Simplify y^3 square root of 108x^2y^2

$y^{3} \sqrt{108 x^{2} y^{2}}$

Rewrite $108 x^{2} y^{2}$ as ${(6 x y)}^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$y^{3} \sqrt{36 (3) x^{2} y^{2}}$

Rewrite $36$ as $6^{2}$ .

$y^{3} \sqrt{6^{2} \cdot 3 x^{2} y^{2}}$

Move $3$ .

$y^{3} \sqrt{6^{2} x^{2} y^{2} \cdot 3}$

Rewrite $6^{2} x^{2} y^{2}$ as ${(6 x y)}^{2}$ .

$y^{3} \sqrt{{(6 x y)}^{2} \cdot 3}$

Pull terms out from under the radical.

$y^{3} (6 x y \sqrt{3})$

Rewrite using the commutative property of multiplication.

$6 \sqrt{3} y^{3} (x y)$

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

Move $y$ .

$6 \sqrt{3} (y \cdot y^{3}) x$

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

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

$6 \sqrt{3} (y^{1} y^{3}) x$

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

$6 \sqrt{3} y^{1 + 3} x$

Add $1$ and $3$ .

$6 \sqrt{3} y^{4} x$