8.3 Simplify Rational Exponents

Learning objectives

Learning objectives

• 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

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.

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

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

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

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, sothe index of the radical is 2. We do not show theindex when it is 2. · $\,\sqrt{x}$

ⓑ

$\,{y}^{\frac{1}{3}}$

The denominator of the exponent is 3, so theindex is 3. · $\,\sqrt[3]{y}$

ⓒ

$\,{z}^{\frac{1}{4}}$

The denominator of the exponent is 4, so theindex is 4. · $\,\sqrt[4]{z}$

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

Solution

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

ⓐ

$\,\sqrt{5y}$

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

Put parentheses around the entireexpression $5y.$

ⓑ

$\,\sqrt[3]{4x}$

The index is 3, so the denominator of theexponent is 3. Include parentheses $(4x).$ · $\,{(4x)}^{\frac{1}{3}}$

ⓒ

$\,3\,\sqrt[4]{5z}$

The index is 4, so the denominator of theexponent is 4. Put parentheses only aroundthe $5z$ since 3 is not under the radical sign. · $\,3{(5z)}^{\frac{1}{4}}$

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

Example 8.28

Solution

ⓐ

$\,{25}^{\frac{1}{2}}$

Rewrite as a square root. · $\,\sqrt{25}$ Simplify. · $\,5$

ⓑ

$\,{64}^{\frac{1}{3}}$

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

ⓒ

$\,{256}^{\frac{1}{4}}$

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

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

Solution

ⓐ

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

Rewrite as a fourth root. · $\,\sqrt[4]{-16}$

$\,\sqrt[4]{{(-2)}^{4}}$

Simplify. · $\,\mathrm{No real solution.}$

ⓑ

$\,-{16}^{\frac{1}{4}}$

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. · $\,-2$

ⓒ

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

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. · $\,\frac{1}{2}$

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

This leads us to the following definition.

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

Solution

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

ⓐ

ⓑ

ⓒ

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

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 theexponent, 3. · $\,{(\sqrt[3]{125})}^{2}$ Simplify. · $\,{(5)}^{2}$

$\,25$

ⓑ 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 thenumerator of the exponent, 3. The index is the denominatorof the exponent, 2. · $\,\frac{1}{{(\sqrt{16})}^{3}}$ Simplify. · $\,\frac{1}{{4}^{3}}$

$\,\frac{1}{64}$

ⓒ

$\,{32}^{-\frac{2}{5}}$

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. · $\,\frac{1}{{2}^{2}}$

$\,\frac{1}{4}$

Example 8.32

Solution

ⓐ

$\,-{25}^{\frac{3}{2}}$

Rewrite in radical form. · $\,-{(\sqrt{25})}^{3}$ Simplify the radical. · $\,-{(5)}^{3}$ Simplify. · $\,-125$

ⓑ

$\,-{25}^{-\frac{3}{2}}$

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. · $\,-\frac{1}{125}$

ⓒ

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

Rewrite in radical form. · $\,{(\sqrt{-25})}^{3}$ There is no real number whose square root$\mathrm{is}\,-25.$ · $\,\mathrm{Not a real number.}$

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 Exponenets here to have them for reference as we simplify expressions.

We will apply these properties in the next example.

Example 8.33

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 theexponents. · $\,{x}^{\frac{1}{2}+\frac{5}{6}}$ Add the fractions. · $\,{x}^{\frac{8}{6}}$ Simplify the exponent. · $\,{x}^{\frac{4}{3}}$

ⓑ 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 multiplythe exponents. · $\,{z}^{9\cdot\frac{2}{3}}$ Simplify. · $\,{z}^{6}$

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

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

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

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

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

Solution

ⓐ

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

First we use the Product to a PowerProperty. · $\,{(27)}^{\frac{2}{3}}{({u}^{\frac{1}{2}})}^{\frac{2}{3}}$ 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 multiplythe exponents. · $\,({3}^{2})({u}^{\frac{1}{3}})$ Simplify. · $\,9{u}^{\frac{1}{3}}$

ⓑ

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

First we use the Product to a PowerProperty. · $\,{({m}^{\frac{2}{3}})}^{\frac{3}{2}}{({n}^{\frac{1}{2}})}^{\frac{3}{2}}$ To raise a power to a power, we multiplythe exponents. · $\,m{n}^{\frac{3}{4}}$

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

Example 8.35

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 theexponents. · $\,{x}^{\frac{8}{4}}$ Simplify. · $\,{x}^{2}$

ⓑ 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 theexponents. · $\,{(\frac{16{x}^{\frac{6}{3}}}{{y}^{\frac{6}{6}}})}^{\frac{1}{2}}$ Simplify. · $\,{(\frac{16{x}^{2}}{y})}^{\frac{1}{2}}$ Use the Product to a Power Property,multiply the exponents. · $\,\frac{4x}{{y}^{\frac{1}{2}}}$

Section 8.3 Exercises

Section 8.3 Exercises

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

In the following exercises, write as a radical expression.

