Finite Math Examples

\begin{matrix} x & P (x) 0 & \frac{0}{15} 1 & \frac{1}{15} 2 & \frac{2}{15} 3 & \frac{3}{15} 4 & \frac{4}{15} 5 & \frac{5}{15} \end{matrix}

$\begin{array}{c} x & P (x) \\ 0 & \frac{0}{15} \\ 1 & \frac{1}{15} \\ 2 & \frac{2}{15} \\ 3 & \frac{3}{15} \\ 4 & \frac{4}{15} \\ 5 & \frac{5}{15} \end{array}$

Step 1

Prove that the given table satisfies the two properties needed for a probability distribution.

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Step 1.1

A discrete random variable

$x$ takes a set of separate values (such as

$0$ ,

$1$ ,

$2$ ...). Its probability distribution assigns a probability

$P (x)$ to each possible value

$x$ . For each

$x$ , the probability

$P (x)$ falls between

$0$ and

$1$ inclusive and the sum of the probabilities for all the possible

$x$ values equals to

$1$ .

1. For each

$x$ ,

$0 \leq P (x) \leq 1$ .

$P (x_{0}) + P (x_{1}) + P (x_{2}) + \dots + P (x_{n}) = 1$ .

Step 1.2

$\frac{0}{15}$ is between

$0$ and

$1$ inclusive, which meets the first property of the probability distribution.

$\frac{0}{15}$ is between

$0$ and

$1$ inclusive

Step 1.3

$\frac{1}{15}$ is between

$0$ and

$1$ inclusive, which meets the first property of the probability distribution.

$\frac{1}{15}$ is between

$0$ and

$1$ inclusive

Step 1.4

$\frac{2}{15}$ is between

$0$ and

$1$ inclusive, which meets the first property of the probability distribution.

$\frac{2}{15}$ is between

$0$ and

$1$ inclusive

Step 1.5

$\frac{3}{15}$ is between

$0$ and

$1$ inclusive, which meets the first property of the probability distribution.

$\frac{3}{15}$ is between

$0$ and

$1$ inclusive

Step 1.6

$\frac{4}{15}$ is between

$0$ and

$1$ inclusive, which meets the first property of the probability distribution.

$\frac{4}{15}$ is between

$0$ and

$1$ inclusive

Step 1.7

$\frac{5}{15}$ is between

$0$ and

$1$ inclusive, which meets the first property of the probability distribution.

$\frac{5}{15}$ is between

$0$ and

$1$ inclusive

Step 1.8

For each

$x$ , the probability

$P (x)$ falls between

$0$ and

$1$ inclusive, which meets the first property of the probability distribution.

$0 \leq P (x) \leq 1$ for all x values

Step 1.9

Find the sum of the probabilities for all the possible

$x$ values.

$\frac{0}{15} + \frac{1}{15} + \frac{2}{15} + \frac{3}{15} + \frac{4}{15} + \frac{5}{15}$

Step 1.10

The sum of the probabilities for all the possible

$x$ values is

$\frac{0}{15} + \frac{1}{15} + \frac{2}{15} + \frac{3}{15} + \frac{4}{15} + \frac{5}{15} = 1$ .

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Step 1.10.1

Combine the numerators over the common denominator.

$\frac{1 + 2 + 3 + 4 + 5}{15}$

Step 1.10.2

Simplify the expression.

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Step 1.10.2.1

Add

$1$ and

$2$ .

$\frac{3 + 3 + 4 + 5}{15}$

Step 1.10.2.2

Add

$3$ and

$3$ .

$\frac{6 + 4 + 5}{15}$

Step 1.10.2.3

Add

$6$ and

$4$ .

$\frac{10 + 5}{15}$

Step 1.10.2.4

Add

$10$ and

$5$ .

$\frac{15}{15}$

Step 1.10.2.5

Divide

$15$ by

$15$ .

$1$

Step 1.11

For each

$x$ , the probability of

$P (x)$ falls between

$0$ and

$1$ inclusive. In addition, the sum of the probabilities for all the possible

$x$ equals

$1$ , which means that the table satisfies the two properties of a probability distribution.

