Roots and Radicals

# 79 Rational Exponents

### Learning Objectives

By the end of this section, you will be able to:

- Simplify expressions with
- Simplify expressions with
- Use the Laws of Exponents to simply expressions with rational exponents

Before you get started, take this readiness quiz.

### Simplify Expressions with

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 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 . We will use the Power Property of Exponents to find the value of *p*.

But we know also . Then it must be that .

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

If is a real number and , .

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.

Write as a radical expression: ⓐ ⓑ ⓒ .

We want to write each expression in the form .

ⓐ

ⓑ

ⓒ

Write as a radical expression: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Write as a radial expression: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Write with a rational exponent: ⓐ ⓑ ⓒ .

We want to write each radical in the form .

ⓐ

ⓑ

ⓒ

Write with a rational exponent: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Write with a rational exponent: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Write with a rational exponent: ⓐ ⓑ ⓒ .

We want to write each radical in the form .

- ⓐ
- ⓑ
- ⓒ

Write with a rational exponent: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Write with a rational exponent: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

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

Simplify: ⓐ ⓑ ⓒ .

ⓐ

ⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐ 6 ⓑ 2 ⓒ 2

Simplify: ⓐ ⓑ ⓒ .

ⓐ 10 ⓑ 3 ⓒ 3

Be careful of the placement of the negative signs in the next example. We will need to use the property in one case.

Simplify: ⓐ ⓑ ⓒ .

ⓐ

ⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐ

ⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

### Simplify Expressions with

Let’s work with the Power Property for Exponents some more.

Suppose we raise to the power *m.*

Now suppose we take to the power.

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.

For any positive integers *m* and *n*,

Write with a rational exponent: ⓐ ⓑ ⓒ .

We want to use to write each radical in the form .

- ⓐ
- ⓑ
- ⓒ

Write with a rational exponent: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Write with a rational exponent: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Simplify: ⓐ ⓑ ⓒ .

We will rewrite each expression as a radical first using the property, . 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.

- ⓐ
- ⓑ
- ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐ 8 ⓑ 9 ⓒ 125

Simplify: ⓐ ⓑ ⓒ .

ⓐ 32 ⓑ 729 ⓒ 8

Remember that . The negative sign in the exponent does not change the sign of the expression.

Simplify: ⓐ ⓑ ⓒ .

We will rewrite each expression first using and then change to radical form.

ⓐ

ⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐ

ⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ not a real number

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ not a real number

### 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.

If are real numbers and are rational numbers, then

When we multiply the same base, we add the exponents.

Simplify: ⓐ ⓑ ⓒ .

ⓐ

ⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐ 9 ⓑ ⓒ *m*

Simplify: ⓐ ⓑ ⓒ .

ⓐ 25 ⓑ *z* ⓒ *n*

We will use the Power Property in the next example.

Simplify: ⓐ ⓑ ⓒ .

ⓐ

ⓑ

ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

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

Simplify: ⓐ ⓑ ⓒ .

- ⓐ
- ⓑ
- ⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐ*u*ⓑⓒ

Simplify: ⓐ ⓑ ⓒ .

ⓐⓑⓒ

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.

Simplify: ⓐ ⓑ .

- ⓐ
- ⓑ

Simplify: ⓐ ⓑ .

ⓐⓑ

Simplify: ⓐ ⓑ .

ⓐⓑ

Simplify: ⓐ ⓑ .

- ⓐ
- ⓑ

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

Simplify: ⓐ ⓑ .

- ⓐ
- ⓑ

Simplify: ⓐ ⓑ .

ⓐⓑ

Simplify: ⓐ ⓑ .

ⓐⓑ

### Key Concepts

**Summary of Exponent Properties**- If are real numbers and are rational numbers, then
**Product Property****Power Property****Product to a Power****Quotient Property**:

**Zero Exponent Definition**,**Quotient to a Power Property**

### Section Exercises

#### Practice Makes Perfect

**Simplify Expressions with **

In the following exercises, write as a radical expression.

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

In the following exercises, write with a rational exponent.

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

In the following exercises, simplify.

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ 5 ⓑ 3 ⓒ 2

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ 6 ⓑ 2 ⓒ 3

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ not a real number ⓑ ⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ not a real number ⓑ ⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑ

ⓒ

**Simplify Expressions with **

In the following exercises, write with a rational exponent.

