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

1 Answer
Mar 15, 2018

Answer:

#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

#(3^(-2)x^4y^(-2))/(4^(-1)x^(-1)y^2)#

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

#((3^-2)/(4^-1))((x^4)/(x^-1))((y^-2)/(y^2))# 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

#((3^-2)/(4^-1))(x^(4-(-1)))(y^(-2-(2)))#

#rArr ((3^-2)/(4^-1))(x^(4+1))(y^(-2-2))#

#rArr ((3^-2)/(4^-1))(x^5)(y^-4)# Expression 3

#color(green)(Step" 3"#

Rule 3

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

Using this rule, we can simplify Expression 3 as

#((1/3^2)/(1/4^1))(x^5)(y^-4)#

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

#rArr (1/3^2)(4^1/1)(x^5)(y^-4)#

#rArr (4/9)(x^5)(1/(y^4))#

#rArr (4x^5)/(9y^4)#

Hence,

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

Hope you find this solution useful.