Finite Math Examples

$\begin{array}{c} x & P (x) \\ 2 & \frac{2}{10} \\ 3 & \frac{3}{10} \\ 5 & \frac{5}{10} \end{array}$

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

$\frac{2}{10}$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$\frac{2}{10}$ is between $0$ and $1$ inclusive

Step 1.3

$\frac{3}{10}$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$\frac{3}{10}$ is between $0$ and $1$ inclusive

Step 1.4

$\frac{5}{10}$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$\frac{5}{10}$ is between $0$ and $1$ inclusive

Step 1.5

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 1.6

Find the sum of the probabilities for all the possible $x$ values.

$\frac{2}{10} + \frac{3}{10} + \frac{5}{10}$

Step 1.7

The sum of the probabilities for all the possible $x$ values is $\frac{2}{10} + \frac{3}{10} + \frac{5}{10} = 1$ .

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Step 1.7.1

Combine the numerators over the common denominator.

$\frac{2 + 3 + 5}{10}$

Step 1.7.2

Simplify the expression.

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Step 1.7.2.1

Add $2$ and $3$ .

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

Step 1.7.2.2

Add $5$ and $5$ .

$\frac{10}{10}$

Step 1.7.2.3

Divide $10$ by $10$ .

$1$

Step 1.8

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: $\frac{2}{10} + \frac{3}{10} + \frac{5}{10} = 1$

The table satisfies the two properties of a probability distribution:

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

Property 2: $\frac{2}{10} + \frac{3}{10} + \frac{5}{10} = 1$

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.

$u = 2 \cdot \frac{2}{10} + 3 \cdot \frac{3}{10} + 5 \cdot \frac{5}{10}$

Step 3

Simplify each term.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Factor $2$ out of $10$ .

$u = 2 \cdot \frac{2}{2 (5)} + 3 \cdot \frac{3}{10} + 5 \cdot \frac{5}{10}$

Step 3.1.2

Cancel the common factor.

$u = 2 \cdot \frac{2}{2 \cdot 5} + 3 \cdot \frac{3}{10} + 5 \cdot \frac{5}{10}$

Step 3.1.3

Rewrite the expression.

$u = \frac{2}{5} + 3 \cdot \frac{3}{10} + 5 \cdot \frac{5}{10}$

Step 3.2

Multiply $3 (\frac{3}{10})$ .

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Step 3.2.1

Combine $3$ and $\frac{3}{10}$ .

$u = \frac{2}{5} + \frac{3 \cdot 3}{10} + 5 \cdot \frac{5}{10}$

Step 3.2.2

Multiply $3$ by $3$ .

$u = \frac{2}{5} + \frac{9}{10} + 5 \cdot \frac{5}{10}$

Step 3.3

Cancel the common factor of $5$ .

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Step 3.3.1

Factor $5$ out of $10$ .

$u = \frac{2}{5} + \frac{9}{10} + 5 \cdot \frac{5}{5 (2)}$

Step 3.3.2

Cancel the common factor.

$u = \frac{2}{5} + \frac{9}{10} + 5 \cdot \frac{5}{5 \cdot 2}$

Step 3.3.3

Rewrite the expression.

$u = \frac{2}{5} + \frac{9}{10} + \frac{5}{2}$

Step 4

Find the common denominator.

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Step 4.1

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

$u = \frac{2}{5} \cdot \frac{2}{2} + \frac{9}{10} + \frac{5}{2}$

Step 4.2

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

$u = \frac{2 \cdot 2}{5 \cdot 2} + \frac{9}{10} + \frac{5}{2}$

Step 4.3

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

$u = \frac{2 \cdot 2}{5 \cdot 2} + \frac{9}{10} + \frac{5}{2} \cdot \frac{5}{5}$

Step 4.4

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

$u = \frac{2 \cdot 2}{5 \cdot 2} + \frac{9}{10} + \frac{5 \cdot 5}{2 \cdot 5}$

Step 4.5

Reorder the factors of $5 \cdot 2$ .

