Simplify 8x^3y^2 square root of 12x^2y^5

$8 x^{3} y^{2} \sqrt{12 x^{2} y^{5}}$

Rewrite $12 x^{2} y^{5}$ as ${(2 x y^{2})}^{2} \cdot (3 y)$ .

Factor $4$ out of $12$ .

$8 x^{3} y^{2} \sqrt{4 (3) x^{2} y^{5}}$

Rewrite $4$ as $2^{2}$ .

$8 x^{3} y^{2} \sqrt{2^{2} \cdot 3 x^{2} y^{5}}$

Factor out $y^{4}$ .

$8 x^{3} y^{2} \sqrt{2^{2} \cdot 3 x^{2} (y^{4} y)}$

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

$8 x^{3} y^{2} \sqrt{2^{2} \cdot 3 x^{2} ({(y^{2})}^{2} y)}$

Move $3$ .

$8 x^{3} y^{2} \sqrt{2^{2} x^{2} {(y^{2})}^{2} \cdot 3 y}$

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

$8 x^{3} y^{2} \sqrt{{(2 x y^{2})}^{2} \cdot 3 y}$

Add parentheses.

$8 x^{3} y^{2} \sqrt{{(2 x y^{2})}^{2} \cdot (3 y)}$

Pull terms out from under the radical.

$8 x^{3} y^{2} (2 x y^{2} \sqrt{3 y})$

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

Move $x$ .

$8 (x \cdot x^{3}) y^{2} (2 y^{2} \sqrt{3 y})$

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

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

$8 (x^{1} x^{3}) y^{2} (2 y^{2} \sqrt{3 y})$

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

$8 x^{1 + 3} y^{2} (2 y^{2} \sqrt{3 y})$

Add $1$ and $3$ .

$8 x^{4} y^{2} (2 y^{2} \sqrt{3 y})$

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

Move $y^{2}$ .

$8 x^{4} (y^{2} y^{2}) (2 \sqrt{3 y})$

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

$8 x^{4} y^{2 + 2} (2 \sqrt{3 y})$

Add $2$ and $2$ .

$8 x^{4} y^{4} (2 \sqrt{3 y})$

Multiply $2$ by $8$ .

$16 x^{4} y^{4} \sqrt{3 y}$