Simplify x^2y^2 square root of 108x^4y^3

$x^{2} y^{2} \sqrt{108 x^{4} y^{3}}$

Step 1

Rewrite $108 x^{4} y^{3}$ as ${(6 x^{2} y)}^{2} \cdot (3 y)$ .

Factor $36$ out of $108$ .

$x^{2} y^{2} \sqrt{36 (3) x^{4} y^{3}}$

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

$x^{2} y^{2} \sqrt{6^{2} \cdot 3 x^{4} y^{3}}$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Factor out $y^{2}$ .

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

Move $3$ .

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

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

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

$6 x^{4} y^{2} (y \sqrt{3 y})$

Step 5

Multiply $y^{2}$ by $y$ by adding the exponents.

Move $y$ .

$6 x^{4} (y \cdot y^{2}) \sqrt{3 y}$

Multiply $y$ by $y^{2}$ .

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

$6 x^{4} (y^{1} y^{2}) \sqrt{3 y}$

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

$6 x^{4} y^{1 + 2} \sqrt{3 y}$

Add $1$ and $2$ .

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