Simplify Rational Exponents

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.

when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

This same logic can be used for any positive integer exponent n to show that

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.

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.

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

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}}

Simplify Expressions with

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

in two ways. Remember the Power Property tells us to multiply the exponents and so

{(a^{\frac{1}{n}})}^{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.

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.

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.

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

Key Concepts

Practice Makes Perfect

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

In the following exercises, write as a radical expression.

ⓐ $x^{\frac{1}{2}}$

ⓑ $y^{\frac{1}{3}}$

ⓐ $\sqrt{x}$

ⓑ $\sqrt[3]{y}$

ⓐ $r^{\frac{1}{2}}$

ⓑ $s^{\frac{1}{3}}$

ⓐ $u^{\frac{1}{5}}$

ⓑ $v^{\frac{1}{9}}$

ⓐ $\sqrt[5]{u}$

ⓑ $\sqrt[9]{v}$

ⓐ $g^{\frac{1}{7}}$

ⓑ $h^{\frac{1}{5}}$

In the following exercises, write with a rational exponent.

ⓐ $\sqrt[7]{x}$

ⓑ $\sqrt[9]{y}$

ⓐ $\frac{1}{x^{7}}$

ⓑ $\frac{1}{y^{9}}$

ⓐ $\sqrt[8]{r}$

ⓑ $\sqrt[10]{s}$

ⓐ $\sqrt[3]{7 c}$

ⓑ $\sqrt[7]{12 d}$

ⓐ ${(7 c)}^{\frac{1}{4}}$

ⓑ ${(12 d)}^{\frac{1}{7}}$

ⓐ $\sqrt[4]{5 x}$

ⓑ $\sqrt[8]{9 y}$

ⓐ $\sqrt{21 p}$

ⓑ $\sqrt[4]{8 q}$

ⓐ ${(21 p)}^{\frac{1}{2}}$

ⓑ ${(8 q)}^{\frac{1}{4}}$

ⓐ $\sqrt[3]{25 a}$

ⓑ $\sqrt{3 b}$

In the following exercises, simplify.

ⓐ $81^{\frac{1}{2}}$

ⓑ $125^{\frac{1}{3}}$

ⓐ 9 ⓑ 5 ⓒ 8

ⓐ $625^{\frac{1}{4}}$

ⓑ $243^{\frac{1}{5}}$

ⓐ $16^{\frac{1}{4}}$

ⓑ $16^{\frac{1}{2}}$

ⓐ 2 ⓑ 4 ⓒ 5

ⓐ $64^{\frac{1}{3}}$

ⓑ $32^{\frac{1}{5}}$

ⓐ ${(−216)}^{\frac{1}{3}}$

ⓑ $− 216^{\frac{1}{3}}$

ⓐ $−6$

ⓑ $−6$

ⓐ ${(−1000)}^{\frac{1}{3}}$

ⓑ $− 1000^{\frac{1}{3}}$

ⓐ ${(−81)}^{\frac{1}{4}}$

ⓑ $− 81^{\frac{1}{4}}$

ⓐ not real ⓑ $−3$

ⓐ ${(−49)}^{\frac{1}{2}}$

ⓑ $− 49^{\frac{1}{2}}$

ⓐ ${(−36)}^{\frac{1}{2}}$

ⓑ $− 36^{\frac{1}{2}}$

ⓐ not real ⓑ $−6$

ⓐ ${(−16)}^{\frac{1}{4}}$

ⓑ $− 16^{\frac{1}{4}}$

ⓐ ${(−100)}^{\frac{1}{2}}$

ⓑ $− 100^{\frac{1}{2}}$

ⓐ not real ⓑ $−10$

ⓐ ${(−32)}^{\frac{1}{5}}$

ⓑ ${(243)}^{- \frac{1}{5}}$

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

In the following exercises, write with a rational exponent.

ⓐ $\sqrt{m^{5}}$

ⓑ ${(\sqrt[3]{3 y})}^{7}$

ⓐ $m^{\frac{5}{2}}$

ⓑ ${(3 y)}^{\frac{7}{3}}$

ⓐ $\sqrt[4]{r^{7}}$

ⓑ ${(\sqrt[5]{2 p q})}^{3}$

ⓐ $\sqrt[5]{u^{2}}$

ⓑ ${(\sqrt[3]{6 x})}^{5}$

ⓐ $u^{\frac{2}{5}}$

ⓑ ${(6 x)}^{\frac{5}{3}}$

ⓐ $\sqrt[3]{a}$

ⓑ ${(\sqrt[4]{21 v})}^{3}$

In the following exercises, simplify.

ⓐ $64^{\frac{5}{2}}$

ⓑ $81^{\frac{−3}{2}}$

ⓐ 32,768 ⓑ $\frac{1}{729}$

ⓒ 9

ⓐ $25^{\frac{3}{2}}$

ⓑ $9^{- \frac{3}{2}}$

ⓐ $32^{\frac{2}{5}}$

ⓑ $27^{- \frac{2}{3}}$

ⓐ 4 ⓑ $\frac{1}{9}$

ⓐ $100^{\frac{3}{2}}$

ⓑ $49^{- \frac{5}{2}}$

ⓐ $− 9^{\frac{3}{2}}$

ⓑ $− 9^{- \frac{3}{2}}$

ⓐ $−27$

ⓑ $- \frac{1}{27}$

ⓐ $− 64^{\frac{3}{2}}$

ⓑ $− 64^{- \frac{3}{2}}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $c^{\frac{1}{4}} \cdot c^{\frac{5}{8}}$

ⓑ ${(p^{12})}^{\frac{3}{4}}$

ⓐ $c^{\frac{7}{8}}$

ⓑ $p^{9}$

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$

ⓑ ${(b^{15})}^{\frac{3}{5}}$

ⓐ $y^{\frac{1}{2}} \cdot y^{\frac{3}{4}}$

ⓑ ${(x^{12})}^{\frac{2}{3}}$

ⓐ $y^{\frac{5}{4}}$

ⓑ $x^{8}$

ⓐ $q^{\frac{2}{3}} \cdot q^{\frac{5}{6}}$

ⓑ ${(h^{6})}^{\frac{4}{3}}$

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

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

ⓐ $81 q^{2}$

ⓑ $a^{\frac{1}{2}} b$

ⓐ ${(64 s^{\frac{3}{7}})}^{\frac{1}{6}}$

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

ⓐ ${(16 u^{\frac{1}{3}})}^{\frac{3}{4}}$

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

ⓐ $8 u^{\frac{1}{4}}$

ⓑ $8 p^{\frac{1}{2}} q^{\frac{3}{4}}$

ⓐ ${(625 n^{\frac{8}{3}})}^{\frac{3}{4}}$

ⓑ ${(9 x^{\frac{2}{5}} y^{\frac{3}{5}})}^{\frac{5}{2}}$

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

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

ⓐ $r^{\frac{7}{2}}$

ⓑ $\frac{6 s}{t}$

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

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

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

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

ⓐ $c^{2}$

ⓑ $\frac{2 x}{3 y}$

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

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

Writing Exercises

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

Explain all your steps.

Answers will vary.

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?

Simplify Expressions with a1n

Simplify Expressions with amn

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

Key Concepts

Practice Makes Perfect

Writing Exercises

Self Check

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

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