Quotient of Powers Property

When you multiply two powers with the same base, you add the exponents. So when you divide two powers with the same base, you subtract the exponents. In other words, for all real numbers $a$ , $b$ and $c$ , where $a\ne 0$ ,

$\frac{{a}^{b}}{{a}^{c}}={a}^{\mathit{b - c}}$

What you're really doing here is cancelling common factors from the numerator and denominator. Example:

$\frac{{9}^{5}}{{9}^{2}}=\frac{9\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}9\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}9\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\overline{)9}\cdot \overline{)9}}{\overline{)9}\cdot \overline{)9}}=9\cdot 9\cdot 9={9}^{3}$