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

$x$ ,

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

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

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

$18$ .

$\frac{39}{25}$