (27u^(1/2))^(2/3) to 9u^(1/3) using exponent rules.

ⓑ

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

This table demonstrates the step-by-step simplification of an algebraic expression involving rational exponents.

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

ⓐ

	${(m^{3} n^{9})}^{\frac{1}{3}}$
First we use the Product to a Power Property.	${(m^{3})}^{\frac{1}{3}} {(n^{9})}^{\frac{1}{3}}$
To raise a power to a power, we multiply the exponents.	$m n^{3}$

Step-by-step simplification of a mathematical expression involving powers and roots using the Product to a Power Property.

ⓑ

	${(p^{4} q^{8})}^{\frac{1}{4}}$
First we use the Product to a Power Property.	${(p^{4})}^{\frac{1}{4}} {(q^{8})}^{\frac{1}{4}}$
To raise a power to a power, we multiply the exponents.	$p q^{2}$

Step-by-step example illustrating the simplification of an exponential expression (p^4q^8)^(1/4) using exponent properties.

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

ⓐ

	$\frac{x^{\frac{3}{4}} \cdot x^{- \frac{1}{4}}}{x^{- \frac{6}{4}}}$
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.	$x^{2}$

This table illustrates the step-by-step simplification of an algebraic expression involving fractional exponents using properties of exponents.

ⓑ

	$\frac{y^{\frac{4}{3}} \cdot y}{y^{- \frac{2}{3}}}$
Use the Product Property in the numerator, add the exponents.	$\frac{y^{\frac{7}{3}}}{y^{- \frac{2}{3}}}$
Use the Quotient Property, subtract the exponents.	$y^{\frac{9}{3}}$
Simplify.	$y^{3}$

Step-by-step simplification of an algebraic expression using product and quotient properties of exponents.

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.

540.

541.

542.

543.

544.

545.

546.

547.

548.

549.

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

In the following exercises, write with a rational exponent.

550.

551.

552.

553.

In the following exercises, simplify.

554.

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

Learning Objectives

Be Prepared 9.21

Be Prepared 9.22

Be Prepared 9.23

Simplify Expressions with a1na1n

Rational Exponent a 1 n a 1 n

Example 9.103

Solution

Try It 9.205

Try It 9.206

Example 9.104

Solution

Try It 9.207

Try It 9.208

Example 9.105

Solution

Try It 9.209

Try It 9.210

Example 9.106

Solution

Try It 9.211

Try It 9.212

Example 9.107

Solution

Try It 9.213

Try It 9.214

Example 9.108

Solution

Try It 9.215

Try It 9.216

Simplify Expressions with amnamn

Rational Exponent a m n a m n

Example 9.109

Solution

Try It 9.217

Try It 9.218

Example 9.110

Solution

Try It 9.219

Try It 9.220

Example 9.111

Solution

Try It 9.221

Try It 9.222

Example 9.112

Solution

Try It 9.223

Try It 9.224

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

Summary of Exponent Properties

Example 9.113

Solution

Try It 9.225

Try It 9.226

Example 9.114

Solution

Try It 9.227

Try It 9.228

Example 9.115

Solution

Try It 9.229

Try It 9.230

Example 9.116

Solution

Try It 9.231

Try It 9.232

Example 9.117

Solution

Example 9.118

Solution

Try It 9.233

Try It 9.234

Section 9.8 Exercises

Practice Makes Perfect

Everyday Math

Writing Exercises

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

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

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

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