The table satisfies the two properties of a probability distribution:

Property 1:

$0 \leq P (x) \leq 1$ for all

$x$ values

Property 2:

$\frac{0}{15} + \frac{1}{15} + \frac{2}{15} + \frac{3}{15} + \frac{4}{15} + \frac{5}{15} = 1$

The table satisfies the two properties of a probability distribution:

Property 1:

$0 \leq P (x) \leq 1$ for all

$x$ values

Property 2:

$\frac{0}{15} + \frac{1}{15} + \frac{2}{15} + \frac{3}{15} + \frac{4}{15} + \frac{5}{15} = 1$

Step 2

The expectation mean of a distribution is the value expected if trials of the distribution could continue indefinitely. This is equal to each value multiplied by its discrete probability.

$0 \cdot \frac{0}{15} + 1 \cdot \frac{1}{15} + 2 \cdot \frac{2}{15} + 3 \cdot \frac{3}{15} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3

Simplify each term.

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Step 3.1

Divide

$0$ by

$15$ .

$0 \cdot 0 + 1 \cdot \frac{1}{15} + 2 \cdot \frac{2}{15} + 3 \cdot \frac{3}{15} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3.2

Multiply

$0$ by

$0$ .

$0 + 1 \cdot \frac{1}{15} + 2 \cdot \frac{2}{15} + 3 \cdot \frac{3}{15} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3.3

Multiply

$\frac{1}{15}$ by

$1$ .

$0 + \frac{1}{15} + 2 \cdot \frac{2}{15} + 3 \cdot \frac{3}{15} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3.4

Multiply

$2 (\frac{2}{15})$ .

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Step 3.4.1

Combine

$2$ and

$\frac{2}{15}$ .

$0 + \frac{1}{15} + \frac{2 \cdot 2}{15} + 3 \cdot \frac{3}{15} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3.4.2

Multiply

$2$ by

$2$ .

$0 + \frac{1}{15} + \frac{4}{15} + 3 \cdot \frac{3}{15} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3.5

Cancel the common factor of

$3$ .

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Step 3.5.1

Factor

$3$ out of

$15$ .

$0 + \frac{1}{15} + \frac{4}{15} + 3 \cdot \frac{3}{3 (5)} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3.5.2

Cancel the common factor.

$0 + \frac{1}{15} + \frac{4}{15} + 3 \cdot \frac{3}{3 \cdot 5} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3.5.3

Rewrite the expression.

$0 + \frac{1}{15} + \frac{4}{15} + \frac{3}{5} + 4 \cdot \frac{4}{15} + 5 \cdot \frac{5}{15}$

Step 3.6

Multiply

$4 (\frac{4}{15})$ .

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Step 3.6.1

Combine

$4$ and

$\frac{4}{15}$ .

$0 + \frac{1}{15} + \frac{4}{15} + \frac{3}{5} + \frac{4 \cdot 4}{15} + 5 \cdot \frac{5}{15}$

Step 3.6.2

Multiply

$4$ by

$4$ .

$0 + \frac{1}{15} + \frac{4}{15} + \frac{3}{5} + \frac{16}{15} + 5 \cdot \frac{5}{15}$

Step 3.7

Cancel the common factor of

$5$ .

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Step 3.7.1

Factor

$5$ out of

$15$ .

$0 + \frac{1}{15} + \frac{4}{15} + \frac{3}{5} + \frac{16}{15} + 5 \cdot \frac{5}{5 (3)}$

Step 3.7.2

Cancel the common factor.

$0 + \frac{1}{15} + \frac{4}{15} + \frac{3}{5} + \frac{16}{15} + 5 \cdot \frac{5}{5 \cdot 3}$

Step 3.7.3

Rewrite the expression.

$0 + \frac{1}{15} + \frac{4}{15} + \frac{3}{5} + \frac{16}{15} + \frac{5}{3}$

Step 4

Combine fractions.

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Step 4.1

Combine the numerators over the common denominator.

$\frac{1 + 4 + 16}{15} + \frac{3}{5} + \frac{5}{3}$

Step 4.2

Simplify by adding numbers.

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Step 4.2.1

Add

$1$ and

$4$ .

$\frac{5 + 16}{15} + \frac{3}{5} + \frac{5}{3}$

Step 4.2.2

Add

$5$ and

$16$ .

$\frac{21}{15} + \frac{3}{5} + \frac{5}{3}$

Step 5

Find the common denominator.

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Step 5.1

Multiply

$\frac{3}{5}$ by

$\frac{3}{3}$ .

