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Exponents

Raising One Power to Another

When one power is raised to another, we multiply exponents:

(b^m)^n=b^(m*n).

This is true for all kinds of exponents, positive and negative (and as we will see later, fractional).

Examples

Raising a Positive Power to a Positive Power

Long solution: (b^3)^2=b^3*b^3=(b*b*b)*(b*b*b)=b^6=b^(3*2).

Short solution: (y^4)^5=y^(4*5)=y^20.

Have a look at some more worked examples:

(   ) = =   =  
x     x   x  

 

Exercise

Now practice some of the following exercises:

 
   
(
x      
)

=

 
x   


Examples

Raising a Positve Power to a Negative Power

Long solution:
(y^3)^(-4)=1/(y^3)^4=1/y^12=y^(-12)=y^(3*(-4)).

Short solution: y to the power 3 raised to the power negative 4 equals y to the power negative 12

(x^2)^(-5)=x^(2*(-5))=x^(-10).

Study some more worked examples showing the longer solution:

           

1


 

1


   

(

 

)

=

(

 

)

=

 

=

 

x

   

x

   

x

 

x

 

Exercise

Now practice by solving some of the following exercises. Here you can use the short solution and just enter the final answer:

(
  
x      
)
=
 
x   
 


Examples

Raising a Negative Power to a Positive Power

Long solution: (a^(-3))^2=(1/a^3)^2=(1)^2/(a^3)^2=1/a^6=a^(-6)=a^(-3*2).

Short solution:a to the power negative 3 raised to the power 2 equals a to the power negative 6 (x^(-3))^8=x^(-3*8)=x^(-24).

Study some more worked examples:

         

=

       

=

       

=

   

=

   

=

   
(   ) leftbracket

1


rightbracket  

1


1


1


 

s

        (   )    

s

 
         

s

   

s

   

s

 

s

     

Exercise

Practice by solving some of the following exercises:

 
   
(
  
x      
)

 = 

  
x   

Example

Finally just one long solution for a negative exponent raised to a negative exponent.

r raised to the power -2 raised to the power -3 equals r raised to the power positive 6

Exercise

Solve some of these mixed exercises:

 
   
(
  
m      
)

=

  
m   

<< Dividing Terms with the Same Base | Exponents Index | Raising a Product to a Power >>

 

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