Simplify square root of 12* square root of y^3* square root of 6y

$\sqrt{12} \cdot \sqrt{y^{3}} \cdot \sqrt{6 y}$

Step 1

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

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

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

$\sqrt{2^{2} \cdot 3} \cdot \sqrt{y^{3}} \cdot \sqrt{6 y}$

Step 2

Pull terms out from under the radical.

$2 \sqrt{3} \cdot \sqrt{y^{3}} \cdot \sqrt{6 y}$

Step 3

Factor out $y^{2}$ .

$2 \sqrt{3} \cdot \sqrt{y^{2} y} \cdot \sqrt{6 y}$

Step 4

Pull terms out from under the radical.

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

Step 5

Combine using the product rule for radicals.

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

Step 6

Multiply $2 y \sqrt{3 y} \sqrt{6 y}$ .

Combine using the product rule for radicals.

$2 y \sqrt{6 y (3 y)}$

Multiply $3$ by $6$ .

$2 y \sqrt{18 y \cdot y}$

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

$2 y \sqrt{18 (y^{1} y)}$

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

$2 y \sqrt{18 (y^{1} y^{1})}$

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

$2 y \sqrt{18 y^{1 + 1}}$

Add $1$ and $1$ .

$2 y \sqrt{18 y^{2}}$

Step 7

Rewrite $18 y^{2}$ as ${(3 y)}^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$2 y \sqrt{9 (2) y^{2}}$

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

$2 y \sqrt{3^{2} \cdot 2 y^{2}}$

Move $2$ .

$2 y \sqrt{3^{2} y^{2} \cdot 2}$

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

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

Step 8

Pull terms out from under the radical.

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

Step 9

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

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

Multiply $y$ by $y$ .

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

Step 10

Multiply $3$ by $2$ .

$6 y^{2} \sqrt{2}$

Simplify square root of 12* square root of y^3* square root of 6y

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