Basic Math Examples

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

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

Step 1

Multiply

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

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

\frac{\sqrt{13 a}}{\sqrt{13 a}}

$\frac{\sqrt{13 a}}{\sqrt{13 a}}$ .

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

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

\frac{\sqrt{13 a}}{\sqrt{13 a}}

$\frac{\sqrt{13 a}}{\sqrt{13 a}}$ .

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

$\sqrt{13 a}$ to the power of

1

$1$ .

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

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

Step 2.3

Raise

\sqrt{13 a}

$\sqrt{13 a}$ to the power of

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 2.5

Add

1

$1$ and

1

$1$ .

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

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

Step 2.6

Rewrite

{\sqrt{13 a}}^{2}

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

13 a

$13 a$ .

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Step 2.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{13 a}

$\sqrt{13 a}$ as

{(13 a)}^{\frac{1}{2}}

${(13 a)}^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

$2$ .

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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.

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