8.2: 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 , what would be the volume of the cube? What would be its base area? Knowing that you could find the height of the cube by dividing its volume by its base area, what expression could you write to represent this quotient? In this Concept, you'll learn about exponential properties involving quotients so that you can perform division problems such as this one.
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 .
Solution: To simplify , repeated multiplication can be used.
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 .
Vocabulary
Quotient of Powers Property: For all real numbers ,
.
Power of a Quotient Property:
Guided Practice
Simplify the following expression.
Solution:
Practice
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.
Simplify the following expressions.
Mixed Review
- Evaluate when and .
- Graph the solution set to the system .
- Evaluate .
- Make up a situation that can be solved by 4!.
- Write the following as an algebraic sentence: A number cubed is 8.
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
Description
Learning Objectives
Here you'll learn how to use the quotient rule and how to work with powers of quotients.