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.
