Simplify 8y^2 square root of 12x^7y^5

$8 y^{2} \sqrt{12 x^{7} y^{5}}$

Step 1

Rewrite $12 x^{7} y^{5}$ as ${(2 x^{3} y^{2})}^{2} \cdot (3 x y)$ .

Factor $4$ out of $12$ .

$8 y^{2} \sqrt{4 (3) x^{7} y^{5}}$

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

$8 y^{2} \sqrt{2^{2} \cdot 3 x^{7} y^{5}}$

Factor out $x^{6}$ .

$8 y^{2} \sqrt{2^{2} \cdot 3 (x^{6} x) y^{5}}$

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

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

Factor out $y^{4}$ .

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

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

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

Move $x$ .

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

Move $3$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

Move $y^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Step 4

Multiply $2$ by $8$ .

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

Step 5

Reorder factors in $16 y^{4} (x^{3} \sqrt{3 x y})$ .

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