# 7.2: Rational Exponents and Roots

**At Grade**Created by: CK-12

**Practice**Fractional Exponents

A planet's maximum distance from the sun (in astronomical units) is given by the formula *p* is the period (in years) of the planet's orbit around the sun. If a planet's orbit around the sun is 27 years, what is its distance from the sun?

### Guidance

Now that you are familiar with nth roots, we will convert them into exponents. Let’s look at the square root and see if we can use the properties of exponents to determine what exponential number it is equivalent to.

#### Investigation: Writing the Square Root as an Exponent

1. Evaluate

- The
√ and the2 cancel each other out,(x√2)=x .

2. Recall that when a power is raised to another power, we multiply the exponents. Therefore, we can rewrite the exponents and root as an equation,

n⋅22n=12=12

3. From #2, we can conclude that

(x√)2=(x12)2=x(12)⋅2=x1=x

From this investigation, we see that

#### Example A

Find

**Solution:** Rewrite this expression in terms of roots. A number to the one-fourth power is the same as the fourth root.

Therefore,

#### Example B

Find

**Solution:** This problem is the same as the ones in the previous concept. However, now, the root is written in the exponent. Rewrite the problem.

From the previous concept, we know that it is easier to evaluate the second option above.

**The Rational Exponent Theorem:** For any real number

#### Example C

Find

**Solution:** To type this into a calculator, the keystrokes would probably look like:

Other calculators might have a

**Intro Problem Revisit** Substitute 27 for *p* and solve.

Rewrite the problem.

Therefore, the planet's distance from the sun is 9 astronomical units.

### Guided Practice

1. Rewrite

2. Rewrite

Evaluate without a calculator.

3.

4.

5.

#### Answers

1. Using rational exponents, the

2. Using roots, the 9 in the denominator of the exponent is the root;**MATH** and select **5**:

3.

4.

5.

### Explore More

Write the following expressions using rational exponents and then evaluate using a calculator. Answers should be rounded to the nearest hundredth.

45−−√5 140−−−√9 50−−√83

Write the following expressions using roots and then evaluate using a calculator. Answers should be rounded to the nearest hundredth.

7253 9523 12534

Evaluate the following without a calculator.

6423 2743 1654 253−−−√ 9√25 322−−−√5

For the following problems, rewrite the expressions with rational exponents and then simplify the exponent and evaluate without a calculator.

(23)8−−−−−√4 72−−√36 (16)12−−−−−√6

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.2.

### Image Attributions

## Description

## Learning Objectives

Here you'll learn about rational exponents and relate them to

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## Date Created:

Mar 12, 2013## Last Modified:

Jun 04, 2015## Vocabulary

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