Algebra Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Factor $4$ out of $12$ .

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

Step 1.2

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

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 4

Factor out $y^{2}$ .

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

Step 5

Pull terms out from under the radical.

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

Step 6

Combine using the product rule for radicals.

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

Step 7

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

Tap for more steps...

Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $3$ by $6$ .

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

Step 7.3

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

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

Step 7.4

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

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

Step 7.5

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

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

Step 7.6

Add $1$ and $1$ .

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

Step 8

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

Tap for more steps...

Step 8.1

Factor $9$ out of $18$ .

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

Step 8.2

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

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

Step 8.3

Move $2$ .

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

Step 8.4

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

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

Step 9

Pull terms out from under the radical.

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

Step 10

Remove non-negative terms from the absolute value.

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

Step 11

Multiply $3$ by $2$ .

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

Step 12

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

Tap for more steps...

Step 12.1

To multiply absolute values, multiply the terms inside each absolute value.

$6 | y \cdot y | \sqrt{2}$

Step 12.2

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

$6 | y^{1} y | \sqrt{2}$

Step 12.3

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

$6 | y^{1} y^{1} | \sqrt{2}$

Step 12.4

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

$6 | y^{1 + 1} | \sqrt{2}$

Step 12.5

Add $1$ and $1$ .

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

Step 13

Remove non-negative terms from the absolute value.

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