Basic Math Examples

$8$ , $9 \frac{6}{5}$

Step 1

Convert $9 \frac{6}{5}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$8, 9 + \frac{6}{5}$

Step 1.2

Add $9$ and $\frac{6}{5}$ .

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Step 1.2.1

To write $9$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$8, 9 \cdot \frac{5}{5} + \frac{6}{5}$

Step 1.2.2

Combine $9$ and $\frac{5}{5}$ .

$8, \frac{9 \cdot 5}{5} + \frac{6}{5}$

Step 1.2.3

Combine the numerators over the common denominator.

$8, \frac{9 \cdot 5 + 6}{5}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $9$ by $5$ .

$8, \frac{45 + 6}{5}$

Step 1.2.4.2

Add $45$ and $6$ .

$8, \frac{51}{5}$

Step 2

Use the formula to find the geometric mean.

$\sqrt{8 \cdot \frac{51}{5}}$

Step 3

Combine $8$ and $\frac{51}{5}$ .

$\sqrt{\frac{8 \cdot 51}{5}}$

Step 4

Multiply $8$ by $51$ .

$\sqrt{\frac{408}{5}}$

Step 5

Rewrite $\sqrt{\frac{408}{5}}$ as $\frac{\sqrt{408}}{\sqrt{5}}$ .

$\frac{\sqrt{408}}{\sqrt{5}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $408$ as $2^{2} \cdot 102$ .

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Step 6.1.1

Factor $4$ out of $408$ .

$\frac{\sqrt{4 (102)}}{\sqrt{5}}$

Step 6.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} \cdot 102}}{\sqrt{5}}$

Step 6.2

Pull terms out from under the radical.

$\frac{2 \sqrt{102}}{\sqrt{5}}$

Step 7

Multiply $\frac{2 \sqrt{102}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{102}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{2 \sqrt{102}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{102} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 8.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{102} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 8.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \sqrt{102} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 8.4

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

$\frac{2 \sqrt{102} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{2 \sqrt{102} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 8.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{2 \sqrt{102} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{2 \sqrt{102} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{2 \sqrt{102} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{102} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{102} \sqrt{5}}{5^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{2 \sqrt{102} \sqrt{5}}{5}$

Step 9

Simplify the numerator.

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Step 9.1

Combine using the product rule for radicals.

$\frac{2 \sqrt{5 \cdot 102}}{5}$

Step 9.2

Multiply $5$ by $102$ .

$\frac{2 \sqrt{510}}{5}$

Step 10

Approximate the result.

$9.03327183$

Step 11

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.

$9$