Basic Math Examples

${(\sqrt{5 v^{2} w^{5}})}^{2}$

Step 1

Rewrite $5 v^{2} w^{5}$ as ${(v w^{2})}^{2} \cdot (5 w)$ .

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Step 1.1

Factor out $w^{4}$ .

${\sqrt{5 v^{2} (w^{4} w)}}^{2}$

Step 1.2

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

${\sqrt{5 v^{2} ({(w^{2})}^{2} w)}}^{2}$

Step 1.3

Move $5$ .

${\sqrt{v^{2} {(w^{2})}^{2} \cdot 5 w}}^{2}$

Step 1.4

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

${\sqrt{{(v w^{2})}^{2} \cdot 5 w}}^{2}$

Step 1.5

Add parentheses.

${\sqrt{{(v w^{2})}^{2} \cdot (5 w)}}^{2}$

Step 2

Pull terms out from under the radical.

${(v w^{2} \sqrt{5 w})}^{2}$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $v w^{2} \sqrt{5 w}$ .

${(v w^{2})}^{2} {\sqrt{5 w}}^{2}$

Step 3.2

Apply the product rule to $v w^{2}$ .

$v^{2} {(w^{2})}^{2} {\sqrt{5 w}}^{2}$

Step 4

Multiply the exponents in ${(w^{2})}^{2}$ .

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Step 4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$v^{2} w^{2 \cdot 2} {\sqrt{5 w}}^{2}$

Step 4.2

Multiply $2$ by $2$ .

$v^{2} w^{4} {\sqrt{5 w}}^{2}$

Step 5

Rewrite ${\sqrt{5 w}}^{2}$ as $5 w$ .

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 w}$ as ${(5 w)}^{\frac{1}{2}}$ .

$v^{2} w^{4} {({(5 w)}^{\frac{1}{2}})}^{2}$

Step 5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$v^{2} w^{4} {(5 w)}^{\frac{1}{2} \cdot 2}$

Step 5.3

Combine $\frac{1}{2}$ and $2$ .

$v^{2} w^{4} {(5 w)}^{\frac{2}{2}}$

Step 5.4

Cancel the common factor of $2$ .

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Step 5.4.1

Cancel the common factor.

$v^{2} w^{4} {(5 w)}^{\frac{2}{2}}$

Step 5.4.2

Rewrite the expression.

$v^{2} w^{4} {(5 w)}^{1}$

Step 5.5

Simplify.

$v^{2} w^{4} (5 w)$

Step 6

Rewrite using the commutative property of multiplication.

$5 v^{2} w^{4} w$

Step 7

Multiply $w^{4}$ by $w$ by adding the exponents.

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Step 7.1

Move $w$ .

$5 v^{2} (w \cdot w^{4})$

Step 7.2

Multiply $w$ by $w^{4}$ .

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Step 7.2.1

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

$5 v^{2} (w^{1} w^{4})$

Step 7.2.2

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

$5 v^{2} w^{1 + 4}$

Step 7.3

Add $1$ and $4$ .

$5 v^{2} w^{5}$