(xy)^1/4(x^2y^2)^1/2 over(x^2y)^3/4
x^1/2y^1/2
(xy)^1/4(x^2y^2)^1/2 over(x^2y)^3/4
since (x^2y^2)^1/2 = xy,
= x^(5/4) y^(5/4) / x^(6/4) y^(3/4)
= y^(2/4)/x^(1/4)
= √y/∜x
Better double check the algebra. either Alyonka or I (or both) made a mistake.
To simplify the expression (xy)^(1/4)(x^2y^2)^(1/2) / (x^2y)^(3/4), let's break it down step by step:
Step 1: Distribute the exponents within each term.
(xy)^(1/4) can be expanded as x^(1/4) * y^(1/4).
(x^2y^2)^(1/2) can be expanded as (x^2)^(1/2) * (y^2)^(1/2), which simplifies to x * y.
(x^2y)^(3/4) can be expanded as (x^2)^(3/4) * (y)^(3/4), which simplifies to x^(3/2) * y^(3/4).
Step 2: Simplify the exponents by multiplying them and bringing them outside the parentheses.
The expression becomes:
x^(1/4) * y^(1/4) * x * y / (x^(3/2) * y^(3/4)).
Step 3: Combine like terms and simplify.
Combining the x terms (x^(1/4) * x / x^(3/2)), we subtract the exponents: x^((1/4) - (3/2)) = x^(-5/4) = 1/x^(5/4).
Combining the y terms (y^(1/4) * y / y^(3/4)), we subtract the exponents: y^((1/4) - (3/4)) = y^(-2/4) = 1/y^(2/4) = 1/y^(1/2).
The simplified expression is:
1 / (x^(5/4) * y^(1/2)).
So the simplified form of (xy)^(1/4)(x^2y^2)^(1/2) / (x^2y)^(3/4) is 1 / (x^(5/4) * y^(1/2)).