8.3: Exponential Properties Involving Quotients
What if you had a fractional expression like
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CK12 Foundation: 0803S Quotient of Powers
Guidance
The rules for simplifying quotients of exponents are a lot like the rules for simplifying products.
Example A
Let’s look at what happens when we divide
You can see that when we divide two powers of
Quotient Rule for Exponents:
When we have expressions with more than one base, we apply the quotient rule separately for each base:
Now let’s see what happens if the exponent in the denominator is bigger than the exponent in the numerator. For example, what happens when we apply the quotient rule to
The quotient rule tells us to subtract the exponents. 4 minus 7 is 3, so our answer is
Example B
OR
Well, let’s look at what we get when we do the division longhand by writing each term in factored form:
Even when the exponent in the denominator is bigger than the exponent in the numerator, we can still subtract the powers. The
Example C
Simplify the following expressions, leaving all exponents positive.
a)
b)
Solution
a) Subtract the exponent in the numerator from the exponent in the denominator and leave the
b) Apply the rule to each variable separately:
Watch this video for help with the Examples above.
CK12 Foundation: Quotient of Powers
Vocabulary
Quotient of Powers Property: For all real numbers
Guided Practice
Simplify each of the following expressions using the quotient rule.
a)
b)
c)
Solution
a)
b)
c)
Review Questions
Evaluate the following expressions.

5652 
6763 
34310 
22⋅3252 
33⋅5237
Simplify the following expressions.

a3a2 
x5x9 
x6y2x2y5 
6a32a2 
15x55x 
25yx620y5x2
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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 simplify one exponential expression that is divided by another.