# How do you simplify and write (a^4b^-3)/( ab^-2) with positive exponents?

Jun 14, 2017

See a solution process below:

#### Explanation:

First, rewrite this expression as:

$\left({a}^{4} / a\right) \left({b}^{-} \frac{3}{b} ^ - 2\right)$

Use these rules for exponents to simplify the $a$ terms:

$a = {a}^{\textcolor{b l u e}{1}}$ and ${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} - \textcolor{b l u e}{b}}$

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

${a}^{3} \left({b}^{-} \frac{3}{b} ^ - 2\right)$

Next, use these rules of exponents to simplify the $b$ terms:

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

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

${a}^{3} \left(\frac{1}{b} ^ \textcolor{red}{1}\right) \implies$

${a}^{3} / b$