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Multiplying and Dividing Powers


Powers can be multiplied and divided. In the example below, we will use powers of 2 to form a rule about multiplying exponents. The table shows several powers of 2 and their values.

You can use the table to substitute exponents for the factors of multiplication equations. What do you notice about the exponents in the following products?

These examples suggest that you can multiply powers with the same
base by adding the exponents. Think about a2
a3.

Example

Simplify each expression.

 

1.     43 45

 

Alternative Solutions:

 

 



2.     x3 x4

 

Alternative Solutions:

 

 


3.     (4y2)(3y)

 

Alternative Solutions:

 

 



4.     (a3b2)(a2b4)

 

Alternative Solutions:

 

 



You can use powers of 2 to help find a rule for dividing powers. Study each quotient in the table. What do you notice about the exponents?

These examples suggest that you can divide powers with the same base by subtracting the exponents. Think about a5 ÷ a2. Remember that you can write a division expression as a fraction.

Example

Simplify each expression.

 

 


Alternative Solutions:

 

 


 




Alternative Solutions:

 




 


Alternative Solutions:

 

 




A special case results when you divide a power by itself. Consider the following two ways to simplify , where b ≠ 0.

Since  cannot have two different values, you can conclude that b0 = 1. Therefore, any nonzero number raised to the zero power is equal to 1.

 

Example

 

8.     Simplify 

 

Alternative Solutions:

 

 





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Labels: Mathematician

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