Activity
Product Property for Exponents
If is a real number and and are integers, then
To multiply with like bases, add the exponents.
Complete problems 1 - 6 using the product property for exponents.
1.
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2.
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3.
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4.
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5.
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6.
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Product Quotient for Exponents
If is a real number, , and and are integers, then
, and ,
To divide with like bases, subtract the exponents.
Complete problems 7 - 12 using the quotient property for exponents.
7.
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8.
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9.
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10.
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11.
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12.
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Self Check
Additional Resources
Using Product and Quotient Properties for Exponents
When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.
First, we will look at an example that leads to the Product Property.
What does mean?
Notice that 5 is the sum of the exponents, 2 and 3. We see is or .
The base stayed the same, and we added the exponents. This leads to the Product Property for Exponents.
Product Property for Exponents
If is a real number and and are integers, then
To multiply with like bases, add the exponents.
Example 1
Example 2
Example 3
Example 4
Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.
1. Consider
Step 1 - What does it mean?
Step 2 - Use the Equivalent Fractions Property.
Step 3 - Simplify.
2. Consider
Step 1 - What does it mean?
Step 2 - Use the Equivalent Fractions Property.
Step 3 - Simplify.
Notice that in each case, the bases were the same and we subtracted exponents. We see is or . We see is . When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, so we need a 1 in the numerator. . This leads to the Quotient Property for Exponents.
Quotient Property for Exponents
If is a real number, , and and are integers, then
, and ,
Example 5
Example 6
Example 7
Try it
Try It: Using Product and Quotient Properties for Exponents
Simplify each expression.
Compare your answers:
Here is how to simplify the expressions:
Video: Understanding Product and Quotient Properties for Exponents
Watch the following video to learn more about applying the Product and Quotient Properties of Exponents.