Question

What is `((x^(1/2)y^(-7/4))/(y^(-5/4)))^-4`?

Answer 1

`y^2/x^2`

Explanation:

Let's get rid of all the negative indices first:

Recall: `(a/b)^-m = (b/a)^m`

`((x^(1/2)y^(-7/4))/(y^(-5/4)))^color(blue)(-4)`

`= ((y^(-5/4))/(x^(1/2)y^(-7/4)))^color(blue)(4)" "larr`invert the fraction

Recall: `x^-m = 1/x^m and 1/x^-n = x^n`

`= ((y^(7/4))/(x^(1/2)y^(5/4)))^4`

Recall: `x^m/x^n = x^(m-n)`

`= ((y^(7/4 -5/4))/(x^(1/2)))^4`

`=(y^(1/2)/x^(1/2))^4color(white)(xxxxxxxxxxx)(y^(2/4) = y^(1/2))`

Multiply the indices

`=y^2/x^2`