Finite Math Examples

$\frac{\sqrt{\frac{3}{13}}}{\sqrt{18}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $\sqrt{\frac{3}{13}}$ as $\frac{\sqrt{3}}{\sqrt{13}}$ .

$\frac{\frac{\sqrt{3}}{\sqrt{13}}}{\sqrt{18}}$

Step 1.2

Multiply $\frac{\sqrt{3}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{\frac{\sqrt{3}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}}{\sqrt{18}}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\frac{\frac{\sqrt{3} \sqrt{13}}{\sqrt{13} \sqrt{13}}}{\sqrt{18}}$

Step 1.3.2

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

$\frac{\frac{\sqrt{3} \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}}{\sqrt{18}}$

Step 1.3.3

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

$\frac{\frac{\sqrt{3} \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}}{\sqrt{18}}$

Step 1.3.4

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

$\frac{\frac{\sqrt{3} \sqrt{13}}{{\sqrt{13}}^{1 + 1}}}{\sqrt{18}}$

Step 1.3.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{3} \sqrt{13}}{{\sqrt{13}}^{2}}}{\sqrt{18}}$

Step 1.3.6

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

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Step 1.3.6.1

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

$\frac{\frac{\sqrt{3} \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}}{\sqrt{18}}$

Step 1.3.6.2

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

$\frac{\frac{\sqrt{3} \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}}{\sqrt{18}}$

Step 1.3.6.3

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

$\frac{\frac{\sqrt{3} \sqrt{13}}{13^{\frac{2}{2}}}}{\sqrt{18}}$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{3} \sqrt{13}}{13^{\frac{2}{2}}}}{\sqrt{18}}$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{3} \sqrt{13}}{13^{1}}}{\sqrt{18}}$

Step 1.3.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{3} \sqrt{13}}{13}}{\sqrt{18}}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{3 \cdot 13}}{13}}{\sqrt{18}}$

Step 1.4.2

Multiply $3$ by $13$ .

$\frac{\frac{\sqrt{39}}{13}}{\sqrt{18}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 2.1.1

Factor $9$ out of $18$ .

$\frac{\frac{\sqrt{39}}{13}}{\sqrt{9 (2)}}$

Step 2.1.2

Rewrite $9$ as $3^{2}$ .

$\frac{\frac{\sqrt{39}}{13}}{\sqrt{3^{2} \cdot 2}}$

Step 2.2

Pull terms out from under the radical.

$\frac{\frac{\sqrt{39}}{13}}{3 \sqrt{2}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{39}}{13} \cdot \frac{1}{3 \sqrt{2}}$

Step 4

Multiply $\frac{1}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{39}}{13} (\frac{1}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{1}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 \sqrt{2} \sqrt{2}}$

Step 5.2

Move $\sqrt{2}$ .

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 (\sqrt{2} \sqrt{2})}$

Step 5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 5.4

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 5.5

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

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 {\sqrt{2}}^{2}}$

Step 5.7

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

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Step 5.7.1

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

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}}$

Step 5.7.2

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

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 5.7.3

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

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 5.7.4

Cancel the common factor of $2$ .

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Step 5.7.4.1

Cancel the common factor.

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 5.7.4.2

Rewrite the expression.

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 \cdot 2^{1}}$

Step 5.7.5

Evaluate the exponent.

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 \cdot 2}$

Step 6

Multiply $3$ by $2$ .

$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{6}$

Step 7

Multiply $\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{6}$ .

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Step 7.1

Multiply $\frac{\sqrt{39}}{13}$ by $\frac{\sqrt{2}}{6}$ .

$\frac{\sqrt{39} \sqrt{2}}{13 \cdot 6}$

Step 7.2

Combine using the product rule for radicals.

$\frac{\sqrt{39 \cdot 2}}{13 \cdot 6}$

Step 7.3

Multiply $39$ by $2$ .

$\frac{\sqrt{78}}{13 \cdot 6}$

Step 7.4

Multiply $13$ by $6$ .

$\frac{\sqrt{78}}{78}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{78}}{78}$

Decimal Form:

$0.11322770 \dots$