Simplify 4x^3 square root of 12x^2y^7

$4 x^{3} \sqrt{12 x^{2} y^{7}}$

Rewrite $12 x^{2} y^{7}$ as ${(2 x y^{3})}^{2} \cdot (3 y)$ .

Factor $4$ out of $12$ .

$4 x^{3} \sqrt{4 (3) x^{2} y^{7}}$

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

$4 x^{3} \sqrt{2^{2} \cdot 3 x^{2} y^{7}}$

Factor out $y^{6}$ .

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

Rewrite $y^{6}$ as ${(y^{3})}^{2}$ .

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

Move $3$ .

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

Rewrite $2^{2} x^{2} {(y^{3})}^{2}$ as ${(2 x y^{3})}^{2}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $3$ .

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

Multiply $2$ by $4$ .

$8 x^{4} y^{3} \sqrt{3 y}$