# How do you simplify (\frac { 2^ { - 3} m ^ { 5} p } { 4^ { - 2} m ^ { 9} p ^ { - 4} } ) ^ { - 3}?

Aug 22, 2017

${m}^{12} / \left(8 {p}^{15}\right)$

#### Explanation:

There are several laws of indices we can apply here. The order in which you apply them does not matter, one law is not more important than another.

Recall the law that inverts a fraction: ${\left(\frac{a}{b}\right)}^{-} m = {\left(\frac{b}{a}\right)}^{m}$

$\therefore {\left(\setminus \frac{{2}^{- 3} {m}^{5} p}{{4}^{- 2} {m}^{9} {p}^{- 4}}\right)}^{\textcolor{b l u e}{- 3}} = {\left(\frac{{4}^{-} 2 {m}^{9} {p}^{-} 4}{{2}^{-} 3 {m}^{5} p}\right)}^{\textcolor{b l u e}{3}}$

Recall the law with negative exponents: ${x}^{-} m = \frac{1}{x} ^ m$

${\left(\frac{\textcolor{red}{{4}^{-} 2} {m}^{9} \textcolor{red}{{p}^{-} 4}}{\textcolor{red}{{2}^{-} 3} {m}^{5} p}\right)}^{3} = {\left(\frac{\textcolor{red}{{2}^{3}} {m}^{9}}{\textcolor{red}{{4}^{2}} {m}^{5} p \textcolor{red}{{p}^{4}}}\right)}^{3}$

Simplify inside the bracket - when dividing, subtract indices.

${\left(\frac{\textcolor{red}{8} \textcolor{\lim e}{{m}^{9}}}{\textcolor{red}{16} \textcolor{\lim e}{{m}^{5}} \textcolor{red}{{p}^{5}}}\right)}^{3} = {\left(\frac{\cancel{8} \textcolor{\lim e}{{m}^{4}}}{{\cancel{16}}^{2} {p}^{5}}\right)}^{3}$

${\left({m}^{4} / \left(2 {p}^{5}\right)\right)}^{3} = {m}^{12} / \left(8 {p}^{15}\right)$