Algebra Examples

$\sqrt{8 y^{5}} \cdot \sqrt{40 y^{2}}$

Step 1

Rewrite $8 y^{5}$ as ${(2 y^{2})}^{2} \cdot (2 y)$ .

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Step 1.1

Factor $4$ out of $8$ .

$\sqrt{4 (2) y^{5}} \cdot \sqrt{40 y^{2}}$

Step 1.2

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

$\sqrt{2^{2} \cdot 2 y^{5}} \cdot \sqrt{40 y^{2}}$

Step 1.3

Factor out $y^{4}$ .

$\sqrt{2^{2} \cdot 2 (y^{4} y)} \cdot \sqrt{40 y^{2}}$

Step 1.4

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

$\sqrt{2^{2} \cdot 2 ({(y^{2})}^{2} y)} \cdot \sqrt{40 y^{2}}$

Step 1.5

Move $2$ .

$\sqrt{2^{2} {(y^{2})}^{2} \cdot 2 y} \cdot \sqrt{40 y^{2}}$

Step 1.6

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

$\sqrt{{(2 y^{2})}^{2} \cdot 2 y} \cdot \sqrt{40 y^{2}}$

Step 1.7

Add parentheses.

$\sqrt{{(2 y^{2})}^{2} \cdot (2 y)} \cdot \sqrt{40 y^{2}}$

Step 2

Pull terms out from under the radical.

$2 y^{2} \sqrt{2 y} \cdot \sqrt{40 y^{2}}$

Step 3

Rewrite $40 y^{2}$ as ${(2 y)}^{2} \cdot 10$ .

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Step 3.1

Factor $4$ out of $40$ .

$2 y^{2} \sqrt{2 y} \cdot \sqrt{4 (10) y^{2}}$

Step 3.2

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

$2 y^{2} \sqrt{2 y} \cdot \sqrt{2^{2} \cdot 10 y^{2}}$

Step 3.3

Move $10$ .

$2 y^{2} \sqrt{2 y} \cdot \sqrt{2^{2} y^{2} \cdot 10}$

Step 3.4

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

$2 y^{2} \sqrt{2 y} \cdot \sqrt{{(2 y)}^{2} \cdot 10}$

Step 4

Pull terms out from under the radical.

$2 y^{2} \sqrt{2 y} \cdot (2 y \sqrt{10})$

Step 5

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

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Step 5.1

Move $y$ .

$2 (y \cdot y^{2}) \sqrt{2 y} \cdot (2 \sqrt{10})$

Step 5.2

Multiply $y$ by $y^{2}$ .

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Step 5.2.1

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

$2 (y^{1} y^{2}) \sqrt{2 y} \cdot (2 \sqrt{10})$

Step 5.2.2

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

$2 y^{1 + 2} \sqrt{2 y} \cdot (2 \sqrt{10})$

Step 5.3

Add $1$ and $2$ .

$2 y^{3} \sqrt{2 y} \cdot (2 \sqrt{10})$

Step 6

Multiply $2 y^{3} \sqrt{2 y} (2 \sqrt{10})$ .

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Step 6.1

Multiply $2$ by $2$ .

$4 y^{3} \sqrt{2 y} \sqrt{10}$

Step 6.2

Combine using the product rule for radicals.

$4 y^{3} \sqrt{10 (2 y)}$

Step 6.3

Multiply $10$ by $2$ .

$4 y^{3} \sqrt{20 y}$

Step 7

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

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Step 7.1

Factor $4$ out of $20$ .

$4 y^{3} \sqrt{4 (5) y}$

Step 7.2

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

$4 y^{3} \sqrt{2^{2} \cdot 5 y}$

Step 7.3

Add parentheses.

$4 y^{3} \sqrt{2^{2} \cdot (5 y)}$

Step 8

Pull terms out from under the radical.

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

Step 9

Multiply $2$ by $4$ .

$8 y^{3} \sqrt{5 y}$