$\frac{21}{15} + \frac{3}{5} \cdot \frac{3}{3} + \frac{5}{3}$

Step 5.2

Multiply

$\frac{3}{5}$ by

$\frac{3}{3}$ .

$\frac{21}{15} + \frac{3 \cdot 3}{5 \cdot 3} + \frac{5}{3}$

Step 5.3

Multiply

$\frac{5}{3}$ by

$\frac{5}{5}$ .

$\frac{21}{15} + \frac{3 \cdot 3}{5 \cdot 3} + \frac{5}{3} \cdot \frac{5}{5}$

Step 5.4

Multiply

$\frac{5}{3}$ by

$\frac{5}{5}$ .

$\frac{21}{15} + \frac{3 \cdot 3}{5 \cdot 3} + \frac{5 \cdot 5}{3 \cdot 5}$

Step 5.5

Reorder the factors of

$5 \cdot 3$ .

$\frac{21}{15} + \frac{3 \cdot 3}{3 \cdot 5} + \frac{5 \cdot 5}{3 \cdot 5}$

Step 5.6

Multiply

$3$ by

$5$ .

$\frac{21}{15} + \frac{3 \cdot 3}{15} + \frac{5 \cdot 5}{3 \cdot 5}$

Step 5.7

Multiply

$3$ by

$5$ .

$\frac{21}{15} + \frac{3 \cdot 3}{15} + \frac{5 \cdot 5}{15}$

Step 6

Combine the numerators over the common denominator.

$\frac{21 + 3 \cdot 3 + 5 \cdot 5}{15}$

Step 7

Simplify each term.

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Step 7.1

Multiply

$3$ by

$3$ .

$\frac{21 + 9 + 5 \cdot 5}{15}$

Step 7.2

Multiply

$5$ by

$5$ .

$\frac{21 + 9 + 25}{15}$

Step 8

Reduce the expression by cancelling the common factors.

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Step 8.1

Add

$21$ and

$9$ .

$\frac{30 + 25}{15}$

Step 8.2

Add

$30$ and

$25$ .

$\frac{55}{15}$

Step 8.3

Cancel the common factor of

$55$ and

$15$ .

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Step 8.3.1

Factor

$5$ out of

$55$ .

$\frac{5 (11)}{15}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor

$5$ out of

$15$ .

$\frac{5 \cdot 11}{5 \cdot 3}$

Step 8.3.2.2

Cancel the common factor.

$\frac{5 \cdot 11}{5 \cdot 3}$

Step 8.3.2.3

Rewrite the expression.

$\frac{11}{3}$

Step 9

The standard deviation of a distribution is a measure of the dispersion and is equal to the square root of the variance.

$s = \sqrt{\sum_{}^{} {(x - u)}^{2} \cdot (P (x))}$

Step 10

Fill in the known values.

$\sqrt{{(0 - (\frac{11}{3}))}^{2} \cdot \frac{0}{15} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11

Simplify the expression.

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Step 11.1

Subtract

$\frac{11}{3}$ from

$0$ .

$\sqrt{{(- \frac{11}{3})}^{2} \cdot \frac{0}{15} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.2

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.2.1

Apply the product rule to

$- \frac{11}{3}$ .

$\sqrt{{(- 1)}^{2} {(\frac{11}{3})}^{2} \cdot \frac{0}{15} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.2.2

Apply the product rule to

$\frac{11}{3}$ .

$\sqrt{{(- 1)}^{2} \frac{11^{2}}{3^{2}} \cdot \frac{0}{15} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.3

Simplify the expression.

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Step 11.3.1

Raise

$- 1$ to the power of

$2$ .

$\sqrt{1 \frac{11^{2}}{3^{2}} \cdot \frac{0}{15} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.3.2

Multiply

$\frac{11^{2}}{3^{2}}$ by

$1$ .

$\sqrt{\frac{11^{2}}{3^{2}} \cdot \frac{0}{15} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.4

Combine.

$\sqrt{\frac{11^{2} \cdot 0}{3^{2} \cdot 15} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.5

Cancel the common factor of

$0$ and

$15$ .

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Step 11.5.1

Factor

$15$ out of

$11^{2} \cdot 0$ .

$\sqrt{\frac{15 (11^{2} \cdot 0)}{3^{2} \cdot 15} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.5.2

Cancel the common factors.

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Step 11.5.2.1

Factor

$15$ out of

$3^{2} \cdot 15$ .

