# How do you simplify the expression (12a^-3b^9)/(21a^2b^-5) using the properties?

Apr 23, 2017

See the entire solution process below:

#### Explanation:

First, rewrite this expression as:

$\left(\frac{12}{21}\right) \left({a}^{-} \frac{3}{a} ^ 2\right) \left({b}^{9} / {b}^{-} 5\right) \implies \left(\frac{3 \times 4}{3 \times 7}\right) \left({a}^{-} \frac{3}{a} ^ 2\right) \left({b}^{9} / {b}^{-} 5\right) \implies$

$\left(\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \times 4}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \times 7}\right) \left({a}^{-} \frac{3}{a} ^ 2\right) \left({b}^{9} / {b}^{-} 5\right) \implies \frac{4}{7} \left({a}^{-} \frac{3}{a} ^ 2\right) \left({b}^{9} / {b}^{-} 5\right)$

Now, use these two rules of exponents to simplify the $a$ and $b$ terms:

${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} - \textcolor{b l u e}{b}}$ and ${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = \frac{1}{x} ^ \left(\textcolor{b l u e}{b} - \textcolor{red}{a}\right)$

$\frac{4}{7} \left({a}^{\textcolor{red}{- 3}} / {a}^{\textcolor{b l u e}{2}}\right) \left({b}^{\textcolor{red}{9}} / {b}^{\textcolor{b l u e}{- 5}}\right) \implies \frac{4}{7} \left(\frac{1}{a} ^ \left(\textcolor{b l u e}{2} - \textcolor{red}{- 3}\right)\right) \left({b}^{\textcolor{red}{9} - \textcolor{b l u e}{- 5}}\right) \implies$

$\frac{4}{7} \left(\frac{1}{a} ^ \left(\textcolor{b l u e}{2} + \textcolor{red}{3}\right)\right) \left({b}^{\textcolor{red}{9} + \textcolor{b l u e}{5}}\right) \implies \frac{4}{7} \left(\frac{1}{a} ^ 5\right) \left({b}^{14}\right) \implies \frac{4 \cdot 1 \cdot {b}^{14}}{7 \cdot {a}^{5}} \implies$

$\frac{4 {b}^{14}}{7 {a}^{5}}$