Laws of 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.

 $2^{-4\ }=\ \frac{1}{2^4}$   =

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

10⁰ = 1, 5⁰ = 1, -7⁰ = 1

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.

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

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

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  $\frac{1}{27.}$   $\frac{1}{9}\left(\frac{1}{27}\right)\ =\ \frac{1}{243.}$