Power to a Power Lesson

Power to a Power Lesson

Exponent Inside Parenthesis with an Exponent Outside the Parenthesis

Power to a Power

• Power to a Power means that we have a power (base with an exponent) inside the parenthesis and an exponent outside the parenthesis

• An example of a power to a power looks like

•  $\left(3^2\right)^4$  

What a Power to a Power Really Means

 $\left(4^2\right)^3\ \ \ \ \ really\ means\ \ \left(4^2\right)\left(4^2\right)\left(4^2\right)$  
This is really the same as  $\left(4\right)\left(4\right)\left(4\right)\left(4\right)\left(4\right)\left(4\right)\ \ because\ every\ \left(4^2\right)\ \ means\ \left(4\right)\left(4\right)$  So we could rewrite this as  $4^{6\ }\ \ \ because\ the\ 4\ is\ multiplied\ by\ itself\ 6\ times$  

Power to a Power Rule

 If we have a power inside the parenthesis and a an exponent outside the parenthesis -  We multiply the Exponents

Example

 $\left(5^3\right)^{4\ }\ =\ 5^{3\times4}\ \ =\ \ 5^{12}$  

Power to a Power Examples

Rewrite each as a single power with a positive exponent

Example  $\left(6^2\right)^4\ \ =\ \ 6^{2\times4}\ \ =\ \ 6^8$  

Example       $\left(a^7\right)^3\ \ =\ \ a^{7\times3}\ \ =\ \ a^{21}$  

Example       $\left(m^{-2}\right)^5\ =\ \ m^{-2\times5}\ \ =\ \ m^{-10}\ \ \ =\ \ \frac{1}{m^{10}}$  

Multiple Choice

Rewrite

 $\left(a^7\right)^3$  as a single power

•  $a^{10}$  

•  $a^{21}$  

•  $a^4$  

•  $a^{14}$  

Multiple Choice

Rewrite

 $\left(10^5\right)^5$  as a single power.

•  $a^{25}$  

•  $a^{10}$  

•  $a^0$  

•  $a^{125}$  

Multiple Choice

Rewrite

 $5^2\ \times\ 5^8$  as a single power.
Remember: That when you multiply powers with the same base, you add the exponents

•  $5^6$  

•  $5^{10}$  

•  $5^{16}$  

•  $5^{64}$  

Multiple Choice

Rewrite

 $\left(\left(-2\right)^4\right)^3$  as a single power 

•  $\left(-2\right)^1$  

•  $\left(-2\right)^7$  

•  $\left(-2\right)^{12}$  

•  $4^{12}$