Algebra Examples

$5 \sqrt{13^{3}}$

Step 1

Multiply to rationalize the numerator.

$\frac{5 (\sqrt{13^{3}} \sqrt{13^{3}})}{\sqrt{13^{3}}}$

Step 2

Simplify.

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Step 2.1

Raise $\sqrt{13^{3}}$ to the power of $1$ .

$\frac{5 ({\sqrt{13^{3}}}^{1} \sqrt{13^{3}})}{\sqrt{13^{3}}}$

Step 2.2

Raise $\sqrt{13^{3}}$ to the power of $1$ .

$\frac{5 ({\sqrt{13^{3}}}^{1} {\sqrt{13^{3}}}^{1})}{\sqrt{13^{3}}}$

Step 2.3

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

$\frac{5 {\sqrt{13^{3}}}^{1 + 1}}{\sqrt{13^{3}}}$

Step 2.4

Add $1$ and $1$ .

$\frac{5 {\sqrt{13^{3}}}^{2}}{\sqrt{13^{3}}}$

Step 2.5

Rewrite ${\sqrt{13^{3}}}^{2}$ as $13^{3}$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13^{3}}$ as $13^{\frac{3}{2}}$ .

$\frac{5 {(13^{\frac{3}{2}})}^{2}}{\sqrt{13^{3}}}$

Step 2.5.2

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

$\frac{5 \cdot 13^{\frac{3}{2} \cdot 2}}{\sqrt{13^{3}}}$

Step 2.5.3

Combine $\frac{3}{2}$ and $2$ .

$\frac{5 \cdot 13^{\frac{3 \cdot 2}{2}}}{\sqrt{13^{3}}}$

Step 2.5.4

Multiply $3$ by $2$ .

$\frac{5 \cdot 13^{\frac{6}{2}}}{\sqrt{13^{3}}}$

Step 2.5.5

Cancel the common factor of $6$ and $2$ .

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Step 2.5.5.1

Factor $2$ out of $6$ .

$\frac{5 \cdot 13^{\frac{2 \cdot 3}{2}}}{\sqrt{13^{3}}}$

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $2$ out of $2$ .

$\frac{5 \cdot 13^{\frac{2 \cdot 3}{2 (1)}}}{\sqrt{13^{3}}}$

Step 2.5.5.2.2

Cancel the common factor.

$\frac{5 \cdot 13^{\frac{2 \cdot 3}{2 \cdot 1}}}{\sqrt{13^{3}}}$

Step 2.5.5.2.3

Rewrite the expression.

$\frac{5 \cdot 13^{\frac{3}{1}}}{\sqrt{13^{3}}}$

Step 2.5.5.2.4

Divide $3$ by $1$ .

$\frac{5 \cdot 13^{3}}{\sqrt{13^{3}}}$

Step 3

Raise $13$ to the power of $3$ .

$\frac{5 \cdot 13^{3}}{\sqrt{2197}}$

Step 4

Rewrite $2197$ as $13^{2} \cdot 13$ .

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Step 4.1

Factor $169$ out of $2197$ .

$\frac{5 \cdot 13^{3}}{\sqrt{169 (13)}}$

Step 4.2

Rewrite $169$ as $13^{2}$ .

$\frac{5 \cdot 13^{3}}{\sqrt{13^{2} \cdot 13}}$

Step 5

Pull terms out from under the radical.

$\frac{5 \cdot 13^{3}}{13 \sqrt{13}}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \cdot 13^{3}}{13 \sqrt{13}}$

Decimal Form:

$234.36083290 \dots$