Calculus Examples

$f (x) = x^{2} + 4 x + 4$ , $[- 4, 2]$ , $n = 7$

Step 1

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- 4}^{2} 7 d x - \int_{- 4}^{2} x^{2} + 4 x + 4 d x$

Step 2

Integrate to find the area between $- 4$ and $2$ .

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Step 2.1

Integrate to find the area between $- 4$ and $2$ .

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Step 2.1.1

Combine the integrals into a single integral.

$\int_{- 4}^{2} x^{2} + 4 x + 4 - (7) d x$

Step 2.1.2

Multiply $- 1$ by $7$ .

$\int_{- 4}^{2} x^{2} + 4 x + 4 - 7 d x$

Step 2.1.3

Subtract $7$ from $4$ .

$\int_{- 4}^{2} x^{2} + 4 x - 3 d x$

Step 2.1.4

Split the single integral into multiple integrals.

$\int_{- 4}^{2} x^{2} d x + \int_{- 4}^{2} 4 x d x + \int_{- 4}^{2} - 3 d x$

Step 2.1.5

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

$\frac{1}{3} x^{3}]_{- 4}^{2} + \int_{- 4}^{2} 4 x d x + \int_{- 4}^{2} - 3 d x$

Step 2.1.6

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$\frac{1}{3} x^{3}]_{- 4}^{2} + 4 \int_{- 4}^{2} x d x + \int_{- 4}^{2} - 3 d x$

Step 2.1.7

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

$\frac{1}{3} x^{3}]_{- 4}^{2} + 4 (\frac{1}{2} x^{2}]_{- 4}^{2}) + \int_{- 4}^{2} - 3 d x$

Step 2.1.8

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

$\frac{1}{3} x^{3}]_{- 4}^{2} + 4 (\frac{x^{2}}{2}]_{- 4}^{2}) + \int_{- 4}^{2} - 3 d x$

Step 2.1.9

Apply the constant rule.

$\frac{1}{3} x^{3}]_{- 4}^{2} + 4 (\frac{x^{2}}{2}]_{- 4}^{2}) + - 3 x]_{- 4}^{2}$

Step 2.1.10

Simplify the answer.

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Step 2.1.10.1

Combine $\frac{1}{3} x^{3}]_{- 4}^{2}$ and $- 3 x]_{- 4}^{2}$ .

$4 (\frac{x^{2}}{2}]_{- 4}^{2}) + \frac{1}{3} x^{3} - 3 x]_{- 4}^{2}$

Step 2.1.10.2

Substitute and simplify.

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Step 2.1.10.2.1

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $- 4$ .

$4 ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2}) + \frac{1}{3} x^{3} - 3 x]_{- 4}^{2}$

Step 2.1.10.2.2

Evaluate $\frac{1}{3} x^{3} - 3 x$ at $2$ and at $- 4$ .

$4 ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3

Simplify.

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Step 2.1.10.2.3.1

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

$4 (\frac{4}{2} - \frac{{(- 4)}^{2}}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.2

Cancel the common factor of $4$ and $2$ .

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Step 2.1.10.2.3.2.1

Factor $2$ out of $4$ .

$4 (\frac{2 \cdot 2}{2} - \frac{{(- 4)}^{2}}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.2.2

Cancel the common factors.

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Step 2.1.10.2.3.2.2.1

Factor $2$ out of $2$ .

$4 (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 4)}^{2}}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.2.2.2

Cancel the common factor.

$4 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.2.2.3

Rewrite the expression.

$4 (\frac{2}{1} - \frac{{(- 4)}^{2}}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.2.2.4

Divide $2$ by $1$ .

$4 (2 - \frac{{(- 4)}^{2}}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.3

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

$4 (2 - \frac{16}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.4

Cancel the common factor of $16$ and $2$ .

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Step 2.1.10.2.3.4.1

Factor $2$ out of $16$ .

$4 (2 - \frac{2 \cdot 8}{2}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.4.2

Cancel the common factors.

