Algebra Examples

$6 \sqrt{15 y^{4}} \cdot 2 \sqrt{20 y^{2}}$

Step 1

Rewrite $15 y^{4}$ as ${(y^{2})}^{2} \cdot 15$ .

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Step 1.1

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

$6 \sqrt{15 {(y^{2})}^{2}} \cdot 2 \sqrt{20 y^{2}}$

Step 1.2

Reorder $15$ and ${(y^{2})}^{2}$ .

$6 \sqrt{{(y^{2})}^{2} \cdot 15} \cdot 2 \sqrt{20 y^{2}}$

Step 2

Pull terms out from under the radical.

$6 (| y^{2} | \sqrt{15}) \cdot 2 \sqrt{20 y^{2}}$

Step 3

Remove non-negative terms from the absolute value.

$6 (y^{2} \sqrt{15}) \cdot 2 \sqrt{20 y^{2}}$

Step 4

Multiply $2$ by $6$ .

$12 y^{2} \sqrt{15} \sqrt{20 y^{2}}$

Step 5

Rewrite $20 y^{2}$ as ${(2 y)}^{2} \cdot 5$ .

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Step 5.1

Factor $4$ out of $20$ .

$12 y^{2} \sqrt{15} \sqrt{4 (5) y^{2}}$

Step 5.2

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

$12 y^{2} \sqrt{15} \sqrt{2^{2} \cdot 5 y^{2}}$

Step 5.3

Move $5$ .

$12 y^{2} \sqrt{15} \sqrt{2^{2} y^{2} \cdot 5}$

Step 5.4

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

$12 y^{2} \sqrt{15} \sqrt{{(2 y)}^{2} \cdot 5}$

Step 6

Pull terms out from under the radical.

$12 y^{2} \sqrt{15} (| 2 y | \sqrt{5})$

Step 7

Remove non-negative terms from the absolute value.

$12 y^{2} \sqrt{15} (2 | y | \sqrt{5})$

Step 8

Multiply $12 y^{2} \sqrt{15} (2 | y | \sqrt{5})$ .

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Step 8.1

Multiply $2$ by $12$ .

$24 y^{2} \sqrt{15} (| y | \sqrt{5})$

Step 8.2

Combine using the product rule for radicals.

$24 y^{2} \sqrt{5 \cdot 15} | y |$

Step 8.3

Multiply $5$ by $15$ .

$24 y^{2} \sqrt{75} | y |$

Step 9

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 9.1

Factor $25$ out of $75$ .

$24 y^{2} \sqrt{25 (3)} | y |$

Step 9.2

Rewrite $25$ as $5^{2}$ .

$24 y^{2} \sqrt{5^{2} \cdot 3} | y |$

Step 10

Pull terms out from under the radical.

$24 y^{2} (| 5 | \sqrt{3}) | y |$

Step 11

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$24 y^{2} (5 \sqrt{3}) | y |$

Step 12

Multiply $5$ by $24$ .

$120 y^{2} \sqrt{3} | y |$