# How do you simplify ((3x^-2 y)^-2)/( 4xy^-2)^-1 ?

Mar 15, 2018

color(red)(((3x^-2 y)^-2)/( 4xy^-2)^-1  = color(blue)((4x^5)/(9y^4)

#### Explanation:

Given:

color(red)(((3x^-2 y)^-2)/( 4xy^-2)^-1  Expression 1

We can simplify the above exponent problem as follows:

color(green)(Step" 1"

Rule 1

color(blue)((a^m)^n = a^(mn)

Using this rule, we can write Expression 1 as

$\frac{{3}^{- 2} {x}^{4} {y}^{- 2}}{{4}^{- 1} {x}^{- 1} {y}^{2}}$

We can rewrite the above expression, combining the like terms as

$\left(\frac{{3}^{-} 2}{{4}^{-} 1}\right) \left(\frac{{x}^{4}}{{x}^{-} 1}\right) \left(\frac{{y}^{-} 2}{{y}^{2}}\right)$ Expression 2

color(green)(Step" 2"

Rule 2

color(blue)(a^m/a^n = a^(m-n)

Using this rule, we can simplify Expression 2 as

$\left(\frac{{3}^{-} 2}{{4}^{-} 1}\right) \left({x}^{4 - \left(- 1\right)}\right) \left({y}^{- 2 - \left(2\right)}\right)$

$\Rightarrow \left(\frac{{3}^{-} 2}{{4}^{-} 1}\right) \left({x}^{4 + 1}\right) \left({y}^{- 2 - 2}\right)$

$\Rightarrow \left(\frac{{3}^{-} 2}{{4}^{-} 1}\right) \left({x}^{5}\right) \left({y}^{-} 4\right)$ Expression 3

color(green)(Step" 3"

Rule 3

color(blue)(a^-m = 1/a^m

Using this rule, we can simplify Expression 3 as

$\left(\frac{\frac{1}{3} ^ 2}{\frac{1}{4} ^ 1}\right) \left({x}^{5}\right) \left({y}^{-} 4\right)$

Use the rule color(brown)((1/m)/(1/n)=(1/m)(n/1)

$\Rightarrow \left(\frac{1}{3} ^ 2\right) \left({4}^{1} / 1\right) \left({x}^{5}\right) \left({y}^{-} 4\right)$

$\Rightarrow \left(\frac{4}{9}\right) \left({x}^{5}\right) \left(\frac{1}{{y}^{4}}\right)$

$\Rightarrow \frac{4 {x}^{5}}{9 {y}^{4}}$

Hence,

color(red)(((3x^-2 y)^-2)/( 4xy^-2)^-1  = color(blue)((4x^5)/(9y^4)

Hope you find this solution useful.