$u = \frac{2 \cdot 2}{2 \cdot 5} + \frac{9}{10} + \frac{5 \cdot 5}{2 \cdot 5}$

Step 4.6

Multiply $2$ by $5$ .

$u = \frac{2 \cdot 2}{10} + \frac{9}{10} + \frac{5 \cdot 5}{2 \cdot 5}$

Step 4.7

Multiply $2$ by $5$ .

$u = \frac{2 \cdot 2}{10} + \frac{9}{10} + \frac{5 \cdot 5}{10}$

Step 5

Combine the numerators over the common denominator.

$u = \frac{2 \cdot 2 + 9 + 5 \cdot 5}{10}$

Step 6

Simplify each term.

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Step 6.1

Multiply $2$ by $2$ .

$u = \frac{4 + 9 + 5 \cdot 5}{10}$

Step 6.2

Multiply $5$ by $5$ .

$u = \frac{4 + 9 + 25}{10}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Add $4$ and $9$ .

$u = \frac{13 + 25}{10}$

Step 7.2

Add $13$ and $25$ .

$u = \frac{38}{10}$

Step 7.3

Cancel the common factor of $38$ and $10$ .

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Step 7.3.1

Factor $2$ out of $38$ .

$u = \frac{2 (19)}{10}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $2$ out of $10$ .

$u = \frac{2 \cdot 19}{2 \cdot 5}$

Step 7.3.2.2

Cancel the common factor.

$u = \frac{2 \cdot 19}{2 \cdot 5}$

Step 7.3.2.3

Rewrite the expression.

$u = \frac{19}{5}$

Step 8

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

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

Step 9

Fill in the known values.

${(2 - (\frac{19}{5}))}^{2} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10

Simplify the expression.

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Step 10.1

Simplify each term.

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Step 10.1.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(2 \cdot \frac{5}{5} - \frac{19}{5})}^{2} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.2

Combine $2$ and $\frac{5}{5}$ .

${(\frac{2 \cdot 5}{5} - \frac{19}{5})}^{2} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.3

Combine the numerators over the common denominator.

${(\frac{2 \cdot 5 - 19}{5})}^{2} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.4

Simplify the numerator.

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Step 10.1.4.1

Multiply $2$ by $5$ .

${(\frac{10 - 19}{5})}^{2} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.4.2

Subtract $19$ from $10$ .

${(\frac{- 9}{5})}^{2} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.5

Move the negative in front of the fraction.

${(- \frac{9}{5})}^{2} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.6

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

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Step 10.1.6.1

Apply the product rule to $- \frac{9}{5}$ .

${(- 1)}^{2} {(\frac{9}{5})}^{2} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.6.2

Apply the product rule to $\frac{9}{5}$ .

${(- 1)}^{2} \frac{9^{2}}{5^{2}} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.7

Raise $- 1$ to the power of $2$ .

$1 \frac{9^{2}}{5^{2}} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.8

Multiply $\frac{9^{2}}{5^{2}}$ by $1$ .

$\frac{9^{2}}{5^{2}} \cdot \frac{2}{10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.9

Combine.

$\frac{9^{2} \cdot 2}{5^{2} \cdot 10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.10

Cancel the common factor of $2$ and $10$ .

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Step 10.1.10.1

Factor $2$ out of $9^{2} \cdot 2$ .

$\frac{2 \cdot 9^{2}}{5^{2} \cdot 10} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.10.2

Cancel the common factors.

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Step 10.1.10.2.1

Factor $2$ out of $5^{2} \cdot 10$ .

$\frac{2 \cdot 9^{2}}{2 (5^{2} \cdot 5)} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.10.2.2

Cancel the common factor.

$\frac{2 \cdot 9^{2}}{2 (5^{2} \cdot 5)} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.10.2.3

Rewrite the expression.

$\frac{9^{2}}{5^{2} \cdot 5} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.11

Multiply $5^{2}$ by $5$ by adding the exponents.

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Step 10.1.11.1

Multiply $5^{2}$ by $5$ .

