Dividing Exponents and Negative Exponents

Dividing with Exponents

Objective: Divide Expressions with exponents.

Multiple Select

REVIEW: Choose all the expressions that would require you to multiply the exponents to simplify them.

• $$x^2\cdot x^8$$  

• $$\left(a^4\right)^3$$  

• $$g^2h^6\times h^7$$  

• $$\left(3y^4\right)^{22}$$  

• $$\frac{b^8}{b^3}$$  

Dividing with Exponents

Going to explain WHY we do what we do to answer a division problem with exponents.

Example 1:   Simplify:  $\frac{5^5}{5^3}$  

Expand:  $\frac{5\times5\times5\times5\times5}{5\times5\times5}$  

Because they are all being multiplied we can separate them into individual fractions.

Separate:  $\frac{5}{5}\times\frac{5}{5}\times\frac{5}{5}\times\frac{5}{1}\times\frac{5}{1}$  

Example 1 (continued)

Separated:  $\frac{5}{5}\times\frac{5}{5}\times\frac{5}{5}\times\frac{5}{1}\times\frac{5}{1}$  

 $\frac{5}{5}=1$  so...

Rewrite:  $1\times1\times1\times\frac{5}{1}\times\frac{5}{1}$  

Evaluate:  $\frac{5}{1}\times\frac{5}{1}=\frac{25}{1}=25$  
 

Example 1 (continued)

We can also see it works mathematically

 $\frac{5^5}{5^3}$  =  $\frac{3125}{125}$  =  $25$  

BUT what if we are dealing with variables?

Well, we can see that on the next slide.

Example 2:   Simplify:  $\frac{y^2}{y^6}$  

Expand:  $\frac{y\times y}{y\times y\times y\times y\times y\times y}$  


Separate:  $\frac{y}{y}\times\frac{y}{y}\times\frac{1}{y}\times\frac{1}{y}\times\frac{1}{y}\times\frac{1}{y}$  

Rewrite:  $1\times1\times\frac{1}{y}\times\frac{1}{y}\times\frac{1}{y}\times\frac{1}{y}$  

Example 2 (continued)

Rewritten:  $1\times1\times\frac{1}{y}\times\frac{1}{y}\times\frac{1}{y}\times\frac{1}{y}$  

Rewrite as one fraction:  $\frac{1\times1\times1\times1}{y\times y\times y\times y}$  

Simplify:  $\frac{1}{y^4}$  

NOW...is there a shortcut?

YES

But I am going to ask you a question first to see how you do.

Multiple Choice

Simplify:  $$\frac{m^4}{m^2}$$  

• $$m^2$$  

• $$m^6$$  

• $$\frac{1}{m^2}$$  

• $$\frac{1}{m^6}$$  

Multiple Choice

Simplify:  $$\frac{p^3}{p^9}$$  

• $$\frac{1}{p^6}$$  

• $$p^6$$  
  

• $$p^{12}$$  
  

• $$\frac{1}{p^{12}}$$  
  

SO WHAT IS THE SHORTCUT?

You may already know.

The shortcut is to subtract the top exponent from the bottom.

(the only issue is you may end up with a negative exponent which we have not covered yet)

So I like to word it this way...

Ask yourself where are there more?

Then ask how many more?

I'll give an example on the next slide.

Example 3:   Simplify:  $\frac{f^{18}}{f^7}$   

ASK: Where are there more f's?
ANSWER: on top

ASK: How many more?
ANSWER: 11

So there is  $f^{11}$  on top and  $1$  on the bottom.

So the answer is just  $f^{11}$  

Example 4:   Simplify:  $\frac{w^8}{w^{15}}$   

ASK: Where are there more w's?
ANSWER: on the bottom

ASK: How many more?
ANSWER: 7

So there is  $w^7$  on the bottom and  $1$  on top.

So the answer is  $\frac{1}{w^7}$  

Multiple Choice

Simplify:  $$\frac{n^4}{n^7}$$  

• $$n^3$$  
  

• $$\frac{1}{n^3}$$  

• $$n^{11}$$  
  

• $$\frac{1}{n^{11}}$$  
  

Multiple Choice

Simplify:  $$\frac{x^5}{x^3}$$  

• $$x^2$$  
  

• $$\frac{1}{x^2}$$  
  

• $$x^8$$  
  

• $$\frac{1}{x^8}$$  
  

Multiple Choice

Simplify:  $$\frac{y^{23}}{y^3}$$  

• $$y^{20}$$  
  

• $$y^{26}$$  
  

• $$\frac{1}{y^{20}}$$  
  

• $$\frac{1}{y^{26}}$$