Calculus Examples

$\int_{0}^{7} | x^{2} - 6 x + 5 | d x$

Step 1

Split up the integral depending on where $x^{2} - 6 x + 5$ is positive and negative.

$\int_{0}^{1} x^{2} - 6 x + 5 d x + \int_{1}^{5} - x^{2} + 6 x - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{1} x^{2} d x + \int_{0}^{1} - 6 x d x + \int_{0}^{1} 5 d x + \int_{1}^{5} - x^{2} + 6 x - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 3

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{0}^{1} + \int_{0}^{1} - 6 x d x + \int_{0}^{1} 5 d x + \int_{1}^{5} - x^{2} + 6 x - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 4

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$\frac{1}{3} x^{3}]_{0}^{1} - 6 \int_{0}^{1} x d x + \int_{0}^{1} 5 d x + \int_{1}^{5} - x^{2} + 6 x - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{1}{2} x^{2}]_{0}^{1}) + \int_{0}^{1} 5 d x + \int_{1}^{5} - x^{2} + 6 x - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 6

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

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + \int_{0}^{1} 5 d x + \int_{1}^{5} - x^{2} + 6 x - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 7

Apply the constant rule.

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} + \int_{1}^{5} - x^{2} + 6 x - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 8

Split the single integral into multiple integrals.

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} + \int_{1}^{5} - x^{2} d x + \int_{1}^{5} 6 x d x + \int_{1}^{5} - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 9

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - \int_{1}^{5} x^{2} d x + \int_{1}^{5} 6 x d x + \int_{1}^{5} - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 10

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - (\frac{1}{3} x^{3}]_{1}^{5}) + \int_{1}^{5} 6 x d x + \int_{1}^{5} - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 11

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - (\frac{x^{3}}{3}]_{1}^{5}) + \int_{1}^{5} 6 x d x + \int_{1}^{5} - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 12

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - (\frac{x^{3}}{3}]_{1}^{5}) + 6 \int_{1}^{5} x d x + \int_{1}^{5} - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 13

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{1}{2} x^{2}]_{1}^{5}) + \int_{1}^{5} - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 14

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

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + \int_{1}^{5} - 5 d x + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 15

Apply the constant rule.

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + - 5 x]_{1}^{5} + \int_{5}^{7} x^{2} - 6 x + 5 d x$

Step 16

Split the single integral into multiple integrals.

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + - 5 x]_{1}^{5} + \int_{5}^{7} x^{2} d x + \int_{5}^{7} - 6 x d x + \int_{5}^{7} 5 d x$

Step 17

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) + 5 x]_{0}^{1} - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + - 5 x]_{1}^{5} + \frac{1}{3} x^{3}]_{5}^{7} + \int_{5}^{7} - 6 x d x + \int_{5}^{7} 5 d x$

Step 18

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

Step 19

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

Step 20

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

Step 21

Apply the constant rule.

Step 22

Simplify the answer.

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Step 22.1

Simplify.

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Step 22.1.1

Combine $\frac{1}{3} x^{3}]_{0}^{1}$ and $5 x]_{0}^{1}$ .

$\frac{1}{3} x^{3} + 5 x]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + - 5 x]_{1}^{5} + \frac{1}{3} x^{3}]_{5}^{7} - 6 (\frac{x^{2}}{2}]_{5}^{7}) + 5 x]_{5}^{7}$

Step 22.1.2

Combine $\frac{1}{3} x^{3}]_{5}^{7}$ and $5 x]_{5}^{7}$ .

$\frac{1}{3} x^{3} + 5 x]_{0}^{1} - 6 (\frac{x^{2}}{2}]_{0}^{1}) - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + - 5 x]_{1}^{5} + \frac{1}{3} x^{3} + 5 x]_{5}^{7} - 6 (\frac{x^{2}}{2}]_{5}^{7})$

Step 22.2

Substitute and simplify.

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Step 22.2.1

Evaluate $\frac{1}{3} x^{3} + 5 x$ at $1$ and at $0$ .

$(\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 (\frac{x^{2}}{2}]_{0}^{1}) - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + - 5 x]_{1}^{5} + \frac{1}{3} x^{3} + 5 x]_{5}^{7} - 6 (\frac{x^{2}}{2}]_{5}^{7})$

Step 22.2.2

Evaluate $\frac{x^{2}}{2}$ at $1$ and at $0$ .

