Simplify the radical expression sqrt root(125x^3y^4) Show all work and steps.

Answer 1

First, we need to find perfect square factors of the expression under the square root.

Factor the expression into prime factors:
125x^3y^4 = 5^3 * x^3 * y^4

Next, group the prime factors into pairs to find perfect square factors:
5^3 = 5^2 * 5 = 25 * 5
x^3 = x^2 * x
y^4 = y^2 * y^2

Now, rewrite the expression with the perfect square factors outside the square root:
sqrt(125x^3y^4) = sqrt(25 * 5 * x^2 * x * y^2 * y^2)

Split the square root using the product property:
sqrt(25) * sqrt(5) * sqrt(x^2) * sqrt(x) * sqrt(y^2) * sqrt(y^2)

Simplify the perfect square roots:
5 * sqrt(5)* x * sqrt(x) * y^2

Final answer:
5xy^2sqrt(5x)