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Step 10.1.11.1.1

Raise $5$ to the power of $1$ .

$\frac{9^{2}}{5^{2} \cdot 5^{1}} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.11.1.2

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

$\frac{9^{2}}{5^{2 + 1}} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.11.2

Add $2$ and $1$ .

$\frac{9^{2}}{5^{3}} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.12

Raise $9$ to the power of $2$ .

$\frac{81}{5^{3}} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.13

Raise $5$ to the power of $3$ .

$\frac{81}{125} + {(3 - (\frac{19}{5}))}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.14

To write $3$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{81}{125} + {(3 \cdot \frac{5}{5} - \frac{19}{5})}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.15

Combine $3$ and $\frac{5}{5}$ .

$\frac{81}{125} + {(\frac{3 \cdot 5}{5} - \frac{19}{5})}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.16

Combine the numerators over the common denominator.

$\frac{81}{125} + {(\frac{3 \cdot 5 - 19}{5})}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.17

Simplify the numerator.

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Step 10.1.17.1

Multiply $3$ by $5$ .

$\frac{81}{125} + {(\frac{15 - 19}{5})}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.17.2

Subtract $19$ from $15$ .

$\frac{81}{125} + {(\frac{- 4}{5})}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.18

Move the negative in front of the fraction.

$\frac{81}{125} + {(- \frac{4}{5})}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.19

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

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Step 10.1.19.1

Apply the product rule to $- \frac{4}{5}$ .

$\frac{81}{125} + {(- 1)}^{2} {(\frac{4}{5})}^{2} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.19.2

Apply the product rule to $\frac{4}{5}$ .

$\frac{81}{125} + {(- 1)}^{2} \frac{4^{2}}{5^{2}} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.20

Raise $- 1$ to the power of $2$ .

$\frac{81}{125} + 1 \frac{4^{2}}{5^{2}} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.21

Multiply $\frac{4^{2}}{5^{2}}$ by $1$ .

$\frac{81}{125} + \frac{4^{2}}{5^{2}} \cdot \frac{3}{10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.22

Combine.

$\frac{81}{125} + \frac{4^{2} \cdot 3}{5^{2} \cdot 10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.23

Raise $4$ to the power of $2$ .

$\frac{81}{125} + \frac{16 \cdot 3}{5^{2} \cdot 10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.24

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

$\frac{81}{125} + \frac{16 \cdot 3}{25 \cdot 10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.25

Multiply $16$ by $3$ .

$\frac{81}{125} + \frac{48}{25 \cdot 10} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.26

Multiply $25$ by $10$ .

$\frac{81}{125} + \frac{48}{250} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.27

Cancel the common factor of $48$ and $250$ .

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Step 10.1.27.1

Factor $2$ out of $48$ .

$\frac{81}{125} + \frac{2 (24)}{250} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.27.2

Cancel the common factors.

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Step 10.1.27.2.1

Factor $2$ out of $250$ .

$\frac{81}{125} + \frac{2 \cdot 24}{2 \cdot 125} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.27.2.2

Cancel the common factor.

$\frac{81}{125} + \frac{2 \cdot 24}{2 \cdot 125} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.27.2.3

Rewrite the expression.

$\frac{81}{125} + \frac{24}{125} + {(5 - (\frac{19}{5}))}^{2} \cdot \frac{5}{10}$

Step 10.1.28

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

$\frac{81}{125} + \frac{24}{125} + {(5 \cdot \frac{5}{5} - \frac{19}{5})}^{2} \cdot \frac{5}{10}$

Step 10.1.29

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

$\frac{81}{125} + \frac{24}{125} + {(\frac{5 \cdot 5}{5} - \frac{19}{5})}^{2} \cdot \frac{5}{10}$

Step 10.1.30

Combine the numerators over the common denominator.

$\frac{81}{125} + \frac{24}{125} + {(\frac{5 \cdot 5 - 19}{5})}^{2} \cdot \frac{5}{10}$

Step 10.1.31

Simplify the numerator.

