Finite Math Examples

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

Step 1

Combine $\frac{13}{3}$ and $\sqrt{\frac{3}{13}}$ .

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Combine and simplify the denominator.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Add $1$ and $1$ .

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

Step 2.3.6

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

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Step 2.3.6.1

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

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

Step 2.3.6.2

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

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

Step 2.3.6.3

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

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

Step 2.3.6.4

Cancel the common factor of $2$ .

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Step 2.3.6.4.1

Cancel the common factor.

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

Step 2.3.6.4.2

Rewrite the expression.

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

Step 2.3.6.5

Evaluate the exponent.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Combine using the product rule for radicals.

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

Step 2.4.2

Multiply $3$ by $13$ .

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

Step 3

Combine $13$ and $\frac{\sqrt{39}}{13}$ .

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

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Reduce the expression $\frac{13 \sqrt{39}}{13}$ by cancelling the common factors.

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

$\sqrt{\frac{\frac{\sqrt{39}}{1}}{3}}$

Step 4.2

Divide $\sqrt{39}$ by $1$ .

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

Step 5

Rewrite $\sqrt{\frac{\sqrt{39}}{3}}$ as $\frac{\sqrt{\sqrt{39}}}{\sqrt{3}}$ .

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

Step 6

Rewrite $\sqrt{\sqrt{39}}$ as $\sqrt[4]{39}$ .

$\frac{\sqrt[4]{39}}{\sqrt{3}}$

Step 7

Multiply $\frac{\sqrt[4]{39}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt[4]{39}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{\sqrt[4]{39}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{\sqrt[4]{39} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.2

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

$\frac{\sqrt[4]{39} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 8.3

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

$\frac{\sqrt[4]{39} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 8.4

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

$\frac{\sqrt[4]{39} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{\sqrt[4]{39} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.6

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

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{39} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{\sqrt[4]{39} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{\sqrt[4]{39} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{\sqrt[4]{39} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{\sqrt[4]{39} \sqrt{3}}{3^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt[4]{39} \sqrt{3}}{3}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite the expression using the least common index of $4$ .

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Step 9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{39} \cdot 3^{\frac{1}{2}}}{3}$

Step 9.1.2

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{39} \cdot 3^{\frac{2}{4}}}{3}$

Step 9.1.3

Rewrite $3^{\frac{2}{4}}$ as $\sqrt[4]{3^{2}}$ .

$\frac{\sqrt[4]{39} \sqrt[4]{3^{2}}}{3}$

Step 9.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{39 \cdot 3^{2}}}{3}$

Step 9.3

Raise $3$ to the power of $2$ .

$\frac{\sqrt[4]{39 \cdot 9}}{3}$

Step 10

Multiply $39$ by $9$ .

$\frac{\sqrt[4]{351}}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{351}}{3}$

Decimal Form:

$1.44279797 \dots$