Simplify 7y^3 square root of 4x^6y^5

$7 y^{3} \sqrt{4 x^{6} y^{5}}$

Rewrite $4 x^{6} y^{5}$ as ${(2 x^{3} y^{2})}^{2} y$ .

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

$7 y^{3} \sqrt{2^{2} x^{6} y^{5}}$

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

$7 y^{3} \sqrt{2^{2} {(x^{3})}^{2} y^{5}}$

Factor out $y^{4}$ .

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

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

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

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

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

Pull terms out from under the radical.

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

Multiply $y^{3}$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

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

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

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

Add $2$ and $3$ .

$7 y^{5} (2 x^{3} \sqrt{y})$

Multiply $2$ by $7$ .

$14 y^{5} (x^{3} \sqrt{y})$

Reorder factors in $14 y^{5} (x^{3} \sqrt{y})$ .

$14 y^{5} x^{3} \sqrt{y}$