Dividing Exponents (Same Base)

You have probably already learned that when you multiply exponents with the same base, you can just add the exponents (if you haven't, check out the detailed Multiplying Exponents lesson).

Division with exponents, as long as they have the same base, follows the same principle.  If the numbers have the same base, you can subtract the exponents for the final answer. 

Let's start with a simple problem:

$$\eqalign{2^4 \div 2^2 &=&&\text{This problem divides two exponents with the same base: 2}\\(2 \times 2 \times 2 \times 2) \div (2 \times 2)&= &&\text{If you write this all the way out, it's just 2 multiplied by itself 4 times divided by two times two}\\ \dfrac{2\times 2 \times 2 \times 2=16}{2 \times 2 = 4}&=&&\text{Sometimes division is clearest when written as a fraction}\\\dfrac{16}{4}&=4}$$

Is there an easier way to get to this point? Yes, there is. But, if you forget the trick, you can always  just write out all of your exponents, set the up division problem as a fraction, and do the math.

A rule that you can use when dividing exponents with the same base is:

When you divide exponents with the same base, keep the base and subtract the exponents.

$$\eqalign{2^4 \div 2^2 \\&=2^{4-2}\\&=2^2\\&=4}$$

This trick will work anytime you are dividing exponents with the same bases (even when those bases are negative or when the bases are fractions, decimals, or variables):

$$\eqalign{5^3 \div 5^2 \\&=(5 \times 5  \times 5) \div (5 \times 5)\\&=\dfrac{5^3 = 125}{5^2 = 25}\\&=5\\\text{OR}\\&=5^{3-2}\\&=5^1\\&=5}$$

Because exponents work the same way, no matter what the base is, this also works for fractions, decimals, variables, and terms that include exponents:

$$\eqalign{{x^8 \div x^3} \\&=x^{8-3}\\&=x^{5}}$$

This trick also works with negative exponents.  Just follow the rules of adding negative numbers:

$$\eqalign{x^8 \div x^{-3}\\ &=x^{8-(-3)}\\&=x^{11}}$$

For more about how negative exponents work, see Exponents (Negative exponents).