Laws of Exponents

Laws of Exponents

a^-n is the reciprocal of aⁿ. For example, 4² and

4^-2 are inverses or reciprocals.

4²  = 4(4) = 16   4^-2 = 1 / 4(4) =1 / 16

If this is the case, if 3³ = 27, then 3^-3

Multiple Choice

If  $3^{3\ }=\ 27,\ then\ 3\ ^{-3\ }=\ ?$  

•  $\frac{1}{3^3}$  

• -3

• -27

•  $\frac{1}{3^{-3}}$  

Negative Exponents...

represent fractions or numbers less than 1. We can simplify negative exponents by making them positive. To do this, move the negative exponents to the denominator where they become positive.

 $2^{-4\ }=\ \frac{1}{2^4}$   =

Multiple Choice

 $If\ 2^{-4}=\ \frac{1}{2^4},\ then\ 5^{-2\ }=\ ?$   

• -25

•  $\frac{1}{5^2}$  

• -5

•  $\frac{1}{5^{-2}}$  

The power of 0

~ raising a nonzero number to the power of 0 = 1.

10⁰ = 1, 5⁰ = 1, -7⁰ = 1

Multiplying Exponents

When multiplying exponents, you simply add the exponents.

2³(2⁵) expanded out is 2(2)(2) times 2(2)(2)(2)(2)

You keep the base, 2, and add the exponents 3 + 5 = 8. 2³⁺⁵ = 2⁸

This rule applies to variables as well.

Multiple Choice

Simplify a⁴(a⁵).

• a⁴⁺⁵ = a⁹

• a^4-5 = a^-1

• a⁴⁽⁵⁾ = a²⁰

• 2a⁴⁺⁵ = 2a⁹

Dividing Exponents

When dividing exponents, you keep the base and subtract the exponents.

 $\frac{10^5}{10^3}$  =  $10^{\left(5-3\right)}$  =  $10^2$  

Multiple Choice

 $\frac{12^6}{12^4}=\ ?$  

• 12

• 1

• 
   $12^{\left(6+4\right)}\ =\ 12^{10}$  
  

•  $12^{\left(6\ -\ 4\right)}\ =\ 12^2$  

Negative Exponents & Dividing Fractions

When dividing fractions, you multiply by the reciprocal.  The expression $3^{-2}\ \div\ 3^{3\ }=\ \frac{1}{3^2}\div3^3$ .  This is the same as  $\frac{1}{9}\div27$ since  $3^{-2\ }=\ \frac{1}{3^2}\ or\ \frac{1}{3\left(3\right)\ }or\ \frac{1}{9.}$ To divide this, you would rewrite as  $\frac{1}{9\ }\left(\frac{1}{27}\right)$  since the reciprocal of 27 is  $\frac{1}{27.}$   $\frac{1}{9}\left(\frac{1}{27}\right)\ =\ \frac{1}{243.}$