$\sqrt{\frac{15 (11^{2} \cdot 0)}{15 \cdot 3^{2}} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.5.2.2

Cancel the common factor.

Step 11.5.2.3

Rewrite the expression.

$\sqrt{\frac{11^{2} \cdot 0}{3^{2}} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.6

Simplify the expression.

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Step 11.6.1

Multiply

$11^{2}$ by

$0$ .

$\sqrt{\frac{0}{3^{2}} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.6.2

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{0}{9} + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.6.3

Divide

$0$ by

$9$ .

$\sqrt{0 + {(1 - (\frac{11}{3}))}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.6.4

Write

$1$ as a fraction with a common denominator.

$\sqrt{0 + {(\frac{3}{3} - \frac{11}{3})}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.6.5

Combine the numerators over the common denominator.

$\sqrt{0 + {(\frac{3 - 11}{3})}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.6.6

Subtract

$11$ from

$3$ .

$\sqrt{0 + {(\frac{- 8}{3})}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.6.7

Move the negative in front of the fraction.

$\sqrt{0 + {(- \frac{8}{3})}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.7

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.7.1

Apply the product rule to

$- \frac{8}{3}$ .

$\sqrt{0 + {(- 1)}^{2} {(\frac{8}{3})}^{2} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.7.2

Apply the product rule to

$\frac{8}{3}$ .

$\sqrt{0 + {(- 1)}^{2} \frac{8^{2}}{3^{2}} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.8

Simplify the expression.

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Step 11.8.1

Raise

$- 1$ to the power of

$2$ .

$\sqrt{0 + 1 \frac{8^{2}}{3^{2}} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.8.2

Multiply

$\frac{8^{2}}{3^{2}}$ by

$1$ .

$\sqrt{0 + \frac{8^{2}}{3^{2}} \cdot \frac{1}{15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.9

Combine.

$\sqrt{0 + \frac{8^{2} \cdot 1}{3^{2} \cdot 15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.10

Simplify the expression.

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Step 11.10.1

Multiply

$8^{2}$ by

$1$ .

$\sqrt{0 + \frac{8^{2}}{3^{2} \cdot 15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.10.2

Raise

$3$ to the power of

$2$ .

$\sqrt{0 + \frac{8^{2}}{9 \cdot 15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.10.3

Raise

$8$ to the power of

$2$ .

$\sqrt{0 + \frac{64}{9 \cdot 15} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.10.4

Multiply

$9$ by

$15$ .

$\sqrt{0 + \frac{64}{135} + {(2 - (\frac{11}{3}))}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.11

To write

$2$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$\sqrt{0 + \frac{64}{135} + {(2 \cdot \frac{3}{3} - \frac{11}{3})}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.12

Combine

$2$ and

$\frac{3}{3}$ .

$\sqrt{0 + \frac{64}{135} + {(\frac{2 \cdot 3}{3} - \frac{11}{3})}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.13

Combine the numerators over the common denominator.

$\sqrt{0 + \frac{64}{135} + {(\frac{2 \cdot 3 - 11}{3})}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.14

Simplify the numerator.

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Step 11.14.1

Multiply

$2$ by

$3$ .

$\sqrt{0 + \frac{64}{135} + {(\frac{6 - 11}{3})}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.14.2

Subtract

$11$ from

$6$ .

$\sqrt{0 + \frac{64}{135} + {(\frac{- 5}{3})}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.15

Move the negative in front of the fraction.

$\sqrt{0 + \frac{64}{135} + {(- \frac{5}{3})}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.16

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.16.1

Apply the product rule to

$- \frac{5}{3}$ .

$\sqrt{0 + \frac{64}{135} + {(- 1)}^{2} {(\frac{5}{3})}^{2} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.16.2

Apply the product rule to

$\frac{5}{3}$ .

$\sqrt{0 + \frac{64}{135} + {(- 1)}^{2} \frac{5^{2}}{3^{2}} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.17

Simplify the expression.

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Step 11.17.1

Raise

$- 1$ to the power of

$2$ .

$\sqrt{0 + \frac{64}{135} + 1 \frac{5^{2}}{3^{2}} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.17.2

Multiply

$\frac{5^{2}}{3^{2}}$ by

$1$ .

$\sqrt{0 + \frac{64}{135} + \frac{5^{2}}{3^{2}} \cdot \frac{2}{15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.18

Combine.

