Simplify 7x^3y^2 square root of 9x^7y^7

$7 x^{3} y^{2} \sqrt{9 x^{7} y^{7}}$

Rewrite $9 x^{7} y^{7}$ as ${(3 x^{3} y^{3})}^{2} (x y)$ .

Rewrite $9$ as $3^{2}$ .

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

Factor out $x^{6}$ .

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

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

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

Factor out $y^{6}$ .

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

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

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

Move $x$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

Move $x^{3}$ .

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

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

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

Add $3$ and $3$ .

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

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

Move $y^{3}$ .

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

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

$7 x^{6} y^{3 + 2} (3 \sqrt{x y})$

Add $3$ and $2$ .

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

Multiply $3$ by $7$ .

$21 x^{6} y^{5} \sqrt{x y}$