Finite Math Examples

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

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

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

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:

$1.24721912 \dots$