Question

How do you simplify `(x^(-1/2)y^2)^(-5/4)/(x^2y^(1/2)`?

Answer 1

`f(x,y) = frac(1)(x^(11/8)y^3)`

Explanation:

Given:
`f(x,y) =frac((x^(-1/2)y^2)^(-5/4))(x^2y^(1/2))`

We first distribute the power in the numerator using the Power Rule for exponents, since the base terms `x` and `y` are being multiplied together and not added.

`f(x,y) =frac(x^((-1/2)(-5/4))y^((2)(-5/4)))(x^2y^(1/2)) = frac(x^(5/8)y^(-5/2))(x^2y^(1/2))`

Now we apply the Quotient Rule for Exponents on both `x` and `y`, following:

`x^a/x^b = x^(a-b)`

`f(x,y) = x^(5/8-2)y^(-5/2-1/2) = x^(-11/8)y^(-3)`

Now, we apply the negative exponent rule,

`x^(-a) = 1/x^(a)`

`f(x,y) =1/(x^(11/8)y^(3))`