$\sqrt{0 + \frac{64}{135} + \frac{5^{2} \cdot 2}{3^{2} \cdot 15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.19

Simplify the expression.

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Step 11.19.1

Raise

$5$ to the power of

$2$ .

$\sqrt{0 + \frac{64}{135} + \frac{25 \cdot 2}{3^{2} \cdot 15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.19.2

Raise

$3$ to the power of

$2$ .

$\sqrt{0 + \frac{64}{135} + \frac{25 \cdot 2}{9 \cdot 15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.19.3

Multiply

$25$ by

$2$ .

$\sqrt{0 + \frac{64}{135} + \frac{50}{9 \cdot 15} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.19.4

Multiply

$9$ by

$15$ .

$\sqrt{0 + \frac{64}{135} + \frac{50}{135} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.20

Cancel the common factor of

$50$ and

$135$ .

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Step 11.20.1

Factor

$5$ out of

$50$ .

$\sqrt{0 + \frac{64}{135} + \frac{5 (10)}{135} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.20.2

Cancel the common factors.

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Step 11.20.2.1

Factor

$5$ out of

$135$ .

$\sqrt{0 + \frac{64}{135} + \frac{5 \cdot 10}{5 \cdot 27} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.20.2.2

Cancel the common factor.

Step 11.20.2.3

Rewrite the expression.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(3 - (\frac{11}{3}))}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.21

To write

$3$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(3 \cdot \frac{3}{3} - \frac{11}{3})}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.22

Combine

$3$ and

$\frac{3}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(\frac{3 \cdot 3}{3} - \frac{11}{3})}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.23

Combine the numerators over the common denominator.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(\frac{3 \cdot 3 - 11}{3})}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.24

Simplify the numerator.

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Step 11.24.1

Multiply

$3$ by

$3$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(\frac{9 - 11}{3})}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.24.2

Subtract

$11$ from

$9$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(\frac{- 2}{3})}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.25

Move the negative in front of the fraction.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(- \frac{2}{3})}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.26

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 11.26.1

Apply the product rule to

$- \frac{2}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(- 1)}^{2} {(\frac{2}{3})}^{2} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.26.2

Apply the product rule to

$\frac{2}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + {(- 1)}^{2} \frac{2^{2}}{3^{2}} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.27

Simplify the expression.

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Step 11.27.1

Raise

$- 1$ to the power of

$2$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + 1 \frac{2^{2}}{3^{2}} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.27.2

Multiply

$\frac{2^{2}}{3^{2}}$ by

$1$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{2^{2}}{3^{2}} \cdot \frac{3}{15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.28

Combine.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{2^{2} \cdot 3}{3^{2} \cdot 15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.29

Cancel the common factor of

$3$ and

$3^{2}$ .

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Step 11.29.1

Factor

$3$ out of

$2^{2} \cdot 3$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{3 \cdot 2^{2}}{3^{2} \cdot 15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.29.2

Cancel the common factors.

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Step 11.29.2.1

Factor

$3$ out of

$3^{2} \cdot 15$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{3 \cdot 2^{2}}{3 (3 \cdot 15)} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.29.2.2

Cancel the common factor.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{3 \cdot 2^{2}}{3 (3 \cdot 15)} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.29.2.3

Rewrite the expression.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{2^{2}}{3 \cdot 15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.30

Simplify the expression.

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Step 11.30.1

Raise

$2$ to the power of

$2$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{3 \cdot 15} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.30.2

Multiply

$3$ by

$15$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + {(4 - (\frac{11}{3}))}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.31

To write

$4$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + {(4 \cdot \frac{3}{3} - \frac{11}{3})}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.32

Combine

$4$ and

$\frac{3}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + {(\frac{4 \cdot 3}{3} - \frac{11}{3})}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.33

Combine the numerators over the common denominator.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + {(\frac{4 \cdot 3 - 11}{3})}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.34

Simplify the numerator.

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Step 11.34.1

Multiply

$4$ by

$3$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + {(\frac{12 - 11}{3})}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.34.2

Subtract

$11$ from

$12$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + {(\frac{1}{3})}^{2} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.35

Combine fractions.

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Step 11.35.1

Apply the product rule to

$\frac{1}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{1^{2}}{3^{2}} \cdot \frac{4}{15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.35.2

Combine.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{1^{2} \cdot 4}{3^{2} \cdot 15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.36

Simplify the numerator.