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Step 10.1.31.1

Multiply $5$ by $5$ .

$\frac{81}{125} + \frac{24}{125} + {(\frac{25 - 19}{5})}^{2} \cdot \frac{5}{10}$

Step 10.1.31.2

Subtract $19$ from $25$ .

$\frac{81}{125} + \frac{24}{125} + {(\frac{6}{5})}^{2} \cdot \frac{5}{10}$

Step 10.1.32

Apply the product rule to $\frac{6}{5}$ .

$\frac{81}{125} + \frac{24}{125} + \frac{6^{2}}{5^{2}} \cdot \frac{5}{10}$

Step 10.1.33

Combine.

$\frac{81}{125} + \frac{24}{125} + \frac{6^{2} \cdot 5}{5^{2} \cdot 10}$

Step 10.1.34

Cancel the common factor of $5$ and $5^{2}$ .

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Step 10.1.34.1

Factor $5$ out of $6^{2} \cdot 5$ .

$\frac{81}{125} + \frac{24}{125} + \frac{5 \cdot 6^{2}}{5^{2} \cdot 10}$

Step 10.1.34.2

Cancel the common factors.

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Step 10.1.34.2.1

Factor $5$ out of $5^{2} \cdot 10$ .

$\frac{81}{125} + \frac{24}{125} + \frac{5 \cdot 6^{2}}{5 (5 \cdot 10)}$

Step 10.1.34.2.2

Cancel the common factor.

$\frac{81}{125} + \frac{24}{125} + \frac{5 \cdot 6^{2}}{5 (5 \cdot 10)}$

Step 10.1.34.2.3

Rewrite the expression.

$\frac{81}{125} + \frac{24}{125} + \frac{6^{2}}{5 \cdot 10}$

Step 10.1.35

Raise $6$ to the power of $2$ .

$\frac{81}{125} + \frac{24}{125} + \frac{36}{5 \cdot 10}$

Step 10.1.36

Multiply $5$ by $10$ .

$\frac{81}{125} + \frac{24}{125} + \frac{36}{50}$

Step 10.1.37

Cancel the common factor of $36$ and $50$ .

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Step 10.1.37.1

Factor $2$ out of $36$ .

$\frac{81}{125} + \frac{24}{125} + \frac{2 (18)}{50}$

Step 10.1.37.2

Cancel the common factors.

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Step 10.1.37.2.1

Factor $2$ out of $50$ .

$\frac{81}{125} + \frac{24}{125} + \frac{2 \cdot 18}{2 \cdot 25}$

Step 10.1.37.2.2

Cancel the common factor.

$\frac{81}{125} + \frac{24}{125} + \frac{2 \cdot 18}{2 \cdot 25}$

Step 10.1.37.2.3

Rewrite the expression.

$\frac{81}{125} + \frac{24}{125} + \frac{18}{25}$

Step 10.2

Simplify terms.

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Step 10.2.1

Combine the numerators over the common denominator.

$\frac{81 + 24}{125} + \frac{18}{25}$

Step 10.2.2

Add $81$ and $24$ .

$\frac{105}{125} + \frac{18}{25}$

Step 10.2.3

Cancel the common factor of $105$ and $125$ .

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Step 10.2.3.1

Factor $5$ out of $105$ .

$\frac{5 (21)}{125} + \frac{18}{25}$

Step 10.2.3.2

Cancel the common factors.

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Step 10.2.3.2.1

Factor $5$ out of $125$ .

$\frac{5 \cdot 21}{5 \cdot 25} + \frac{18}{25}$

Step 10.2.3.2.2

Cancel the common factor.

$\frac{5 \cdot 21}{5 \cdot 25} + \frac{18}{25}$

Step 10.2.3.2.3

Rewrite the expression.

$\frac{21}{25} + \frac{18}{25}$

Step 10.2.4

Combine the numerators over the common denominator.

$\frac{21 + 18}{25}$

Step 10.2.5

Add $21$ and $18$ .

$\frac{39}{25}$