Finite Math Examples

$3$ , $5$ , $12$ , $14$ , $18$

Step 1

The quadratic mean (rms) of a set of numbers is the square root of the sum of the squares of the numbers divided by the number of terms.

$\sqrt{\frac{{(3)}^{2} + {(5)}^{2} + {(12)}^{2} + {(14)}^{2} + {(18)}^{2}}{5}}$

Step 2

Simplify the result.

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Step 2.1

Simplify the expression.

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Step 2.1.1

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

$\sqrt{\frac{9 + {(5)}^{2} + {(12)}^{2} + {(14)}^{2} + {(18)}^{2}}{5}}$

Step 2.1.2

Raise $5$ to the power of $2$ .

$\sqrt{\frac{9 + 25 + {(12)}^{2} + {(14)}^{2} + {(18)}^{2}}{5}}$

Step 2.1.3

Raise $12$ to the power of $2$ .

$\sqrt{\frac{9 + 25 + 144 + {(14)}^{2} + {(18)}^{2}}{5}}$

Step 2.1.4

Raise $14$ to the power of $2$ .

$\sqrt{\frac{9 + 25 + 144 + 196 + {(18)}^{2}}{5}}$

Step 2.1.5

Raise $18$ to the power of $2$ .

$\sqrt{\frac{9 + 25 + 144 + 196 + 324}{5}}$

Step 2.1.6

Add $9$ and $25$ .

$\sqrt{\frac{34 + 144 + 196 + 324}{5}}$

Step 2.1.7

Add $34$ and $144$ .

$\sqrt{\frac{178 + 196 + 324}{5}}$

Step 2.1.8

Add $178$ and $196$ .

$\sqrt{\frac{374 + 324}{5}}$

Step 2.1.9

Add $374$ and $324$ .

$\sqrt{\frac{698}{5}}$

Step 2.2

Cancel the common factor of $698$ and $5$ .

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Step 2.2.1

Rewrite $698$ as $1 (698)$ .

$\sqrt{\frac{1 (698)}{5}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Rewrite $5$ as $1 (5)$ .

$\sqrt{\frac{1 \cdot 698}{1 \cdot 5}}$

Step 2.2.2.2

Cancel the common factor.

$\sqrt{\frac{1 \cdot 698}{1 \cdot 5}}$

Step 2.2.2.3

Rewrite the expression.

$\sqrt{\frac{698}{5}}$

Step 2.3

Rewrite $\sqrt{\frac{698}{5}}$ as $\frac{\sqrt{698}}{\sqrt{5}}$ .

$\frac{\sqrt{698}}{\sqrt{5}}$

Step 2.4

Multiply $\frac{\sqrt{698}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{698}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{\sqrt{698}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt{698} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 2.5.2

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

$\frac{\sqrt{698} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 2.5.3

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

$\frac{\sqrt{698} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 2.5.4

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

$\frac{\sqrt{698} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 2.5.5

Add $1$ and $1$ .

$\frac{\sqrt{698} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 2.5.6

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

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Step 2.5.6.1

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

$\frac{\sqrt{698} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 2.5.6.2

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

$\frac{\sqrt{698} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 2.5.6.3

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

$\frac{\sqrt{698} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.6.4.1

Cancel the common factor.

$\frac{\sqrt{698} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.5.6.4.2

Rewrite the expression.

$\frac{\sqrt{698} \sqrt{5}}{5^{1}}$

Step 2.5.6.5

Evaluate the exponent.

$\frac{\sqrt{698} \sqrt{5}}{5}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Combine using the product rule for radicals.

$\frac{\sqrt{698 \cdot 5}}{5}$

Step 2.6.2

Multiply $698$ by $5$ .

$\frac{\sqrt{3490}}{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3490}}{5}$

Decimal Form:

$11.81524439 \dots$

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