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The Division Rule

The rule for dividing powers is the opposite of multiplying: If the two powers have the same base, you subtract their exponents from each other. Here is an example to illustrate:

Example 1

Calculate $\frac{2^{3}}{2^{2}}$

You split up the power, cancel equal factors, and get:

\frac{2^{3}}{2^{2}} = \frac{2 \times 2 \times 2}{2 \times 2} = \frac{2 \times 2 \times 2}{2 \times 2} = 2

You see that what has happened is that you have subtracted the exponents:

\frac{2^{3}}{2^{2}} = 2^{3 - 2} = 2^{1} = 2

Rule

Division of Powers with the Same Base

\frac{a^{n}}{a^{m}} = a^{n - m}

Example 2

Calculate $\frac{2^{5}}{2^{- 2}}$

Using the rule above and that $- (-) = +$ , you get

\frac{2^{5}}{2^{- 2}} = 2^{5 - (- 2)} = 2^{5 + 2} = 2^{7}

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