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Step 2.1.10.2.3.4.2.1

Factor $2$ out of $2$ .

$4 (2 - \frac{2 \cdot 8}{2 (1)}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.4.2.2

Cancel the common factor.

$4 (2 - \frac{2 \cdot 8}{2 \cdot 1}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.4.2.3

Rewrite the expression.

$4 (2 - \frac{8}{1}) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.4.2.4

Divide $8$ by $1$ .

$4 (2 - 1 \cdot 8) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.5

Multiply $- 1$ by $8$ .

$4 (2 - 8) + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.6

Subtract $8$ from $2$ .

$4 \cdot - 6 + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.7

Multiply $4$ by $- 6$ .

$- 24 + (\frac{1}{3} \cdot 2^{3} - 3 \cdot 2) - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.8

Raise $2$ to the power of $3$ .

$- 24 + \frac{1}{3} \cdot 8 - 3 \cdot 2 - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.9

Combine $\frac{1}{3}$ and $8$ .

$- 24 + \frac{8}{3} - 3 \cdot 2 - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.10

Multiply $- 3$ by $2$ .

$- 24 + \frac{8}{3} - 6 - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.11

To write $- 6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 24 + \frac{8}{3} - 6 \cdot \frac{3}{3} - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.12

Combine $- 6$ and $\frac{3}{3}$ .

$- 24 + \frac{8}{3} + \frac{- 6 \cdot 3}{3} - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.13

Combine the numerators over the common denominator.

$- 24 + \frac{8 - 6 \cdot 3}{3} - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.14

Simplify the numerator.

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Step 2.1.10.2.3.14.1

Multiply $- 6$ by $3$ .

$- 24 + \frac{8 - 18}{3} - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.14.2

Subtract $18$ from $8$ .

$- 24 + \frac{- 10}{3} - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.15

Move the negative in front of the fraction.

$- 24 - \frac{10}{3} - (\frac{1}{3} {(- 4)}^{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.16

Raise $- 4$ to the power of $3$ .

$- 24 - \frac{10}{3} - (\frac{1}{3} \cdot - 64 - 3 \cdot - 4)$

Step 2.1.10.2.3.17

Combine $\frac{1}{3}$ and $- 64$ .

$- 24 - \frac{10}{3} - (\frac{- 64}{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.18

Move the negative in front of the fraction.

$- 24 - \frac{10}{3} - (- \frac{64}{3} - 3 \cdot - 4)$

Step 2.1.10.2.3.19

Multiply $- 3$ by $- 4$ .

$- 24 - \frac{10}{3} - (- \frac{64}{3} + 12)$

Step 2.1.10.2.3.20

To write $12$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 24 - \frac{10}{3} - (- \frac{64}{3} + 12 \cdot \frac{3}{3})$

Step 2.1.10.2.3.21

Combine $12$ and $\frac{3}{3}$ .

$- 24 - \frac{10}{3} - (- \frac{64}{3} + \frac{12 \cdot 3}{3})$

Step 2.1.10.2.3.22

Combine the numerators over the common denominator.

$- 24 - \frac{10}{3} - \frac{- 64 + 12 \cdot 3}{3}$

Step 2.1.10.2.3.23

Simplify the numerator.

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Step 2.1.10.2.3.23.1

Multiply $12$ by $3$ .

$- 24 - \frac{10}{3} - \frac{- 64 + 36}{3}$

Step 2.1.10.2.3.23.2

Add $- 64$ and $36$ .

$- 24 - \frac{10}{3} - \frac{- 28}{3}$

Step 2.1.10.2.3.24

Move the negative in front of the fraction.

$- 24 - \frac{10}{3} - - \frac{28}{3}$

Step 2.1.10.2.3.25

Multiply $- 1$ by $- 1$ .

$- 24 - \frac{10}{3} + 1 (\frac{28}{3})$

Step 2.1.10.2.3.26

Multiply $\frac{28}{3}$ by $1$ .

$- 24 - \frac{10}{3} + \frac{28}{3}$

Step 2.1.10.2.3.27

Combine the numerators over the common denominator.

