Calculus Examples

$\int_{- 2}^{1} (- 3 x^{\frac{8}{3}} - 5 x^{2} + 2 x - 3) d x$

Step 1

Remove parentheses.

$\int_{- 2}^{1} - 3 x^{\frac{8}{3}} - 5 x^{2} + 2 x - 3 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{- 2}^{1} - 3 x^{\frac{8}{3}} d x + \int_{- 2}^{1} - 5 x^{2} d x + \int_{- 2}^{1} 2 x d x + \int_{- 2}^{1} - 3 d x$

Step 3

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$- 3 \int_{- 2}^{1} x^{\frac{8}{3}} d x + \int_{- 2}^{1} - 5 x^{2} d x + \int_{- 2}^{1} 2 x d x + \int_{- 2}^{1} - 3 d x$

Step 4

By the Power Rule, the integral of $x^{\frac{8}{3}}$ with respect to $x$ is $\frac{3}{11} x^{\frac{11}{3}}$ .

$- 3 (\frac{3}{11} x^{\frac{11}{3}}]_{- 2}^{1}) + \int_{- 2}^{1} - 5 x^{2} d x + \int_{- 2}^{1} 2 x d x + \int_{- 2}^{1} - 3 d x$

Step 5

Combine $\frac{3}{11}$ and $x^{\frac{11}{3}}$ .

$- 3 (\frac{3 x^{\frac{11}{3}}}{11}]_{- 2}^{1}) + \int_{- 2}^{1} - 5 x^{2} d x + \int_{- 2}^{1} 2 x d x + \int_{- 2}^{1} - 3 d x$

Step 6

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$- 3 (\frac{3 x^{\frac{11}{3}}}{11}]_{- 2}^{1}) - 5 \int_{- 2}^{1} x^{2} d x + \int_{- 2}^{1} 2 x d x + \int_{- 2}^{1} - 3 d x$

Step 7

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

$- 3 (\frac{3 x^{\frac{11}{3}}}{11}]_{- 2}^{1}) - 5 (\frac{1}{3} x^{3}]_{- 2}^{1}) + \int_{- 2}^{1} 2 x d x + \int_{- 2}^{1} - 3 d x$

Step 8

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

$- 3 (\frac{3 x^{\frac{11}{3}}}{11}]_{- 2}^{1}) - 5 (\frac{x^{3}}{3}]_{- 2}^{1}) + \int_{- 2}^{1} 2 x d x + \int_{- 2}^{1} - 3 d x$

Step 9

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$- 3 (\frac{3 x^{\frac{11}{3}}}{11}]_{- 2}^{1}) - 5 (\frac{x^{3}}{3}]_{- 2}^{1}) + 2 \int_{- 2}^{1} x d x + \int_{- 2}^{1} - 3 d x$

Step 10

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

$- 3 (\frac{3 x^{\frac{11}{3}}}{11}]_{- 2}^{1}) - 5 (\frac{x^{3}}{3}]_{- 2}^{1}) + 2 (\frac{1}{2} x^{2}]_{- 2}^{1}) + \int_{- 2}^{1} - 3 d x$

Step 11

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

$- 3 (\frac{3 x^{\frac{11}{3}}}{11}]_{- 2}^{1}) - 5 (\frac{x^{3}}{3}]_{- 2}^{1}) + 2 (\frac{x^{2}}{2}]_{- 2}^{1}) + \int_{- 2}^{1} - 3 d x$

Step 12

Apply the constant rule.

$- 3 (\frac{3 x^{\frac{11}{3}}}{11}]_{- 2}^{1}) - 5 (\frac{x^{3}}{3}]_{- 2}^{1}) + 2 (\frac{x^{2}}{2}]_{- 2}^{1}) + - 3 x]_{- 2}^{1}$

Step 13

Simplify the answer.

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Step 13.1

Substitute and simplify.

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Step 13.1.1

Evaluate $\frac{3 x^{\frac{11}{3}}}{11}$ at $1$ and at $- 2$ .

$- 3 ((\frac{3 \cdot 1^{\frac{11}{3}}}{11}) - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{x^{3}}{3}]_{- 2}^{1}) + 2 (\frac{x^{2}}{2}]_{- 2}^{1}) + - 3 x]_{- 2}^{1}$

Step 13.1.2

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

$- 3 (\frac{3 \cdot 1^{\frac{11}{3}}}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + 2 (\frac{x^{2}}{2}]_{- 2}^{1}) + - 3 x]_{- 2}^{1}$

Step 13.1.3

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

$- 3 (\frac{3 \cdot 1^{\frac{11}{3}}}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + 2 ((\frac{1^{2}}{2}) - \frac{{(- 2)}^{2}}{2}) + - 3 x]_{- 2}^{1}$

Step 13.1.4

Evaluate $- 3 x$ at $1$ and at $- 2$ .

$- 3 (\frac{3 \cdot 1^{\frac{11}{3}}}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) + - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5

Simplify.

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Step 13.1.5.1

One to any power is one.

$- 3 (\frac{3 \cdot 1}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.2

Multiply $3$ by $1$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1^{3}}{3} - \frac{{(- 2)}^{3}}{3}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.3

One to any power is one.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1}{3} - \frac{{(- 2)}^{3}}{3}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.4

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

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1}{3} - \frac{- 8}{3}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.5

Move the negative in front of the fraction.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1}{3} - - \frac{8}{3}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.6

Multiply $- 1$ by $- 1$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1}{3} + 1 (\frac{8}{3})) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.7

Multiply $\frac{8}{3}$ by $1$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{1}{3} + \frac{8}{3}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.8

Combine the numerators over the common denominator.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 \frac{1 + 8}{3} + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.9

Add $1$ and $8$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{9}{3}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.10

Cancel the common factor of $9$ and $3$ .

