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.