$(\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - (\frac{x^{3}}{3}]_{1}^{5}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + - 5 x]_{1}^{5} + \frac{1}{3} x^{3} + 5 x]_{5}^{7} - 6 (\frac{x^{2}}{2}]_{5}^{7})$

Step 22.2.3

Evaluate $\frac{x^{3}}{3}$ at $5$ and at $1$ .

$(\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 (\frac{x^{2}}{2}]_{1}^{5}) + - 5 x]_{1}^{5} + \frac{1}{3} x^{3} + 5 x]_{5}^{7} - 6 (\frac{x^{2}}{2}]_{5}^{7})$

Step 22.2.4

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $1$ .

$(\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + - 5 x]_{1}^{5} + \frac{1}{3} x^{3} + 5 x]_{5}^{7} - 6 (\frac{x^{2}}{2}]_{5}^{7})$

Step 22.2.5

Evaluate $- 5 x$ at $5$ and at $1$ .

$(\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + \frac{1}{3} x^{3} + 5 x]_{5}^{7} - 6 (\frac{x^{2}}{2}]_{5}^{7})$

Step 22.2.6

Evaluate $\frac{1}{3} x^{3} + 5 x$ at $7$ and at $5$ .

$(\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 (\frac{x^{2}}{2}]_{5}^{7})$

Step 22.2.7

Evaluate $\frac{x^{2}}{2}$ at $7$ and at $5$ .

$(\frac{1}{3} \cdot 1^{3} + 5 \cdot 1) - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8

Simplify.

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Step 22.2.8.1

One to any power is one.

$\frac{1}{3} \cdot 1 + 5 \cdot 1 - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.2

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} + 5 \cdot 1 - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.3

Multiply $5$ by $1$ .

$\frac{1}{3} + 5 - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.4

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

$\frac{1}{3} + 5 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.5

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

$\frac{1}{3} + \frac{5 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.6

Combine the numerators over the common denominator.

$\frac{1 + 5 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.7

Simplify the numerator.

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Step 22.2.8.7.1

Multiply $5$ by $3$ .

$\frac{1 + 15}{3} - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.7.2

Add $1$ and $15$ .

$\frac{16}{3} - (\frac{1}{3} \cdot 0^{3} + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.8

Raising $0$ to any positive power yields $0$ .

$\frac{16}{3} - (\frac{1}{3} \cdot 0 + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.9

Multiply $\frac{1}{3}$ by $0$ .

$\frac{16}{3} - (0 + 5 \cdot 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.10

Multiply $5$ by $0$ .

$\frac{16}{3} - (0 + 0) - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.11

Add $0$ and $0$ .

$\frac{16}{3} - 0 - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.12

Multiply $- 1$ by $0$ .

$\frac{16}{3} + 0 - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.13

Add $\frac{16}{3}$ and $0$ .

$\frac{16}{3} - 6 ((\frac{1^{2}}{2}) - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.14

One to any power is one.

$\frac{16}{3} - 6 (\frac{1}{2} - \frac{0^{2}}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.15

Raising $0$ to any positive power yields $0$ .

$\frac{16}{3} - 6 (\frac{1}{2} - \frac{0}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.16

Cancel the common factor of $0$ and $2$ .

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Step 22.2.8.16.1

Factor $2$ out of $0$ .

$\frac{16}{3} - 6 (\frac{1}{2} - \frac{2 (0)}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.16.2

Cancel the common factors.

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Step 22.2.8.16.2.1

Factor $2$ out of $2$ .

$\frac{16}{3} - 6 (\frac{1}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.16.2.2

Cancel the common factor.

Step 22.2.8.16.2.3

Rewrite the expression.

$\frac{16}{3} - 6 (\frac{1}{2} - \frac{0}{1}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.16.2.4

Divide $0$ by $1$ .

$\frac{16}{3} - 6 (\frac{1}{2} - 0) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.17

Multiply $- 1$ by $0$ .

$\frac{16}{3} - 6 (\frac{1}{2} + 0) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.18

Add $\frac{1}{2}$ and $0$ .

$\frac{16}{3} - 6 (\frac{1}{2}) - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.19

Combine $- 6$ and $\frac{1}{2}$ .

$\frac{16}{3} + \frac{- 6}{2} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.20

Cancel the common factor of $- 6$ and $2$ .

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Step 22.2.8.20.1

Factor $2$ out of $- 6$ .

$\frac{16}{3} + \frac{2 \cdot - 3}{2} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.20.2

Cancel the common factors.

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Step 22.2.8.20.2.1

Factor $2$ out of $2$ .

