3.3 Simplify Rational Exponents

Domenic Spilotro, MSc

3.3 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

Try It

Before you get started, take this readiness quiz:

1) Add: $\frac{7}{15} + \frac{5}{12}$
2) Simplify: ${(4 x^{2} y^{5})}^{3}$
3) Simplify: $5^{- 3}$

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 $p$ .

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

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 3.3.1

Write as a radical expression:

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

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

Solution

a.

Step 1: 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 $2$ .

$\sqrt{x}$

b.

Step 1: The denominator of the exponent is $3$ , so the index is $3$ .

$\sqrt[3]{y}$

c.

Step 1: The denominator of the exponent is $4$ , so the index is $4$ .

$\sqrt[4]{z}$

Try It

4) Write as a radical expression:

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

Solution

a. $\sqrt{t}$
b. $\sqrt[3]{m}$
c. $\sqrt[4]{r}$

Try It

5) Write as a radial expression:

a. $b^{\frac{1}{6}}$
b. $z^{\frac{1}{5}}$
c. $p^{\frac{1}{4}}$

Solution

a. $\sqrt[6]{b}$
b. $\sqrt[5]{z}$
c. $\sqrt[4]{p}$

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 3.3.2

Write with a rational exponent:

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

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

Solution

a.

Step 1: No index is shown, so it is $2$ .

Step 2: The denominator of the exponent will be $2$ .
Put parentheses around the entire expression $5 y$ .

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

b.

Step 1: The index is $3$ , so the denominator of the exponent is $3$ .
Include parentheses $(4 x)$ .

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

c.

Step 1: 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

6) Write with a rational exponent:

a. $\sqrt{10 m}$
b. $\sqrt[5]{3 n}$
c. $3 \sqrt[4]{6 y}$

Solution

a. ${(10 m)}^{\frac{1}{2}}$
b. ${(3 n)}^{\frac{1}{5}}$
c. $3 {(6 y)}^{\frac{1}{4}}$

Try It

7) Write with a rational exponent:

a. $\sqrt[7]{3 k}$
b. $\sqrt[4]{5 j}$
c. $8 \sqrt[3]{2 a}$

Solution

a. ${(3 k)}^{\frac{1}{7}}$
b. ${(5 j)}^{\frac{1}{4}}$
c. $8 {(2 a)}^{\frac{1}{3}}$

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

Example 3.3.3

Simplify:

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

Solution

a.

Step 1: Rewrite as a square root.

$\sqrt{25}$

Step 2: Simplify.

$5$

b.

Step 1: Rewrite as a cube root.

$\sqrt[3]{64}$

Step 2: Recognize $64$ is a perfect cube.

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

Step 3: Simplify.

$4$

c.

Step 1: Rewrite as a fourth root.

$\sqrt[4]{256}$

Step 2: Recognize $256$ is a perfect fourth power.

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

Step 3: Simplify.

$4$

Try It

8) Simplify:

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

Solution

a. $6$
b. $2$
c. $2$

Try It

9) Simplify:

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

Solution

a. $10$
b. $3$
c. $3$

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 3.3.4

Simplify:

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

Solution

a.

Step 1: Rewrite as a fourth root.

$\begin{aligned} \sqrt[4]{- 16} \\ \sqrt[4]{{(- 2)}^{4}} \end{aligned}$

Step 2: Simplify.

No real solution

b.

Step 1: The exponent only applies to the $16$ . Rewrite as a fourth root.

$\sqrt[4]{16}$

Step 2: Rewrite $16$ as $2^{4}$

$\sqrt[4]{2^{4}}$

Step 3: Simplify.

$- 2$

c.

Step 1: Rewrite using the property $a^{- n} = \frac{1}{a^{n}}$

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

Step 2: Rewrite as a fourth root.

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

Step 3: Rewrite $16$ as $2^{4}$

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

Step 4: Simplify.

$\frac{1}{2}$

Try It

10) Simplify:

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

Solution

a. No real solution
b. $- 8$
c. $\frac{1}{8}$

Try It

11) Simplify:

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

Solution

a. No real solution
b. $- 4$
c. $\frac{1}{4}$

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 3.3.5

Write with a rational exponent:

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

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

Solution

a.

$\begin{array}{lll} The numerator of the exponent is the exponent, 3 . & y^{\frac{3}{2}} \\ The denominator of the exponent is the index of the radical, 2 . \end{array}$

b.

$\begin{array}{lll} The numerator of the exponent is the exponent, 4 . & {(2 x)}^{\frac{4}{3}} \\ The denominator of the exponent is the index of the radical, 3 . \\ {(\sqrt[3]{2 x})}^{4} \end{array}$

c.

$\begin{array}{lll} The numerator of the exponent is the exponent, 3 . & {(\frac{3 a}{4 b})}^{\frac{3}{2}} \\ The denominator of the exponent is the index of the radical, 2 . \\ \sqrt{{(\frac{3 a}{4 b})}^{3}} \end{array}$

Try It

12) Write with a rational exponent:

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

Solution

a. $x^{\frac{5}{2}}$
b. ${(3 y)}^{\frac{3}{4}}$
c. ${(\frac{2 m}{3 n})}^{\frac{5}{2}}$

Try It

13) Write with a rational exponent:

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

Solution

a. $a^{\frac{2}{5}}$
b. ${(5 a b)}^{\frac{5}{3}}$
c. ${(\frac{7 x y}{z})}^{\frac{3}{2}}$

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

Example 3.3.6

Simplify:

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

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

Solution

a.