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

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

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

Writing Exercises

ⓐ 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?

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

2. Write as a radial expression: ⓐ ${b}^{\frac{1}{6}}$ ⓑ ${z}^{\frac{1}{5}}$ ⓒ ${p}^{\frac{1}{4}}.$

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

1. Write with a rational exponent: ⓐ $\sqrt[7]{3k}$ ⓑ $\sqrt[4]{5j}$ ⓒ $8\sqrt[3]{2a}.$

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

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

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

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

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

1. Write with a rational exponent: ⓐ $\sqrt[5]{{a}^{2}}$ ⓑ ${(\sqrt[3]{5ab})}^{5}$ ⓒ $\sqrt{{(\frac{7xy}{z})}^{3}}.$

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

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

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

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

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

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

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

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

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

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

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

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

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

3. ⓐ $\sqrt[3]{7c}$ ⓑ $\sqrt[7]{12d}$ ⓒ $2\sqrt[4]{6b}$

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

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

3. ⓐ ${16}^{\frac{1}{4}}$ ⓑ ${16}^{\frac{1}{2}}$ ⓒ ${625}^{\frac{1}{4}}$

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

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

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

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

2. ⓐ $\sqrt{{m}^{5}}$ ⓑ ${(\sqrt[3]{3y})}^{7}$ ⓒ $\sqrt[5]{{(\frac{4x}{5y})}^{3}}$

3. ⓐ $\sqrt[5]{{u}^{2}}$ ⓑ ${(\sqrt[3]{6x})}^{5}$ ⓒ $\sqrt[4]{{(\frac{18a}{5b})}^{7}}$

1. ⓐ ${64}^{\frac{5}{2}}$ ⓑ ${81}^{\frac{-3}{2}}$ ⓒ ${(-27)}^{\frac{2}{3}}$

2. ⓐ ${32}^{\frac{2}{5}}$ ⓑ ${27}^{-\frac{2}{3}}$ ⓒ ${(-25)}^{\frac{1}{2}}$

3. ⓐ $-{9}^{\frac{3}{2}}$ ⓑ $-{9}^{-\frac{3}{2}}$ ⓒ ${(-9)}^{\frac{3}{2}}$

1. ⓐ ${c}^{\frac{1}{4}}\cdot{c}^{\frac{5}{8}}$ ⓑ ${({p}^{12})}^{\frac{3}{4}}$ ⓒ $\frac{{r}^{\frac{4}{5}}}{{r}^{\frac{9}{5}}}$

2. ⓐ ${y}^{\frac{1}{2}}\cdot{y}^{\frac{3}{4}}$ ⓑ ${({x}^{12})}^{\frac{2}{3}}$ ⓒ $\frac{{m}^{\frac{5}{8}}}{{m}^{\frac{13}{8}}}$

3. ⓐ ${(27{q}^{\frac{3}{2}})}^{\frac{4}{3}}$ ⓑ ${({a}^{\frac{1}{3}}{b}^{\frac{2}{3}})}^{\frac{3}{2}}$

1. ⓐ ${(16\,{u}^{\frac{1}{3}})}^{\frac{3}{4}}$ ⓑ ${(4\,{p}^{\frac{1}{3}}{q}^{\frac{1}{2}})}^{\frac{3}{2}}$

2. ⓐ $\frac{{r}^{\frac{5}{2}}\cdot{r}^{-\frac{1}{2}}}{{r}^{-\frac{3}{2}}}$ ⓑ ${(\frac{36\,{s}^{\frac{1}{5}}{t}^{-\frac{3}{2}}}{{s}^{-\frac{9}{5}}{t}^{\frac{1}{2}}})}^{\frac{1}{2}}$

3. ⓐ $\frac{{c}^{\frac{5}{3}}\cdot{c}^{-\frac{1}{3}}}{{c}^{-\frac{2}{3}}}$ ⓑ ${(\frac{8\,{x}^{\frac{5}{3}}\,{y}^{-\frac{1}{2}}}{27\,{x}^{-\frac{4}{3}}\,{y}^{\frac{5}{2}}})}^{\frac{1}{3}}$

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