# How do you simplify ((2p m^-1q^0)^-4*2m^-1p^3)/(2pq^2) and write it using only positive exponents?

Jan 27, 2018

See a solution process below:

#### Explanation:

First, rewrite the expression as:

$\left(\frac{2 \cdot 2}{2}\right) \left(\frac{p \cdot {p}^{3}}{p}\right) \left({m}^{-} 1 \cdot {m}^{-} 1\right) \left({q}^{0} / {q}^{2}\right)$

Next, cancel common terms in the numerator and denominator:

$\left(\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \cdot 2}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}}\right) \left(\frac{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{p}}} \cdot {p}^{3}}{\textcolor{b l u e}{\cancel{\textcolor{b l a c k}{p}}}}\right) \left({m}^{-} 1 \cdot {m}^{-} 1\right) \left({q}^{0} / {q}^{2}\right) \implies$

$2 {p}^{3} \left({m}^{-} 1 \cdot {m}^{-} 1\right) \left({q}^{0} / {q}^{2}\right)$

Next, use this rule of exponents to simplify the $q$ term:

${a}^{\textcolor{red}{0}} = 1$

$2 {p}^{3} \left({m}^{-} 1 \cdot {m}^{-} 1\right) \left({q}^{\textcolor{red}{0}} / {q}^{2}\right) \implies$

$2 {p}^{3} \left({m}^{-} 1 \cdot {m}^{-} 1\right) \left(\frac{1}{q} ^ 2\right) \implies$

$\frac{2 {p}^{3}}{{q}^{2}} \left({m}^{-} 1 \cdot {m}^{-} 1\right)$

Now, use these rules of exponents to simplify the $m$ terms:

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

$\frac{2 {p}^{3}}{{q}^{2}} \left({m}^{\textcolor{red}{- 1}} \cdot {m}^{\textcolor{b l u e}{- 1}}\right) \implies$

$\frac{2 {p}^{3}}{{q}^{2}} {m}^{\textcolor{red}{- 1} + \textcolor{b l u e}{- 1}} \implies$

(2p^3)/(q^2)m^(color(red)(-2) =>

(2p^3)/(q^2)(1/m^(color(red)(- -2) =>

$\frac{2 {p}^{3}}{{q}^{2}} \left(\frac{1}{m} ^ 2\right) \implies$

$\frac{2 {p}^{3}}{{q}^{2} {m}^{2}}$