Simplify: (x^(-1/2)y^4)^(1/4)/(x^(2/3)y^(3/2)x^(-3/2)y^(1/2))
First, let's distribute the 1/4 exponent:
(x^((-1/2)^(1/4))y^((4)^(1/4)))/(x^(2/3)y^(3/2)x^(-3/2)y^(1/2))
When you raise a power to a power you multiply them to find the exponent. Let's do that
(x^((-1/2)(1/4))y^((4)(1/4)))/(x^(2/3)y^(3/2)x^(-3/2)y^(1/2))
(x^(-1/8)y^1)/(x^(2/3)y^(3/2)x^(-3/2)y^(1/2))
(x^(-1/8)y)/(x^(2/3)y^(3/2)x^(-3/2)y^(1/2))
Now let's combine the exponents with like bases in the denominator. When you multiply exponents with like bases you add the exponents together
(x^(-1/8)y)/(x^((2/3)+(-3/2))y^((3/2)+(1/2)))
(x^(-1/8)y)/(x^((4/6)+(-9/6))y^(4/2))
(x^(-1/8)y)/(x^(-5/6)y^(2))
When there is a negative exponent in the numerator we can bring it to the denominator as a positive exponent and vice versa
Let's bring x^(-5/6) to the numerator since it is bigger than x^(-1/8) so when we multiply them and add the exponents it will be positive.
(x^(-1/8)x^(5/6)y)/(y^(2))
(x^((-1/8)+(5/6))y)/(y^(2))
(x^((-3/24)+(20/24))y)/(y^(2))
(x^(17/24)y)/(y^(2))
Now we can cancel out one of the y's on top and bottom
(x^(17/24)cancel(y))/(y^(cancel(2)color(red)(1)))
(x^(17/24))/(y)
The final answer is
(x^(17/24))/(y)