Simplify Rational Exponents - Intermediate Algebra

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.	$\sqrt{x}$

Illustrates the conversion of the rational exponent x^(1/2) to its radical form, sqrt(x), explaining the rule for the radical index.

ⓑ

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

This table illustrates the conversion from fractional exponents to radical form, specifically demonstrating y^(1/3) as the cube root of y.

ⓒ

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

This table illustrates the conversion of a fractional exponent to its equivalent radical form, focusing on how the exponent's denominator determines the radical index.

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 $5 y .$

Shows the conversion of a square root radical expression to its equivalent exponential form, with explanatory notes.

ⓑ

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

This table illustrates converting a radical expression to its equivalent fractional exponent form, with an explanation of the underlying process.

ⓒ

	$3 \sqrt[4]{5 z}$
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.	$3 {(5 z)}^{\frac{1}{4}}$

Illustrates converting a radical expression to its rational exponent form, with detailed explanations.

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.	$5$

Steps to simplify a fractional exponent: rewriting 25^(1/2) as a square root and finding its value.

ⓑ

	$64^{\frac{1}{3}}$
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 64 raised to the power of one-third, demonstrating its conversion to a cube root and final 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.	$4$

This table illustrates the step-by-step simplification of the mathematical expression 256 to the power of 1/4, demonstrating its transformation into a fourth root and final numerical evaluation.

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.	$No real solution.$

This table illustrates the step-by-step simplification of the expression (-16)^(1/4), demonstrating the process to determine there is 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.	$−2$

Step-by-step simplification of the mathematical expression -16^(1/4), detailing each transformation to reach the final result.

ⓒ

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

Demonstrates the step-by-step simplification of (16)^(-1/4), converting it to 1/2 using exponent and root properties.

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

ⓐ

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.

ⓑ

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).

ⓒ

$Illustration showing the relationship between radical form and exponential form, where the numerator of a fractional exponent is the power and the denominator is the radical index.$

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.	${(5)}^{2}$
	$25$

Step-by-step simplification of an expression with a fractional exponent by converting it to radical form.

ⓑ 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.	$\frac{1}{4^{3}}$
	$\frac{1}{64}$

Steps for simplifying the exponential expression 16^(-3/2) using exponent and radical rules.

ⓒ

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

Step-by-step simplification of 32^(-2/5), illustrating conversion from negative fractional exponent to radical form and final evaluation.

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.	$−125$

Step-by-step simplification of a negative number raised to a fractional exponent.

ⓑ

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

Step-by-step simplification of the expression -25^(-3/2) using exponent rules, radical forms, and arithmetic operations.

ⓒ

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

Evaluates the expression (-25)^(3/2), demonstrating the steps to rewrite it in radical form and concluding that it is 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.	$x^{\frac{4}{3}}$

This table demonstrates the step-by-step simplification of an algebraic expression involving fractional exponents by applying the rule for multiplying powers with the same base.

ⓑ 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.	$z^{6}$

Steps for simplifying an exponential expression using the power of a power rule, showing the rule's application and the final result.

	$\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}$
	$\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}$
To divide with the same base, we subtract the exponents.	$\frac{1}{x^{\frac{5}{3} - \frac{1}{3}}}$
Simplify.	$\frac{1}{x^{\frac{4}{3}}}$

This table demonstrates the step-by-step simplification of an exponential expression using the quotient rule for exponents.

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

Step-by-step simplification of an algebraic expression involving fractional exponents using exponent properties.

ⓑ

	${(m^{\frac{2}{3}} n^{\frac{1}{2}})}^{\frac{3}{2}}$
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.	$m n^{\frac{3}{4}}$

This table illustrates the step-by-step simplification of an algebraic expression involving fractional exponents and the product to a power property.

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.	$x^{2}$

Step-by-step simplification of a rational expression using the Product and Quotient Properties of exponents.

ⓑ 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.	$\frac{4 x}{y^{\frac{1}{2}}}$

Step-by-step simplification of an algebraic expression involving fractional exponents using properties of exponents.

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.

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

120.

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

121.

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

122.

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

In the following exercises, write with a rational exponent.

123.

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

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.

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.

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

8.3 Simplify Rational Exponents

Learning Objectives

Be Prepared 8.7

Be Prepared 8.8

Be Prepared 8.9

Simplify Expressions with a1na1n

Rational Exponent a 1 n a 1 n

Example 8.26

Solution

Try It 8.51

Try It 8.52

Example 8.27

Solution

Try It 8.53

Try It 8.54

Example 8.28

Solution

Try It 8.55

Try It 8.56

Example 8.29

Solution

Try It 8.57

Try It 8.58

Simplify Expressions with amnamn

Rational Exponent a m n a m n

Example 8.30

Solution

Try It 8.59

Try It 8.60

Example 8.31

Solution

Try It 8.61

Try It 8.62

Example 8.32

Solution

Try It 8.63

Try It 8.64

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

Properties of Exponents

Example 8.33

Solution

Try It 8.65

Try It 8.66

Example 8.34

Solution

Try It 8.67

Try It 8.68

Example 8.35

Solution

Try It 8.69

Try It 8.70

Media

Section 8.3 Exercises

Practice Makes Perfect

Writing Exercises

Self Check

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

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

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

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