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

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

Step 1

Rewrite $12 x^{4} y^{7}$ as ${(2 x^{2} y^{3})}^{2} \cdot (3 y)$ .

Factor $4$ out of $12$ .

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

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

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

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

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

Factor out $y^{6}$ .

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

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

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

Move $3$ .

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

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

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

Rewrite using the commutative property of multiplication.

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

Step 4

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

Move $y^{3}$ .

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

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

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

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

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

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

Add $3$ and $1$ .

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

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