Simplify (x^(1/3)y)^3 square root of x^2y^2

${(x^{\frac{1}{3}} y)}^{3} \sqrt{x^{2} y^{2}}$

Apply the product rule to $x^{\frac{1}{3}} y$ .

${(x^{\frac{1}{3}})}^{3} y^{3} \sqrt{x^{2} y^{2}}$

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{3} \cdot 3} y^{3} \sqrt{x^{2} y^{2}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} y^{3} \sqrt{x^{2} y^{2}}$

Rewrite the expression.

$x^{1} y^{3} \sqrt{x^{2} y^{2}}$

Simplify.

$x y^{3} \sqrt{x^{2} y^{2}}$

Rewrite $x^{2} y^{2}$ as ${(x y)}^{2}$ .

$x y^{3} \sqrt{{(x y)}^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x y^{3} (x y)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x \cdot x y^{3} y$

Multiply $x$ by $x$ .

$x^{2} y^{3} y$

Multiply $y^{3}$ by $y$ by adding the exponents.

Move $y$ .

$x^{2} (y \cdot y^{3})$

Multiply $y$ by $y^{3}$ .

Raise $y$ to the power of $1$ .

$x^{2} (y^{1} y^{3})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2} y^{1 + 3}$

Add $1$ and $3$ .

$x^{2} y^{4}$