# How do you simplify (-2mn^2)^-3/(4m^-6 n^4)?

Jun 16, 2015

$= \frac{\textcolor{b l u e}{- {m}^{3.} {n}^{-} 10}}{32}$

#### Explanation:

${\left(- 2 m {n}^{2}\right)}^{-} \frac{3}{4 {m}^{-} 6 {n}^{4}}$

Simplifying the numerator:
(-2^1m^1n^2)^color(red)(-3
=-2^(1 . color(red)(-3)) . m^(1. color(red)(-3)) . n^(2 . color(red)(-3)^
$= - {2}^{-} 3 {m}^{-} 3 {n}^{-} 6$
 = color(blue)(-1/2^3m^-3n^-6
$= \frac{\textcolor{b l u e}{- {m}^{-} 3 {n}^{-} 6}}{8}$

The expression can now be written as :
$= \frac{\textcolor{b l u e}{- {m}^{-} 3 {n}^{-} 6}}{8.4 {m}^{-} 6 {n}^{4}}$

$= \frac{\textcolor{b l u e}{- {m}^{-} 3 {n}^{-} 6}}{32 {m}^{-} 6 {n}^{4}}$

Note : color(blue)(a^m /a^n = a^(m-n)
Applying the above to the exponents of $m$ and $n$

$= \frac{- {m}^{- 3 + 6} . {n}^{- 6 - 4}}{32}$
$= \frac{\textcolor{b l u e}{- {m}^{3.} {n}^{-} 10}}{32}$
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