# 7.3: Applying the Laws of Exponents to Rational Exponents

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

**Practice**Operations with Rational Exponents

The period (in seconds) of a pendulum with a length of *L* (in meters) is given by the formula

### Guidance

When simplifying expressions with rational exponents, all the laws of exponents that were learned in the *Polynomial Functions* chapter are still valid. On top of that, the rules of fractions apply as well.

#### Example A

Simplify

**Solution:** Recall from the Product Property of Exponents, that when two numbers with the same base are multiplied we *add* the exponents. Here, the exponents do not have the same base, so we need to find a common denominator and then add the numerators.

This rational exponent does not reduce, so we are done.

#### Example B

Simplify

**Solution:** This problem utilizes the Quotient Property of Exponents. Subtract the exponents with the same base and reduce

If you are writing your answer in terms of positive exponents, your answer would be

If we were to write the answer using roots, then we would take out the whole numbers. For example,

#### Example C

Simplify

**Solution:** On the numerator, the entire expression is raised to the

Combine like terms by subtracting the exponents.

Finally, rewrite the answer with positive exponents by moving the 2 and

**Intro Problem Revisit** Substitute *L* and solve.

Therefore, the period of the pendulum is

### Guided Practice

Simplify each expression. Reduce all rational exponents and write final answers using positive exponents.

1.

2.

3.

#### Answers

1. Change 4 and 8 so that they are powers of 2 and then add exponents with the same base.

2. Subtract the exponents. Change the

3. Distribute the

### Explore More

Simplify each expression. Reduce all rational exponents and write final answer using positive exponents.

15a452532a35 7b434912b−23 m89m23 x47y116x114y53 853r5s34t1324r215s2t79 (a32b45)103 (5x57y4)32 ⎛⎝4x259y45⎞⎠52 ⎛⎝75d1853d35⎞⎠52 ⎛⎝8132a38a92⎞⎠13 2723m45n−32412m−23n85 ⎛⎝3x38y255x14y−310⎞⎠2 - Rewrite your answer from Problem #1 using radicals.
- Rewrite your answer from Problem #4 using radicals.
- Rewrite your answer from Problem #4 using one radical.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.3.

### Image Attributions

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## Learning Objectives

Here you'll learn how to use the laws of exponents with rational exponents.

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## Date Created:

Mar 12, 2013## Last Modified:

Jun 04, 2015## Vocabulary

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