Finite Math Examples

$π$ , $\frac{2 π}{3}$ , $\frac{π}{4}$ , $\frac{2 π}{5}$

Step 1

Use the formula to find the geometric mean.

$\sqrt[4]{π \cdot \frac{2 π}{3} \cdot \frac{π}{4} \cdot \frac{2 π}{5}}$

Step 2

Combine $π$ and $\frac{2 π}{3}$ .

$\sqrt[4]{\frac{π (2 π)}{3} \cdot \frac{π}{4} \frac{2 π}{5}}$

Step 3

Multiply $\frac{π (2 π)}{3}$ by $\frac{π}{4}$ .

$\sqrt[4]{\frac{π (2 π) π}{3 \cdot 4} \cdot \frac{2 π}{5}}$

Step 4

Multiply $\frac{π (2 π) π}{3 \cdot 4}$ by $\frac{2 π}{5}$ .

$\sqrt[4]{\frac{π (2 π) π (2 π)}{3 \cdot 4 \cdot 5}}$

Step 5

Combine exponents.

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Step 5.1

Raise $π$ to the power of $1$ .

$\sqrt[4]{\frac{2 (π^{1} π) π \cdot 2 π}{3 \cdot 4 \cdot 5}}$

Step 5.2

Raise $π$ to the power of $1$ .

$\sqrt[4]{\frac{2 (π^{1} π^{1}) π \cdot 2 π}{3 \cdot 4 \cdot 5}}$

Step 5.3

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

$\sqrt[4]{\frac{2 π^{1 + 1} π \cdot 2 π}{3 \cdot 4 \cdot 5}}$

Step 5.4

Add $1$ and $1$ .

$\sqrt[4]{\frac{2 π^{2} π \cdot 2 π}{3 \cdot 4 \cdot 5}}$

Step 5.5

Raise $π$ to the power of $1$ .

$\sqrt[4]{\frac{2 (π^{1} π^{2}) \cdot 2 π}{3 \cdot 4 \cdot 5}}$

Step 5.6

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

$\sqrt[4]{\frac{2 π^{1 + 2} \cdot 2 π}{3 \cdot 4 \cdot 5}}$

Step 5.7

Add $1$ and $2$ .

$\sqrt[4]{\frac{2 π^{3} \cdot 2 π}{3 \cdot 4 \cdot 5}}$

Step 5.8

Multiply $2$ by $2$ .

$\sqrt[4]{\frac{4 π^{3} π}{3 \cdot 4 \cdot 5}}$

Step 5.9

Raise $π$ to the power of $1$ .

$\sqrt[4]{\frac{4 (π^{1} π^{3})}{3 \cdot 4 \cdot 5}}$

Step 5.10

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

$\sqrt[4]{\frac{4 π^{1 + 3}}{3 \cdot 4 \cdot 5}}$

Step 5.11

Add $1$ and $3$ .

$\sqrt[4]{\frac{4 π^{4}}{3 \cdot 4 \cdot 5}}$

Step 6

Reduce the expression $\frac{4 π^{4}}{3 \cdot 4 \cdot 5}$ by cancelling the common factors.

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Step 6.1

Cancel the common factor.

$\sqrt[4]{\frac{4 π^{4}}{3 \cdot 4 \cdot 5}}$

Step 6.2

Rewrite the expression.

$\sqrt[4]{\frac{π^{4}}{3 \cdot 5}}$

Step 7

Multiply $3$ by $5$ .

$\sqrt[4]{\frac{π^{4}}{15}}$

Step 8

Rewrite $\sqrt[4]{\frac{π^{4}}{15}}$ as $\frac{\sqrt[4]{π^{4}}}{\sqrt[4]{15}}$ .

$\frac{\sqrt[4]{π^{4}}}{\sqrt[4]{15}}$

Step 9

Pull terms out from under the radical, assuming positive real numbers.

$\frac{π}{\sqrt[4]{15}}$

Step 10

Multiply $\frac{π}{\sqrt[4]{15}}$ by $\frac{{\sqrt[4]{15}}^{3}}{{\sqrt[4]{15}}^{3}}$ .

$\frac{π}{\sqrt[4]{15}} \cdot \frac{{\sqrt[4]{15}}^{3}}{{\sqrt[4]{15}}^{3}}$

Step 11

Combine and simplify the denominator.

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Step 11.1

Multiply $\frac{π}{\sqrt[4]{15}}$ by $\frac{{\sqrt[4]{15}}^{3}}{{\sqrt[4]{15}}^{3}}$ .

$\frac{π {\sqrt[4]{15}}^{3}}{\sqrt[4]{15} {\sqrt[4]{15}}^{3}}$

Step 11.2

Raise $\sqrt[4]{15}$ to the power of $1$ .

$\frac{π {\sqrt[4]{15}}^{3}}{{\sqrt[4]{15}}^{1} {\sqrt[4]{15}}^{3}}$

Step 11.3

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

$\frac{π {\sqrt[4]{15}}^{3}}{{\sqrt[4]{15}}^{1 + 3}}$

Step 11.4

Add $1$ and $3$ .

$\frac{π {\sqrt[4]{15}}^{3}}{{\sqrt[4]{15}}^{4}}$

Step 11.5

Rewrite ${\sqrt[4]{15}}^{4}$ as $15$ .

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Step 11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{15}$ as $15^{\frac{1}{4}}$ .

$\frac{π {\sqrt[4]{15}}^{3}}{{(15^{\frac{1}{4}})}^{4}}$

Step 11.5.2

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

$\frac{π {\sqrt[4]{15}}^{3}}{15^{\frac{1}{4} \cdot 4}}$

Step 11.5.3

Combine $\frac{1}{4}$ and $4$ .

$\frac{π {\sqrt[4]{15}}^{3}}{15^{\frac{4}{4}}}$

Step 11.5.4

Cancel the common factor of $4$ .

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Step 11.5.4.1

Cancel the common factor.

$\frac{π {\sqrt[4]{15}}^{3}}{15^{\frac{4}{4}}}$

Step 11.5.4.2

Rewrite the expression.

$\frac{π {\sqrt[4]{15}}^{3}}{15^{1}}$

Step 11.5.5

Evaluate the exponent.

$\frac{π {\sqrt[4]{15}}^{3}}{15}$

Step 12

Simplify the numerator.

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Step 12.1

Rewrite ${\sqrt[4]{15}}^{3}$ as $\sqrt[4]{15^{3}}$ .

$\frac{π \sqrt[4]{15^{3}}}{15}$

Step 12.2

Raise $15$ to the power of $3$ .

$\frac{π \sqrt[4]{3375}}{15}$

Step 13

Approximate the result.

$1.5963461$

Step 14

The geometric mean should be rounded to one more decimal place than the original data. If the original data were mixed, round to one decimal place more than the least precise.

$1.6$