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Step 11.36.1

One to any power is one.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{1 \cdot 4}{3^{2} \cdot 15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.36.2

Multiply

$4$ by

$1$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{3^{2} \cdot 15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.37

Simplify the expression.

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Step 11.37.1

Raise

$3$ to the power of

$2$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{9 \cdot 15} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.37.2

Multiply

$9$ by

$15$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + {(5 - (\frac{11}{3}))}^{2} \cdot \frac{5}{15}}$

Step 11.38

To write

$5$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + {(5 \cdot \frac{3}{3} - \frac{11}{3})}^{2} \cdot \frac{5}{15}}$

Step 11.39

Combine

$5$ and

$\frac{3}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + {(\frac{5 \cdot 3}{3} - \frac{11}{3})}^{2} \cdot \frac{5}{15}}$

Step 11.40

Combine the numerators over the common denominator.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + {(\frac{5 \cdot 3 - 11}{3})}^{2} \cdot \frac{5}{15}}$

Step 11.41

Simplify the numerator.

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Step 11.41.1

Multiply

$5$ by

$3$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + {(\frac{15 - 11}{3})}^{2} \cdot \frac{5}{15}}$

Step 11.41.2

Subtract

$11$ from

$15$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + {(\frac{4}{3})}^{2} \cdot \frac{5}{15}}$

Step 11.42

Apply the product rule to

$\frac{4}{3}$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{4^{2}}{3^{2}} \cdot \frac{5}{15}}$

Step 11.43

Combine.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{4^{2} \cdot 5}{3^{2} \cdot 15}}$

Step 11.44

Cancel the common factor of

$5$ and

$15$ .

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Step 11.44.1

Factor

$5$ out of

$4^{2} \cdot 5$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{5 \cdot 4^{2}}{3^{2} \cdot 15}}$

Step 11.44.2

Cancel the common factors.

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Step 11.44.2.1

Factor

$5$ out of

$3^{2} \cdot 15$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{5 \cdot 4^{2}}{5 (3^{2} \cdot 3)}}$

Step 11.44.2.2

Cancel the common factor.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{5 \cdot 4^{2}}{5 (3^{2} \cdot 3)}}$

Step 11.44.2.3

Rewrite the expression.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{4^{2}}{3^{2} \cdot 3}}$

Step 11.45

Multiply

$3^{2}$ by

$3$ by adding the exponents.

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Step 11.45.1

Multiply

$3^{2}$ by

$3$ .

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Step 11.45.1.1

Raise

$3$ to the power of

$1$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{4^{2}}{3^{2} \cdot 3^{1}}}$

Step 11.45.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{4^{2}}{3^{2 + 1}}}$

Step 11.45.2

Add

$2$ and

$1$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{4^{2}}{3^{3}}}$

Step 11.46

Raise

$4$ to the power of

$2$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{16}{3^{3}}}$

Step 11.47

Raise

$3$ to the power of

$3$ .

$\sqrt{0 + \frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{16}{27}}$

Step 11.48

Add

$0$ and

$\frac{64}{135}$ .

$\sqrt{\frac{64}{135} + \frac{10}{27} + \frac{4}{45} + \frac{4}{135} + \frac{16}{27}}$

Step 11.49

To write

$\frac{10}{27}$ as a fraction with a common denominator, multiply by

$\frac{5}{5}$ .

$\sqrt{\frac{64}{135} + \frac{10}{27} \cdot \frac{5}{5} + \frac{4}{45} + \frac{4}{135} + \frac{16}{27}}$

Step 11.50

Write each expression with a common denominator of

$135$ , by multiplying each by an appropriate factor of

$1$ .

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Step 11.50.1

Multiply

$\frac{10}{27}$ by

$\frac{5}{5}$ .

$\sqrt{\frac{64}{135} + \frac{10 \cdot 5}{27 \cdot 5} + \frac{4}{45} + \frac{4}{135} + \frac{16}{27}}$

Step 11.50.2

Multiply

$27$ by

$5$ .

$\sqrt{\frac{64}{135} + \frac{10 \cdot 5}{135} + \frac{4}{45} + \frac{4}{135} + \frac{16}{27}}$

Step 11.51

Combine the numerators over the common denominator.

$\sqrt{\frac{64 + 10 \cdot 5}{135} + \frac{4}{45} + \frac{4}{135} + \frac{16}{27}}$

Step 11.52

Simplify the numerator.

