Algebra Examples

$\sqrt{8 x y^{3}} \cdot \sqrt{20 y}$

Step 1

Rewrite $8 x y^{3}$ as ${(2 y)}^{2} \cdot (2 x y)$ .

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Step 1.1

Factor $4$ out of $8$ .

$\sqrt{4 (2) x y^{3}} \cdot \sqrt{20 y}$

Step 1.2

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

$\sqrt{2^{2} \cdot 2 x y^{3}} \cdot \sqrt{20 y}$

Step 1.3

Factor out $y^{2}$ .

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

Step 1.4

Move $x$ .

$\sqrt{2^{2} \cdot 2 y^{2} x y} \cdot \sqrt{20 y}$

Step 1.5

Move $2$ .

$\sqrt{2^{2} y^{2} \cdot 2 x y} \cdot \sqrt{20 y}$

Step 1.6

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

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

Step 1.7

Add parentheses.

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

Step 1.8

Add parentheses.

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

Step 2

Pull terms out from under the radical.

$2 y \sqrt{2 x y} \cdot \sqrt{20 y}$

Step 3

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

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Step 3.1

Factor $4$ out of $20$ .

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

Step 3.2

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

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

Step 3.3

Add parentheses.

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

Step 4

Pull terms out from under the radical.

$2 y \sqrt{2 x y} \cdot (2 \sqrt{5 y})$

Step 5

Multiply $2 y \sqrt{2 x y} (2 \sqrt{5 y})$ .

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Step 5.1

Multiply $2$ by $2$ .

$4 y \sqrt{2 x y} \sqrt{5 y}$

Step 5.2

Combine using the product rule for radicals.

$4 y \sqrt{5 y (2 x y)}$

Step 5.3

Multiply $2$ by $5$ .

$4 y \sqrt{10 y (x y)}$

Step 5.4

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

$4 y \sqrt{10 (x (y^{1} y))}$

Step 5.5

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

$4 y \sqrt{10 (x (y^{1} y^{1}))}$

Step 5.6

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

$4 y \sqrt{10 (x y^{1 + 1})}$

Step 5.7

Add $1$ and $1$ .

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

Step 6

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

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Step 6.1

Move $x$ .

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

Step 6.2

Reorder $10$ and $y^{2}$ .

$4 y \sqrt{y^{2} \cdot 10 x}$

Step 6.3

Add parentheses.

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

Step 7

Pull terms out from under the radical.

$4 y (y \sqrt{10 x})$

Step 8

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

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Step 8.1

Move $y$ .

$4 (y \cdot y) \sqrt{10 x}$

Step 8.2

Multiply $y$ by $y$ .

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