Algebra Examples

$f (x) = 6 x^{3} + 8 x^{2} - 8 x - 7$ ; $[1, 2]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $6 x^{3} + 8 x^{2} - 8 x - 7$ with respect to $x$ is $\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [- 7]$ .

$f' (x) = \frac{d}{d x} (6 x^{3}) + \frac{d}{d x} (8 x^{2}) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 7)$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

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Step 1.1.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 6 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (8 x^{2}) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 7)$

Step 1.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 6 (3 x^{2}) + \frac{d}{d x} (8 x^{2}) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 7)$

Step 1.1.1.2.3

Multiply $3$ by $6$ .

$f' (x) = 18 x^{2} + \frac{d}{d x} (8 x^{2}) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 7)$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [8 x^{2}]$ .

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Step 1.1.1.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 18 x^{2} + 8 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 7)$

Step 1.1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = 18 x^{2} + 8 (2 x) + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 7)$

Step 1.1.1.3.3

Multiply $2$ by $8$ .

$f' (x) = 18 x^{2} + 16 x + \frac{d}{d x} (- 8 x) + \frac{d}{d x} (- 7)$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 1.1.1.4.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x$ with respect to $x$ is $- 8 \frac{d}{d x} [x]$ .

$f' (x) = 18 x^{2} + 16 x - 8 \frac{d}{d x} x + \frac{d}{d x} (- 7)$

Step 1.1.1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = 18 x^{2} + 16 x - 8 \cdot 1 + \frac{d}{d x} (- 7)$

Step 1.1.1.4.3

Multiply $- 8$ by $1$ .

$f' (x) = 18 x^{2} + 16 x - 8 + \frac{d}{d x} (- 7)$

Step 1.1.1.5

Differentiate using the Constant Rule.

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Step 1.1.1.5.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$f' (x) = 18 x^{2} + 16 x - 8 + 0$

Step 1.1.1.5.2

Add $18 x^{2} + 16 x - 8$ and $0$ .

$f' (x) = 18 x^{2} + 16 x - 8$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $18 x^{2} + 16 x - 8$ .

$18 x^{2} + 16 x - 8$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $18 x^{2} + 16 x - 8 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$18 x^{2} + 16 x - 8 = 0$

Step 1.2.2

Factor $2$ out of $18 x^{2} + 16 x - 8$ .

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Step 1.2.2.1

Factor $2$ out of $18 x^{2}$ .

$2 (9 x^{2}) + 16 x - 8 = 0$

Step 1.2.2.2

Factor $2$ out of $16 x$ .

$2 (9 x^{2}) + 2 (8 x) - 8 = 0$

Step 1.2.2.3

Factor $2$ out of $- 8$ .

$2 (9 x^{2}) + 2 (8 x) + 2 (- 4) = 0$

Step 1.2.2.4

Factor $2$ out of $2 (9 x^{2}) + 2 (8 x)$ .

$2 (9 x^{2} + 8 x) + 2 (- 4) = 0$

Step 1.2.2.5

Factor $2$ out of $2 (9 x^{2} + 8 x) + 2 (- 4)$ .

$2 (9 x^{2} + 8 x - 4) = 0$

Step 1.2.3

Divide each term in $2 (9 x^{2} + 8 x - 4) = 0$ by $2$ and simplify.

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Step 1.2.3.1

Divide each term in $2 (9 x^{2} + 8 x - 4) = 0$ by $2$ .

$\frac{2 (9 x^{2} + 8 x - 4)}{2} = \frac{0}{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{2 (9 x^{2} + 8 x - 4)}{2} = \frac{0}{2}$

Step 1.2.3.2.1.2

Divide $9 x^{2} + 8 x - 4$ by $1$ .

$9 x^{2} + 8 x - 4 = \frac{0}{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $2$ .

$9 x^{2} + 8 x - 4 = 0$

Step 1.2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.5

Substitute the values $a = 9$ , $b = 8$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (9 \cdot - 4)}}{2 \cdot 9}$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 4}}{2 \cdot 9}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 9 \cdot - 4$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 36 \cdot - 4}}{2 \cdot 9}$

Step 1.2.6.1.2.2

Multiply $- 36$ by $- 4$ .

$x = \frac{- 8 \pm \sqrt{64 + 144}}{2 \cdot 9}$

Step 1.2.6.1.3

Add $64$ and $144$ .

$x = \frac{- 8 \pm \sqrt{208}}{2 \cdot 9}$

Step 1.2.6.1.4

Rewrite $208$ as $4^{2} \cdot 13$ .

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Step 1.2.6.1.4.1

Factor $16$ out of $208$ .

$x = \frac{- 8 \pm \sqrt{16 (13)}}{2 \cdot 9}$

Step 1.2.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 9}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 4 \sqrt{13}}{2 \cdot 9}$

Step 1.2.6.2

Multiply $2$ by $9$ .

$x = \frac{- 8 \pm 4 \sqrt{13}}{18}$

Step 1.2.6.3

Simplify $\frac{- 8 \pm 4 \sqrt{13}}{18}$ .

$x = \frac{- 4 \pm 2 \sqrt{13}}{9}$

Step 1.2.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 4}}{2 \cdot 9}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 9 \cdot - 4$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 36 \cdot - 4}}{2 \cdot 9}$

Step 1.2.7.1.2.2

Multiply $- 36$ by $- 4$ .