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Step 11.52.1

Multiply

$10$ by

$5$ .

$\sqrt{\frac{64 + 50}{135} + \frac{4}{45} + \frac{4}{135} + \frac{16}{27}}$

Step 11.52.2

Add

$64$ and

$50$ .

$\sqrt{\frac{114}{135} + \frac{4}{45} + \frac{4}{135} + \frac{16}{27}}$

Step 11.53

To write

$\frac{4}{45}$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$\sqrt{\frac{114}{135} + \frac{4}{45} \cdot \frac{3}{3} + \frac{4}{135} + \frac{16}{27}}$

Step 11.54

Write each expression with a common denominator of

$135$ , by multiplying each by an appropriate factor of

$1$ .

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Step 11.54.1

Multiply

$\frac{4}{45}$ by

$\frac{3}{3}$ .

$\sqrt{\frac{114}{135} + \frac{4 \cdot 3}{45 \cdot 3} + \frac{4}{135} + \frac{16}{27}}$

Step 11.54.2

Multiply

$45$ by

$3$ .

$\sqrt{\frac{114}{135} + \frac{4 \cdot 3}{135} + \frac{4}{135} + \frac{16}{27}}$

Step 11.55

Combine the numerators over the common denominator.

$\sqrt{\frac{114 + 4 \cdot 3}{135} + \frac{4}{135} + \frac{16}{27}}$

Step 11.56

Simplify the numerator.

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Step 11.56.1

Multiply

$4$ by

$3$ .

$\sqrt{\frac{114 + 12}{135} + \frac{4}{135} + \frac{16}{27}}$

Step 11.56.2

Add

$114$ and

$12$ .

$\sqrt{\frac{126}{135} + \frac{4}{135} + \frac{16}{27}}$

Step 11.57

Combine the numerators over the common denominator.

$\sqrt{\frac{126 + 4}{135} + \frac{16}{27}}$

Step 11.58

Add

$126$ and

$4$ .

$\sqrt{\frac{130}{135} + \frac{16}{27}}$

Step 11.59

To write

$\frac{16}{27}$ as a fraction with a common denominator, multiply by

$\frac{5}{5}$ .

$\sqrt{\frac{130}{135} + \frac{16}{27} \cdot \frac{5}{5}}$

Step 11.60

Write each expression with a common denominator of

$135$ , by multiplying each by an appropriate factor of

$1$ .

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Step 11.60.1

Multiply

$\frac{16}{27}$ by

$\frac{5}{5}$ .

$\sqrt{\frac{130}{135} + \frac{16 \cdot 5}{27 \cdot 5}}$

Step 11.60.2

Multiply

$27$ by

$5$ .

$\sqrt{\frac{130}{135} + \frac{16 \cdot 5}{135}}$

Step 11.61

Combine the numerators over the common denominator.

$\sqrt{\frac{130 + 16 \cdot 5}{135}}$

Step 11.62

Simplify the numerator.

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Step 11.62.1

Multiply

$16$ by

$5$ .

$\sqrt{\frac{130 + 80}{135}}$

Step 11.62.2

Add

$130$ and

$80$ .

$\sqrt{\frac{210}{135}}$

Step 11.63

Cancel the common factor of

$210$ and

$135$ .

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Step 11.63.1

Factor

$15$ out of

$210$ .

$\sqrt{\frac{15 (14)}{135}}$

Step 11.63.2

Cancel the common factors.

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Step 11.63.2.1

Factor

$15$ out of

$135$ .

$\sqrt{\frac{15 \cdot 14}{15 \cdot 9}}$

Step 11.63.2.2

Cancel the common factor.

$\sqrt{\frac{15 \cdot 14}{15 \cdot 9}}$

Step 11.63.2.3

Rewrite the expression.

$\sqrt{\frac{14}{9}}$

Step 11.64

Rewrite

$\sqrt{\frac{14}{9}}$ as

$\frac{\sqrt{14}}{\sqrt{9}}$ .

$\frac{\sqrt{14}}{\sqrt{9}}$

Step 11.65

Simplify the denominator.

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Step 11.65.1

Rewrite

$9$ as

$3^{2}$ .

$\frac{\sqrt{14}}{\sqrt{3^{2}}}$

Step 11.65.2

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

$\frac{\sqrt{14}}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{14}}{3}$

Decimal Form:

$1.24721912 \dots$