$\frac{16}{3} + \frac{2 \cdot - 3}{2 (1)} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.20.2.2

Cancel the common factor.

$\frac{16}{3} + \frac{2 \cdot - 3}{2 \cdot 1} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.20.2.3

Rewrite the expression.

$\frac{16}{3} + \frac{- 3}{1} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.20.2.4

Divide $- 3$ by $1$ .

$\frac{16}{3} - 3 - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.21

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

$\frac{16}{3} - 3 \cdot \frac{3}{3} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.22

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

$\frac{16}{3} + \frac{- 3 \cdot 3}{3} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.23

Combine the numerators over the common denominator.

$\frac{16 - 3 \cdot 3}{3} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.24

Simplify the numerator.

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Step 22.2.8.24.1

Multiply $- 3$ by $3$ .

$\frac{16 - 9}{3} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.24.2

Subtract $9$ from $16$ .

$\frac{7}{3} - ((\frac{5^{3}}{3}) - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.25

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

$\frac{7}{3} - (\frac{125}{3} - \frac{1^{3}}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.26

One to any power is one.

$\frac{7}{3} - (\frac{125}{3} - \frac{1}{3}) + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.27

Combine the numerators over the common denominator.

$\frac{7}{3} - \frac{125 - 1}{3} + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.28

Subtract $1$ from $125$ .

$\frac{7}{3} - \frac{124}{3} + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.29

Combine the numerators over the common denominator.

$\frac{7 - 124}{3} + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.30

Subtract $124$ from $7$ .

$\frac{- 117}{3} + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.31

Cancel the common factor of $- 117$ and $3$ .

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Step 22.2.8.31.1

Factor $3$ out of $- 117$ .

$\frac{3 \cdot - 39}{3} + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.31.2

Cancel the common factors.

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Step 22.2.8.31.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot - 39}{3 (1)} + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.31.2.2

Cancel the common factor.

$\frac{3 \cdot - 39}{3 \cdot 1} + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.31.2.3

Rewrite the expression.

$\frac{- 39}{1} + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.31.2.4

Divide $- 39$ by $1$ .

$- 39 + 6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.32

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

$- 39 + 6 (\frac{25}{2} - \frac{1^{2}}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.33

One to any power is one.

$- 39 + 6 (\frac{25}{2} - \frac{1}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.34

Combine the numerators over the common denominator.

$- 39 + 6 \frac{25 - 1}{2} + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.35

Subtract $1$ from $25$ .

$- 39 + 6 (\frac{24}{2}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.36

Cancel the common factor of $24$ and $2$ .

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Step 22.2.8.36.1

Factor $2$ out of $24$ .

$- 39 + 6 \frac{2 \cdot 12}{2} + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.36.2

Cancel the common factors.

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Step 22.2.8.36.2.1

Factor $2$ out of $2$ .

$- 39 + 6 \frac{2 \cdot 12}{2 (1)} + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.36.2.2

Cancel the common factor.

$- 39 + 6 \frac{2 \cdot 12}{2 \cdot 1} + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.36.2.3

Rewrite the expression.

$- 39 + 6 (\frac{12}{1}) + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.36.2.4

Divide $12$ by $1$ .

$- 39 + 6 \cdot 12 + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.37

Multiply $6$ by $12$ .

$- 39 + 72 + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.38

Add $- 39$ and $72$ .

$33 + (- 5 \cdot 5) + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.39

Multiply $- 5$ by $5$ .

$33 - 25 + 5 \cdot 1 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.40

Multiply $5$ by $1$ .

$33 - 25 + 5 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.41

Add $- 25$ and $5$ .

$33 - 20 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.42

Subtract $20$ from $33$ .

$13 + (\frac{1}{3} \cdot 7^{3} + 5 \cdot 7) - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.43

Raise $7$ to the power of $3$ .

$13 + \frac{1}{3} \cdot 343 + 5 \cdot 7 - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.44

Combine $\frac{1}{3}$ and $343$ .

$13 + \frac{343}{3} + 5 \cdot 7 - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.45

Multiply $5$ by $7$ .

$13 + \frac{343}{3} + 35 - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.46

To write $35$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$13 + \frac{343}{3} + 35 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.47

Combine $35$ and $\frac{3}{3}$ .

$13 + \frac{343}{3} + \frac{35 \cdot 3}{3} - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.48

Combine the numerators over the common denominator.

$13 + \frac{343 + 35 \cdot 3}{3} - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.49

Simplify the numerator.

