Basic Math Examples

$\sqrt{\frac{52 a b^{3}}{\sqrt{13 a}}}$

Step 1

Multiply $\frac{52 a b^{3}}{\sqrt{13 a}}$ by $\frac{\sqrt{13 a}}{\sqrt{13 a}}$ .

$\sqrt{\frac{52 a b^{3}}{\sqrt{13 a}} \cdot \frac{\sqrt{13 a}}{\sqrt{13 a}}}$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{52 a b^{3}}{\sqrt{13 a}}$ by $\frac{\sqrt{13 a}}{\sqrt{13 a}}$ .

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{\sqrt{13 a} \sqrt{13 a}}}$

Step 2.2

Raise $\sqrt{13 a}$ to the power of $1$ .

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{\sqrt{13 a}}^{1} \sqrt{13 a}}}$

Step 2.3

Raise $\sqrt{13 a}$ to the power of $1$ .

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{\sqrt{13 a}}^{1} {\sqrt{13 a}}^{1}}}$

Step 2.4

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

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{\sqrt{13 a}}^{1 + 1}}}$

Step 2.5

Add $1$ and $1$ .

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{\sqrt{13 a}}^{2}}}$

Step 2.6

Rewrite ${\sqrt{13 a}}^{2}$ as $13 a$ .

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Step 2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13 a}$ as ${(13 a)}^{\frac{1}{2}}$ .

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{({(13 a)}^{\frac{1}{2}})}^{2}}}$

Step 2.6.2

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

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{(13 a)}^{\frac{1}{2} \cdot 2}}}$

Step 2.6.3

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

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{(13 a)}^{\frac{2}{2}}}}$

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{(13 a)}^{\frac{2}{2}}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{{(13 a)}^{1}}}$

Step 2.6.5

Simplify.

$\sqrt{\frac{52 a b^{3} \sqrt{13 a}}{13 a}}$

Step 3

Cancel the common factor of $52$ and $13$ .

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Step 3.1

Factor $13$ out of $52 a b^{3} \sqrt{13 a}$ .

$\sqrt{\frac{13 (4 a b^{3} \sqrt{13 a})}{13 a}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $13$ out of $13 a$ .

$\sqrt{\frac{13 (4 a b^{3} \sqrt{13 a})}{13 (a)}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{13 (4 a b^{3} \sqrt{13 a})}{13 a}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{4 a b^{3} \sqrt{13 a}}{a}}$

Step 4

Cancel the common factor of $a$ .

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Step 4.1

Cancel the common factor.

$\sqrt{\frac{4 a b^{3} \sqrt{13 a}}{a}}$

Step 4.2

Divide $4 b^{3} \sqrt{13 a}$ by $1$ .

$\sqrt{4 b^{3} \sqrt{13 a}}$

Step 5

Rewrite $4 b^{3} \sqrt{13 a}$ as ${(2 b)}^{2} (b \sqrt{13 a})$ .

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Step 5.1

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

$\sqrt{2^{2} b^{3} \sqrt{13 a}}$

Step 5.2

Factor out $b^{2}$ .

$\sqrt{2^{2} (b^{2} b) \sqrt{13 a}}$

Step 5.3

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

$\sqrt{{(2 b)}^{2} b \sqrt{13 a}}$

Step 5.4

Add parentheses.

$\sqrt{{(2 b)}^{2} (b \sqrt{13 a})}$

Step 6

Pull terms out from under the radical.

$2 b \sqrt{b \sqrt{13 a}}$