$- 24 + \frac{- 10 + 28}{3}$

Step 2.1.10.2.3.28

Add $- 10$ and $28$ .

$- 24 + \frac{18}{3}$

Step 2.1.10.2.3.29

Cancel the common factor of $18$ and $3$ .

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Step 2.1.10.2.3.29.1

Factor $3$ out of $18$ .

$- 24 + \frac{3 \cdot 6}{3}$

Step 2.1.10.2.3.29.2

Cancel the common factors.

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Step 2.1.10.2.3.29.2.1

Factor $3$ out of $3$ .

$- 24 + \frac{3 \cdot 6}{3 (1)}$

Step 2.1.10.2.3.29.2.2

Cancel the common factor.

$- 24 + \frac{3 \cdot 6}{3 \cdot 1}$

Step 2.1.10.2.3.29.2.3

Rewrite the expression.

$- 24 + \frac{6}{1}$

Step 2.1.10.2.3.29.2.4

Divide $6$ by $1$ .

$- 24 + 6$

Step 2.1.10.2.3.30

Add $- 24$ and $6$ .

$- 18$

Step 2.2

Combine the integrals into a single integral.

$\int_{- 4}^{2} 7 - (x^{2} + 4 x + 4) d x$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$7 - x^{2} - (4 x) - 1 \cdot 4$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $4$ by $- 1$ .

$7 - x^{2} - 4 x - 1 \cdot 4$

Step 2.3.2.2

Multiply $- 1$ by $4$ .

$7 - x^{2} - 4 x - 4$

$\int_{- 4}^{2} 7 - x^{2} - 4 x - 4 d x$

Step 2.4

Subtract $4$ from $7$ .

$\int_{- 4}^{2} - x^{2} - 4 x + 3 d x$

Step 2.5

Split the single integral into multiple integrals.

$\int_{- 4}^{2} - x^{2} d x + \int_{- 4}^{2} - 4 x d x + \int_{- 4}^{2} 3 d x$

Step 2.6

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

$- \int_{- 4}^{2} x^{2} d x + \int_{- 4}^{2} - 4 x d x + \int_{- 4}^{2} 3 d x$

Step 2.7

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

$- (\frac{1}{3} x^{3}]_{- 4}^{2}) + \int_{- 4}^{2} - 4 x d x + \int_{- 4}^{2} 3 d x$

Step 2.8

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

$- (\frac{x^{3}}{3}]_{- 4}^{2}) + \int_{- 4}^{2} - 4 x d x + \int_{- 4}^{2} 3 d x$

Step 2.9

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- (\frac{x^{3}}{3}]_{- 4}^{2}) - 4 \int_{- 4}^{2} x d x + \int_{- 4}^{2} 3 d x$

Step 2.10

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

$- (\frac{x^{3}}{3}]_{- 4}^{2}) - 4 (\frac{1}{2} x^{2}]_{- 4}^{2}) + \int_{- 4}^{2} 3 d x$

Step 2.11

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

$- (\frac{x^{3}}{3}]_{- 4}^{2}) - 4 (\frac{x^{2}}{2}]_{- 4}^{2}) + \int_{- 4}^{2} 3 d x$

Step 2.12

Apply the constant rule.

$- (\frac{x^{3}}{3}]_{- 4}^{2}) - 4 (\frac{x^{2}}{2}]_{- 4}^{2}) + 3 x]_{- 4}^{2}$

Step 2.13

Substitute and simplify.

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Step 2.13.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $- 4$ .

$- ((\frac{2^{3}}{3}) - \frac{{(- 4)}^{3}}{3}) - 4 (\frac{x^{2}}{2}]_{- 4}^{2}) + 3 x]_{- 4}^{2}$

Step 2.13.2

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $- 4$ .

$- (\frac{2^{3}}{3} - \frac{{(- 4)}^{3}}{3}) - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 3 x]_{- 4}^{2}$

Step 2.13.3

Evaluate $3 x$ at $2$ and at $- 4$ .