Step 1: The power of the radical is the numerator of the exponent, $2$ .

Step 2: The index of the radical is the denominator of the exponent, $3$ .

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

Step 3: Simplify.

$\begin{aligned} {(5)}^{2} \\ 25 \end{aligned}$

b.

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

Step 1: Rewrite using $a^{- n} = \frac{1}{a^{n}}$ .

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

Step 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, $2$ .

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

Step 3: Simplify.

$\begin{aligned} \frac{1}{4^{3}} \\ \frac{1}{64} \end{aligned}$

c.

Step 1: Rewrite using $a^{- n} = \frac{1}{a^{n}}$ .

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

Step 2: Change to radical form.

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

Step 3: Rewrite the radicand as a power.

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

Step 4: Simplify.

$\begin{aligned} \frac{1}{2^{2}} \\ \frac{1}{4} \end{aligned}$

Try It

14) Simplify:

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

Solution

a. $9$
b. $\frac{1}{729}$
c. $\frac{1}{8}$

Try It

15) Simplify:

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

Solution

a. $8$
b. $\frac{1}{9}$
c. $\frac{1}{125}$

Example 3.3.7

Simplify:

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

Solution

a.

Step 1: Rewrite in radical form.

${(\sqrt{25})}^{3}$

Step 2: Simplify the radical.

$- {(5)}^{3}$

Step 3: Simplify.

$- 125$

b.

Step 1: Rewrite using $a^{- n} = \frac{1}{a^{n}}$ .

$- (\frac{1}{25^{\frac{3}{2}}})$

Step 2: Rewrite in radical form.

$- (\frac{1}{{(\sqrt{25})}^{3}})$

Step 3: Simplify the radical.

$- (\frac{1}{{(5)}^{3}})$

Step 4: Simplify.

$- \frac{1}{125}$

c.

Step 1: Rewrite in radical form.

${(\sqrt{- 25})}^{3}$

There is no real number whose square root is $- 25$ .

Not a real number.

Try It

16) Simplify:

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

Solution

a. $- 64$
b. $- \frac{1}{64}$
c. Not a real number.

Try It

17) Simplify:

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

Solution

a. $- 729$
b. $- \frac{1}{729}$
c. Not a real number.

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 Exponents 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

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

We will apply these properties in the next example.

Example 3.3.8

Simplify:

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

Solution

a.

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

Step 1: The bases are the same, so we add the exponents.

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

Step 2: Add the fractions.

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

Step 3: Simplify the exponent.

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

b.

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

Step 1: To raise a power to a power, we multiply the exponents.

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

Step 2: Simplify.

$z^{6}$

c.

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

Step 1: To divide with the same base, we subtract the exponents.

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

Step 2: Simplify.

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

Try It

18) Simplify:

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

Solution

a. $x^{\frac{3}{2}}$
b. $x^{8}$
c. $\frac{1}{x}$

Try It

19) Simplify:

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

Solution

a. $y^{\frac{11}{8}}$
b. $m^{2}$
c. $\frac{1}{d}$

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 3.3.9

Simplify:

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

Solution

a.

Step 1: First we use the Product to a Power Property.

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

Step 2: Rewrite $27$ as a power of $3$ .

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

Step 3: To raise a power to a power, we multiply the exponents.

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

Step 4: Simplify.

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

b.

Step 1: First we use the Product to a Power Property.

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

Step 2: To raise a power to a power, we multiply the exponents.

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

Try It

20) Simplify:

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

Solution

a. $8 x^{\frac{1}{5}}$
b. $x^{\frac{1}{2}} y^{\frac{1}{3}}$

Try It

21) Simplify:

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

Solution

a. $729 n^{\frac{3}{5}}$
b. $a^{2} b^{\frac{2}{3}}$

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

Example 3.3.10

Simplify:

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

Solution

a.

Step 1: Use the Product Property in the numerator, add the exponents.

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

Step 2: Use the Quotient Property, subtract the exponents.

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

Step 3: Simplify.

$x^{2}$

b.

Follow the order of operations to simplify inside the parentheses first.

Step 1: Use the Quotient Property, subtract the exponents.

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

Step 2: Simplify.

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

Step 3: Use the Product to a Power Property, multiply the exponents.

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

Try It

22) Simplify:

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

Solution

a. $m^{2}$
b. $\frac{5 n}{m^{\frac{1}{4}}}$

Try It

23) Simplify:

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

Solution

a. $u^{3}$
b. $3 x^{\frac{1}{5}} y^{\frac{1}{3}}$

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

Key Concepts

Rational Exponent $a^{\frac{1}{n}}$
- If $\sqrt[n]{a}$ is a real number and $n \geq 2$ , then $a^{\frac{1}{n}} = \sqrt[n]{a}$
Rational Exponent $a^{\frac{m}{n}}$
- For any positive integers $m$ and $n$ , $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ and $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$
Properties of Exponents
- If $a$ , $b$ are real numbers and $m$ , $n$ are rational numbers, then
  - 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 $a^{− n} = \frac{1}{a^{n}}, a \neq 0$

Self Check

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

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

License

Icon for the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License

Fanshawe Pre-Health Sciences Mathematics 2 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

Simplify Expressions with a1n

Rational Exponent a1n

Simplify Expressions with amn

Rational Exponent amn

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

Properties of Exponents

Self Check

License

Share This Book

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

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

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

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