Dividing Same Bases

When you divide two powers with the same base, you subtract the exponents. You take the exponent on top minus the exponent on the bottom. The rule is called the quotient rule.

To see why it works, write $\frac{x^{5}}{x^{2}}$ out as factors. There are five factors of $x$ on top and two factors of $x$ on the bottom. Each factor on the bottom cancels one factor on top. Two pairs cancel, and three factors of $x$ are left on top, so $\frac{x^{5}}{x^{2}} = x^{3}$.

The rule only works when the two bases are the same. If the bases are different, no factor on the bottom matches a factor on top, so nothing cancels.

The order of the subtraction matters. For $\frac{x^{5}}{x^{2}}$, you work out $5-2$, the exponent on top minus the exponent on the bottom. The subtraction counts how many factors are left after the cancelling.