$- (\frac{2^{3}}{3} - \frac{{(- 4)}^{3}}{3}) - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4

Simplify.

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Step 2.13.4.1

Raise $2$ to the power of $3$ .

$- (\frac{8}{3} - \frac{{(- 4)}^{3}}{3}) - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.2

Raise $- 4$ to the power of $3$ .

$- (\frac{8}{3} - \frac{- 64}{3}) - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.3

Move the negative in front of the fraction.

$- (\frac{8}{3} - - \frac{64}{3}) - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.4

Multiply $- 1$ by $- 1$ .

$- (\frac{8}{3} + 1 (\frac{64}{3})) - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.5

Multiply $\frac{64}{3}$ by $1$ .

$- (\frac{8}{3} + \frac{64}{3}) - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.6

Combine the numerators over the common denominator.

$- \frac{8 + 64}{3} - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.7

Add $8$ and $64$ .

$- \frac{72}{3} - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.8

Cancel the common factor of $72$ and $3$ .

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Step 2.13.4.8.1

Factor $3$ out of $72$ .

$- \frac{3 \cdot 24}{3} - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.8.2

Cancel the common factors.

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Step 2.13.4.8.2.1

Factor $3$ out of $3$ .

$- \frac{3 \cdot 24}{3 (1)} - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.8.2.2

Cancel the common factor.

$- \frac{3 \cdot 24}{3 \cdot 1} - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.8.2.3

Rewrite the expression.

$- \frac{24}{1} - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.8.2.4

Divide $24$ by $1$ .

$- 1 \cdot 24 - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.9

Multiply $- 1$ by $24$ .

$- 24 - 4 (\frac{2^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.10

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

$- 24 - 4 (\frac{4}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.11

Cancel the common factor of $4$ and $2$ .

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Step 2.13.4.11.1

Factor $2$ out of $4$ .

$- 24 - 4 (\frac{2 \cdot 2}{2} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.11.2

Cancel the common factors.

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Step 2.13.4.11.2.1

Factor $2$ out of $2$ .

$- 24 - 4 (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.11.2.2

Cancel the common factor.

$- 24 - 4 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.11.2.3

Rewrite the expression.

$- 24 - 4 (\frac{2}{1} - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.11.2.4

Divide $2$ by $1$ .

$- 24 - 4 (2 - \frac{{(- 4)}^{2}}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.12

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

$- 24 - 4 (2 - \frac{16}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.13

Cancel the common factor of $16$ and $2$ .

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Step 2.13.4.13.1

Factor $2$ out of $16$ .

$- 24 - 4 (2 - \frac{2 \cdot 8}{2}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.13.2

Cancel the common factors.

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Step 2.13.4.13.2.1

Factor $2$ out of $2$ .

$- 24 - 4 (2 - \frac{2 \cdot 8}{2 (1)}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.13.2.2

Cancel the common factor.

$- 24 - 4 (2 - \frac{2 \cdot 8}{2 \cdot 1}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.13.2.3

Rewrite the expression.

$- 24 - 4 (2 - \frac{8}{1}) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.13.2.4

Divide $8$ by $1$ .

$- 24 - 4 (2 - 1 \cdot 8) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.14

Multiply $- 1$ by $8$ .

$- 24 - 4 (2 - 8) + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.15

Subtract $8$ from $2$ .

$- 24 - 4 \cdot - 6 + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.16

Multiply $- 4$ by $- 6$ .

$- 24 + 24 + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.17

Add $- 24$ and $24$ .

$0 + (3 \cdot 2) - 3 \cdot - 4$

Step 2.13.4.18

Multiply $3$ by $2$ .

$0 + 6 - 3 \cdot - 4$

Step 2.13.4.19

Multiply $- 3$ by $- 4$ .

$0 + 6 + 12$

Step 2.13.4.20

Add $6$ and $12$ .

$0 + 18$

Step 2.13.4.21

Add $0$ and $18$ .

$18$

Step 3