# 8.2: Exponential Properties Involving Quotients

**Basic**Created by: CK-12

**Practice**Exponential Properties Involving Quotients

Suppose you wanted to know the volume of a cube and the area of one of its bases. If the length of one of its edges was

### Guidance

In this Concept, you will learn how to simplify quotients of numbers and variables.

**Quotient of Powers Property:** For all real numbers

When dividing expressions with the same base, keep the base and subtract the exponent in the denominator (bottom) from the exponent in the numerator (top). When we have problems with different bases, we apply the rule separately for each base.

#### Example A

*Simplify x7x4.*

**Solution:** To simplify

#### Example B

*Simplify each of the following expressions using the* *quotient rule.*

(a)

(b)

**Solution:**

(a)

(b)

**Power of a Quotient Property:**

The power inside the parenthesis for the numerator and the denominator multiplies with the power outside the parenthesis. The situation below shows why this property is true.

#### Example C

*Simplify (x3y2)4.*

### Video Review

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### Guided Practice

*Simplify the following expression.*

**Solution:**

### Explore More

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Exponent Properties Involving Quotients (9:22)

Evaluate the following expressions.

5652 6763 31034 (2233)3

Simplify the following expressions.

a3a2 x9x5 x10x5 a6a a5b4a3b2 4542 5357 (3452)2 (a3b4a2b)3 x6y5x2y3 6x2y32xy2 (2a3b38a7b)2 (x2)2⋅x6x4 (16a24b5)3⋅b2a16 6a32a2 15x55x (18a1015a4)4 25yx620y5x2 (x6y2x4y4)3 (6a24b4)2⋅5b3a (3ab)2(4a3b4)3(6a2b)4 (2a2bc2)(6abc3)4ab2c

**Mixed Review**

- Evaluate
x|z|−|z| whenx=8 andz=−4 . - Graph the solution set to the system
{y<−x−2y≥−6x+3 . - Evaluate
(84) . - Make up a situation that can be solved by 4!.
- Write the following as an algebraic sentence:
*A number cubed is 8.*

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 8.2.

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Power of a Quotient Property

Quotient of Powers Property

For all real numbers , .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

Here you'll learn how to use the quotient rule and how to work with powers of quotients.

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