# 6.2: Quotient Rules for Exponents

**Advanced**Created by: CK-12

**Practice**Exponential Properties Involving Quotients

Suppose you have the expression:

How could you write this expression in a more concise way?

### Watch This

James Sousa: Simplify Exponential Expressions- Quotient Rule

### Guidance

In the expression **base** and the **exponent**. **Exponents** are often referred to as **powers**. When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:

x3=x⋅x⋅x 24=2⋅2⋅2⋅2=16 .

There are many rules that have to do with exponents (often called the **Laws of Exponents**) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn two rules that have to do with exponents and quotients.

**RULE: To divide two powers with the same base, subtract the exponents.**

m factors↑aman=(a×a×…×a)←→−−−−−−−−−−(a×a×…×a)←→−−−−−−−−−− m>n;a≠0↓ n factorsaman=(a×a×…×a)←→−−−−−−−−−−↓ m−n factorsaman=am−n

**RULE: To raise a quotient to a power, raise both the numerator and the denominator to the power.**

(ab)n=ab×ab×…×ab←→−−−−−−−−−− ↓n factorsn factors ↑(ab)n=(a×a×…×a)←→−−−−−−−−−−(b×b×…×b)←→−−−−−−−−− ↓n factors(ab)n=anbn (b≠0)

#### Example A

Simplify

**Solution:**

The answer can be taken one step further. The base is numerical so the term can be evaluated.

#### Example B

Simplify

**Solution:**

#### Example C

Simplify

**Solution:**

#### Concept Problem Revisited

### Vocabulary

- Base
- In an algebraic expression, the
is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression*base*25 , ‘2’ is the base. In the expression(−3y)4 , ‘−3y ’ is the base.

- Exponent
- In an algebraic expression, the
is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:*exponent*

- In the expression
25 , ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here:25=2×2×2×2×2 . - In the expression
(−3y)4 , ‘4’ is the exponent. It means to multiply−3y times itself 4 times as shown here:(−3y)4=−3y×−3y×−3y×−3y .

- Laws of Exponents
- The
are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.*laws of exponents*

### Guided Practice

Simplify each of the following expressions.

1.

2.

3.

**Answers:**

1.

2.

3.

### Practice

Simplify each of the following expressions, if possible.

(25)6 (47)3 (xy)4 20x4y55x2y4 42x2y8z26xy4z (3x4y)3 72x2y48x2y3 (x4)5 24x14y83x5y7 72x3y924xy6 (7y)3 20x12−5x8

- Simplify using the laws of exponents:
2325 - Evaluate the numerator and denominator separately and then simplify the fraction:
2325 - Use your result from the previous problem to determine the value of
a :2325=12a - Use your results from the previous three problems to help you evaluate
2−4 .

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.2.

Base

When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression , 32 is the base, and 4 is the exponent.Exponent

Exponents are used to describe the number of times that a term is multiplied by itself.Power

The "power" refers to the value of the exponent. For example, is "three to the fourth power".### Image Attributions

## Description

## Learning Objectives

Here you'll learn how to divide two terms with the same base and find the power of a quotient.

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Advanced## Subjects:

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## Date Created:

Apr 30, 2013## Last Modified:

Feb 26, 2015## Vocabulary

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