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Step 22.2.8.49.1

Multiply $35$ by $3$ .

$13 + \frac{343 + 105}{3} - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.49.2

Add $343$ and $105$ .

$13 + \frac{448}{3} - (\frac{1}{3} \cdot 5^{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.50

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

$13 + \frac{448}{3} - (\frac{1}{3} \cdot 125 + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.51

Combine $\frac{1}{3}$ and $125$ .

$13 + \frac{448}{3} - (\frac{125}{3} + 5 \cdot 5) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.52

Multiply $5$ by $5$ .

$13 + \frac{448}{3} - (\frac{125}{3} + 25) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.53

To write $25$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$13 + \frac{448}{3} - (\frac{125}{3} + 25 \cdot \frac{3}{3}) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.54

Combine $25$ and $\frac{3}{3}$ .

$13 + \frac{448}{3} - (\frac{125}{3} + \frac{25 \cdot 3}{3}) - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.55

Combine the numerators over the common denominator.

$13 + \frac{448}{3} - \frac{125 + 25 \cdot 3}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.56

Simplify the numerator.

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Step 22.2.8.56.1

Multiply $25$ by $3$ .

$13 + \frac{448}{3} - \frac{125 + 75}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.56.2

Add $125$ and $75$ .

$13 + \frac{448}{3} - \frac{200}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.57

Combine the numerators over the common denominator.

$13 + \frac{448 - 200}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.58

Subtract $200$ from $448$ .

$13 + \frac{248}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.59

To write $13$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$13 \cdot \frac{3}{3} + \frac{248}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.60

Combine $13$ and $\frac{3}{3}$ .

$\frac{13 \cdot 3}{3} + \frac{248}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.61

Combine the numerators over the common denominator.

$\frac{13 \cdot 3 + 248}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.62

Simplify the numerator.

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Step 22.2.8.62.1

Multiply $13$ by $3$ .

$\frac{39 + 248}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.62.2

Add $39$ and $248$ .

$\frac{287}{3} - 6 ((\frac{7^{2}}{2}) - \frac{5^{2}}{2})$

Step 22.2.8.63

Raise $7$ to the power of $2$ .

$\frac{287}{3} - 6 (\frac{49}{2} - \frac{5^{2}}{2})$

Step 22.2.8.64

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

$\frac{287}{3} - 6 (\frac{49}{2} - \frac{25}{2})$

Step 22.2.8.65

Combine the numerators over the common denominator.

$\frac{287}{3} - 6 \frac{49 - 25}{2}$

Step 22.2.8.66

Subtract $25$ from $49$ .

$\frac{287}{3} - 6 (\frac{24}{2})$

Step 22.2.8.67

Cancel the common factor of $24$ and $2$ .

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Step 22.2.8.67.1

Factor $2$ out of $24$ .

$\frac{287}{3} - 6 \frac{2 \cdot 12}{2}$

Step 22.2.8.67.2

Cancel the common factors.

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Step 22.2.8.67.2.1

Factor $2$ out of $2$ .

$\frac{287}{3} - 6 \frac{2 \cdot 12}{2 (1)}$

Step 22.2.8.67.2.2

Cancel the common factor.

$\frac{287}{3} - 6 \frac{2 \cdot 12}{2 \cdot 1}$

Step 22.2.8.67.2.3

Rewrite the expression.

$\frac{287}{3} - 6 (\frac{12}{1})$

Step 22.2.8.67.2.4

Divide $12$ by $1$ .

$\frac{287}{3} - 6 \cdot 12$

Step 22.2.8.68

Multiply $- 6$ by $12$ .

$\frac{287}{3} - 72$

Step 22.2.8.69

To write $- 72$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{287}{3} - 72 \cdot \frac{3}{3}$

Step 22.2.8.70

Combine $- 72$ and $\frac{3}{3}$ .

$\frac{287}{3} + \frac{- 72 \cdot 3}{3}$

Step 22.2.8.71

Combine the numerators over the common denominator.

$\frac{287 - 72 \cdot 3}{3}$

Step 22.2.8.72

Simplify the numerator.

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Step 22.2.8.72.1

Multiply $- 72$ by $3$ .

$\frac{287 - 216}{3}$

Step 22.2.8.72.2

Subtract $216$ from $287$ .

$\frac{71}{3}$

Step 23

The result can be shown in multiple forms.

Exact Form:

$\frac{71}{3}$

Decimal Form:

$23.\overline{6}$

Mixed Number Form:

$23 \frac{2}{3}$

Step 24