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Step 13.1.5.10.1

Factor $3$ out of $9$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 \frac{3 \cdot 3}{3} + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.10.2

Cancel the common factors.

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Step 13.1.5.10.2.1

Factor $3$ out of $3$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 \frac{3 \cdot 3}{3 (1)} + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.10.2.2

Cancel the common factor.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 \frac{3 \cdot 3}{3 \cdot 1} + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.10.2.3

Rewrite the expression.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 (\frac{3}{1}) + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.10.2.4

Divide $3$ by $1$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 5 \cdot 3 + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.11

Multiply $- 5$ by $3$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1^{2}}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.12

One to any power is one.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - \frac{{(- 2)}^{2}}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.13

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

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - \frac{4}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.14

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

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Step 13.1.5.14.1

Factor $2$ out of $4$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - \frac{2 \cdot 2}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.14.2

Cancel the common factors.

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Step 13.1.5.14.2.1

Factor $2$ out of $2$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - \frac{2 \cdot 2}{2 (1)}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.14.2.2

Cancel the common factor.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - \frac{2 \cdot 2}{2 \cdot 1}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.14.2.3

Rewrite the expression.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - \frac{2}{1}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.14.2.4

Divide $2$ by $1$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - 1 \cdot 2) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.15

Multiply $- 1$ by $2$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - 2) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.16

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

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} - 2 \cdot \frac{2}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.17

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

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{1}{2} + \frac{- 2 \cdot 2}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.18

Combine the numerators over the common denominator.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 \frac{1 - 2 \cdot 2}{2} - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.19

Simplify the numerator.

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Step 13.1.5.19.1

Multiply $- 2$ by $2$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 \frac{1 - 4}{2} - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.19.2

Subtract $4$ from $1$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (\frac{- 3}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.20

Move the negative in front of the fraction.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + 2 (- \frac{3}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.21

Multiply $- 1$ by $2$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 - 2 (\frac{3}{2}) - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.22

Combine $- 2$ and $\frac{3}{2}$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + \frac{- 2 \cdot 3}{2} - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.23

Multiply $- 2$ by $3$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + \frac{- 6}{2} - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.24

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

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Step 13.1.5.24.1

Factor $2$ out of $- 6$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + \frac{2 \cdot - 3}{2} - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.24.2

Cancel the common factors.

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Step 13.1.5.24.2.1

Factor $2$ out of $2$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + \frac{2 \cdot - 3}{2 (1)} - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.24.2.2

Cancel the common factor.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + \frac{2 \cdot - 3}{2 \cdot 1} - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.24.2.3

Rewrite the expression.

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 + \frac{- 3}{1} - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.24.2.4

Divide $- 3$ by $1$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 15 - 3 - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.25

Subtract $3$ from $- 15$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 18 - 3 \cdot 1 + 3 \cdot - 2$

Step 13.1.5.26

Multiply $- 3$ by $1$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 18 - 3 + 3 \cdot - 2$

Step 13.1.5.27

Multiply $3$ by $- 2$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 18 - 3 - 6$

Step 13.1.5.28

Subtract $6$ from $- 3$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 18 - 9$

Step 13.1.5.29

Subtract $9$ from $- 18$ .

$- 3 (\frac{3}{11} - \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 27$

Step 13.2

Simplify.

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Step 13.2.1

Simplify each term.

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Step 13.2.1.1

Apply the distributive property.

$- 3 (\frac{3}{11}) - 3 (- \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 27$

Step 13.2.1.2

Multiply $- 3 (\frac{3}{11})$ .

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Step 13.2.1.2.1

Combine $- 3$ and $\frac{3}{11}$ .

$\frac{- 3 \cdot 3}{11} - 3 (- \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 27$

Step 13.2.1.2.2

Multiply $- 3$ by $3$ .

$\frac{- 9}{11} - 3 (- \frac{3 {(- 2)}^{\frac{11}{3}}}{11}) - 27$

Step 13.2.1.3

Multiply $- 3 (- \frac{3 {(- 2)}^{\frac{11}{3}}}{11})$ .

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Step 13.2.1.3.1

Multiply $- 1$ by $- 3$ .

$\frac{- 9}{11} + 3 \frac{3 {(- 2)}^{\frac{11}{3}}}{11} - 27$

Step 13.2.1.3.2

Combine $3$ and $\frac{3 {(- 2)}^{\frac{11}{3}}}{11}$ .

$\frac{- 9}{11} + \frac{3 (3 {(- 2)}^{\frac{11}{3}})}{11} - 27$

Step 13.2.1.3.3

Multiply $3$ by $3$ .

$\frac{- 9}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11} - 27$

Step 13.2.1.4

Move the negative in front of the fraction.

$- \frac{9}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11} - 27$

Step 13.2.2

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$- \frac{9}{11} - 27 \cdot \frac{11}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11}$

Step 13.2.3

Combine $- 27$ and $\frac{11}{11}$ .

$- \frac{9}{11} + \frac{- 27 \cdot 11}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11}$

Step 13.2.4

Combine the numerators over the common denominator.

$\frac{- 9 - 27 \cdot 11}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11}$

Step 13.2.5

Simplify the numerator.

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Step 13.2.5.1

Multiply $- 27$ by $11$ .

$\frac{- 9 - 297}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11}$

Step 13.2.5.2

Subtract $297$ from $- 9$ .

$\frac{- 306}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11}$

Step 13.2.6

Move the negative in front of the fraction.

$- \frac{306}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$- \frac{306}{11} + \frac{9 {(- 2)}^{\frac{11}{3}}}{11}$

Decimal Form:

$- 38.20844324 \dots$

Step 15