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

In the following exercises, simplify.

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ 100 ⓑ 125 ⓒ 8

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ 64 ⓑ 3125 ⓒ 256

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ 32,768 ⓑ ⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ 1000 ⓑ ⓒ not a real number

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑⓒ not a real number

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ not a real number

**Use the Laws of Exponents to Simplify Expressions with Rational Exponents**

In the following exercises, simplify.

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ 216 ⓑ ⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ 100 ⓑ ⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓒ

ⓐ

ⓑ

ⓒ

ⓐⓑⓒ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

#### Everyday Math

**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 to find the length of each side of his garden.

**Landscaping** Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify to find the length of each side of his patio.Round to the nearest tenth of a foot.

15.6 feet

**Gravity** While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

**Gravity** An airplane dropped a flare from a height of 1024 feet above a lake. Simplify to find how many seconds it took for the flare to reach the water.

8 seconds

#### Writing Exercises

Show two different algebraic methods to simplify Explain all your steps.

Explain why the expression cannot be evaluated.

### Chapter 9 Review Exercises

#### Simplify and Use Square Roots

**Simplify Expressions with Square Roots**

In the following exercises, simplify.

12

not a real number

17

**Estimate Square Roots**

In the following exercises, estimate each square root between two consecutive whole numbers.

**Approximate Square Roots**

In the following exercises, approximate each square root and round to two decimal places.

7.55

**Simplify Variable Expressions with Square Roots**

In the following exercises, simplify.

#### Simplify Square Roots

**Use the Product Property to Simplify Square Roots**

In the following exercises, simplify.

**Use the Quotient Property to Simplify Square Roots**

In the following exercises, simplify.

#### Add and Subtract Square Roots

**Add and Subtract Like Square Roots**

In the following exercises, simplify.

**Add and Subtract Square Roots that Need Simplification**

In the following exercises, simplify.

#### Multiply Square Roots

**Multiply Square Roots**

In the following exercises, simplify.

8

50

**Use Polynomial Multiplication to Multiply Square Roots**

In the following exercises, simplify.

#### Divide Square Roots

**Divide Square Roots**

In the following exercises, simplify.

**Rationalize a One Term Denominator**

In the following exercises, rationalize the denominator.

**Rationalize a Two Term Denominator**

In the following exercises, rationalize the denominator.

#### Solve Equations with Square Roots

**Solve Radical Equations**

In the following exercises, solve the equation.

5

no solution

13

0

11

**Use Square Roots in Applications**

In the following exercises, solve. Round approximations to one decimal place.

A pallet of sod will cover an area of about 600 square feet. Trinh wants to order a pallet of sod to make a square lawn in his backyard. Use the formula to find the length of each side of his lawn.

A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula to find how many seconds it took for the package to reach the hiker.

7.5 seconds

Officer Morales measured the skid marks of one of the cars involved in an accident. The length of the skid marks was 245 feet. Use the formula to find the speed of the car before the brakes were applied.

#### Higher Roots

**Simplify Expressions with Higher Roots**

In the following exercises, simplify.

ⓐ

ⓑ

ⓐ 2 ⓑ 4

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

**Use the Product Property to Simplify Expressions with Higher Roots**

In the following exercises, simplify.

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ not a real number

**Use the Quotient Property to Simplify Expressions with Higher Roots**

In the following exercises, simplify.

**Add and Subtract Higher Roots**

In the following exercises, simplify.

#### Rational Exponents

**Simplify Expressions with **

In the following exercises, write as a radical expression.

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

2

**Simplify Expressions with **

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

32,768

**Use the Laws of Exponents to Simplify Expressions with Rational Exponents**

In the following exercises, simplify.

### Practice Test

In the following exercises, simplify.

ⓐ

ⓑ

ⓐ

ⓑ

ⓐⓑ

ⓐ

ⓑ

343

In the following exercises, rationalize the denominator.

In the following exercises, solve.

42

In the following exercise, solve.

A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.

6.1 seconds

### Glossary

- rational exponents

- If is a real number and , .
- For any positive integers
*m*and*n*, and .