Basic Math Examples

$\sqrt{8 u^{2} w^{3}} \sqrt{2 u^{2} w^{4}}$

Step 1

Rewrite $8 u^{2} w^{3}$ as ${(2 u w)}^{2} \cdot (2 w)$ .

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Step 1.1

Factor $4$ out of $8$ .

$\sqrt{4 (2) u^{2} w^{3}} \sqrt{2 u^{2} w^{4}}$

Step 1.2

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

$\sqrt{2^{2} \cdot 2 u^{2} w^{3}} \sqrt{2 u^{2} w^{4}}$

Step 1.3

Factor out $w^{2}$ .

$\sqrt{2^{2} \cdot 2 u^{2} (w^{2} w)} \sqrt{2 u^{2} w^{4}}$

Step 1.4

Move $2$ .

$\sqrt{2^{2} u^{2} w^{2} \cdot 2 w} \sqrt{2 u^{2} w^{4}}$

Step 1.5

Rewrite $2^{2} u^{2} w^{2}$ as ${(2 u w)}^{2}$ .

$\sqrt{{(2 u w)}^{2} \cdot 2 w} \sqrt{2 u^{2} w^{4}}$

Step 1.6

Add parentheses.

$\sqrt{{(2 u w)}^{2} \cdot (2 w)} \sqrt{2 u^{2} w^{4}}$

Step 2

Pull terms out from under the radical.

$2 u w \sqrt{2 w} \sqrt{2 u^{2} w^{4}}$

Step 3

Rewrite $2 u^{2} w^{4}$ as ${(u w^{2})}^{2} \cdot 2$ .

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Step 3.1

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

$2 u w \sqrt{2 w} \sqrt{2 u^{2} {(w^{2})}^{2}}$

Step 3.2

Move $2$ .

$2 u w \sqrt{2 w} \sqrt{u^{2} {(w^{2})}^{2} \cdot 2}$

Step 3.3

Rewrite $u^{2} {(w^{2})}^{2}$ as ${(u w^{2})}^{2}$ .

$2 u w \sqrt{2 w} \sqrt{{(u w^{2})}^{2} \cdot 2}$

Step 4

Pull terms out from under the radical.

$2 u w \sqrt{2 w} (u w^{2} \sqrt{2})$

Step 5

Multiply $u$ by $u$ by adding the exponents.

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Step 5.1

Move $u$ .

$2 (u \cdot u) w \sqrt{2 w} (w^{2} \sqrt{2})$

Step 5.2

Multiply $u$ by $u$ .

$2 u^{2} w \sqrt{2 w} (w^{2} \sqrt{2})$

Step 6

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

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Step 6.1

Move $w^{2}$ .

$2 u^{2} (w^{2} w) \sqrt{2 w} \sqrt{2}$

Step 6.2

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

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Step 6.2.1

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

$2 u^{2} (w^{2} w^{1}) \sqrt{2 w} \sqrt{2}$

Step 6.2.2

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

$2 u^{2} w^{2 + 1} \sqrt{2 w} \sqrt{2}$

Step 6.3

Add $2$ and $1$ .

$2 u^{2} w^{3} \sqrt{2 w} \sqrt{2}$

Step 7

Multiply $2 u^{2} w^{3} \sqrt{2 w} \sqrt{2}$ .

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Step 7.1

Combine using the product rule for radicals.

$2 u^{2} w^{3} \sqrt{2 (2 w)}$

Step 7.2

Multiply $2$ by $2$ .

$2 u^{2} w^{3} \sqrt{4 w}$

Step 8

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

$2 u^{2} w^{3} \sqrt{2^{2} w}$

Step 9

Pull terms out from under the radical.

$2 u^{2} w^{3} (2 \sqrt{w})$

Step 10

Multiply $2$ by $2$ .

$4 u^{2} w^{3} \sqrt{w}$