$x = \frac{- 8 \pm \sqrt{64 + 144}}{2 \cdot 9}$

Step 1.2.7.1.3

Add $64$ and $144$ .

$x = \frac{- 8 \pm \sqrt{208}}{2 \cdot 9}$

Step 1.2.7.1.4

Rewrite $208$ as $4^{2} \cdot 13$ .

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Step 1.2.7.1.4.1

Factor $16$ out of $208$ .

$x = \frac{- 8 \pm \sqrt{16 (13)}}{2 \cdot 9}$

Step 1.2.7.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 9}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 4 \sqrt{13}}{2 \cdot 9}$

Step 1.2.7.2

Multiply $2$ by $9$ .

$x = \frac{- 8 \pm 4 \sqrt{13}}{18}$

Step 1.2.7.3

Simplify $\frac{- 8 \pm 4 \sqrt{13}}{18}$ .

$x = \frac{- 4 \pm 2 \sqrt{13}}{9}$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = \frac{- 4 + 2 \sqrt{13}}{9}$

Step 1.2.7.5

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 1 \cdot 4 + 2 \sqrt{13}}{9}$

Step 1.2.7.6

Factor $- 1$ out of $2 \sqrt{13}$ .

$x = \frac{- 1 \cdot 4 - (- 2 \sqrt{13})}{9}$

Step 1.2.7.7

Factor $- 1$ out of $- 1 (4) - (- 2 \sqrt{13})$ .

$x = \frac{- 1 (4 - 2 \sqrt{13})}{9}$

Step 1.2.7.8

Move the negative in front of the fraction.

$x = - \frac{4 - 2 \sqrt{13}}{9}$

Step 1.2.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 9 \cdot - 4}}{2 \cdot 9}$

Step 1.2.8.1.2

Multiply $- 4 \cdot 9 \cdot - 4$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{- 8 \pm \sqrt{64 - 36 \cdot - 4}}{2 \cdot 9}$

Step 1.2.8.1.2.2

Multiply $- 36$ by $- 4$ .

$x = \frac{- 8 \pm \sqrt{64 + 144}}{2 \cdot 9}$

Step 1.2.8.1.3

Add $64$ and $144$ .

$x = \frac{- 8 \pm \sqrt{208}}{2 \cdot 9}$

Step 1.2.8.1.4

Rewrite $208$ as $4^{2} \cdot 13$ .

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Step 1.2.8.1.4.1

Factor $16$ out of $208$ .

$x = \frac{- 8 \pm \sqrt{16 (13)}}{2 \cdot 9}$

Step 1.2.8.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 8 \pm \sqrt{4^{2} \cdot 13}}{2 \cdot 9}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 4 \sqrt{13}}{2 \cdot 9}$

Step 1.2.8.2

Multiply $2$ by $9$ .

$x = \frac{- 8 \pm 4 \sqrt{13}}{18}$

Step 1.2.8.3

Simplify $\frac{- 8 \pm 4 \sqrt{13}}{18}$ .

$x = \frac{- 4 \pm 2 \sqrt{13}}{9}$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = \frac{- 4 - 2 \sqrt{13}}{9}$

Step 1.2.8.5

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 1 \cdot 4 - 2 \sqrt{13}}{9}$

Step 1.2.8.6

Factor $- 1$ out of $- 2 \sqrt{13}$ .

$x = \frac{- 1 \cdot 4 - (2 \sqrt{13})}{9}$

Step 1.2.8.7

Factor $- 1$ out of $- 1 (4) - (2 \sqrt{13})$ .

$x = \frac{- 1 (4 + 2 \sqrt{13})}{9}$

Step 1.2.8.8

Move the negative in front of the fraction.

$x = - \frac{4 + 2 \sqrt{13}}{9}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = - \frac{4 - 2 \sqrt{13}}{9}, - \frac{4 + 2 \sqrt{13}}{9}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Step 1.4

Evaluate $6 x^{3} + 8 x^{2} - 8 x - 7$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = - \frac{4 - 2 \sqrt{13}}{9}$ .

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Step 1.4.1.1

Substitute $- \frac{4 - 2 \sqrt{13}}{9}$ for $x$ .

$6 {(- \frac{4 - 2 \sqrt{13}}{9})}^{3} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

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

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Step 1.4.1.2.1.1.1

Apply the product rule to $- \frac{4 - 2 \sqrt{13}}{9}$ .

$6 ({(- 1)}^{3} {(\frac{4 - 2 \sqrt{13}}{9})}^{3}) + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.1.2

Apply the product rule to $\frac{4 - 2 \sqrt{13}}{9}$ .

$6 ({(- 1)}^{3} \frac{{(4 - 2 \sqrt{13})}^{3}}{9^{3}}) + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.2

Raise $- 1$ to the power of $3$ .

$6 (- \frac{{(4 - 2 \sqrt{13})}^{3}}{9^{3}}) + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.3

Raise $9$ to the power of $3$ .

$6 (- \frac{{(4 - 2 \sqrt{13})}^{3}}{729}) + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.4

Cancel the common factor of $3$ .

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Step 1.4.1.2.1.4.1

Move the leading negative in $- \frac{{(4 - 2 \sqrt{13})}^{3}}{729}$ into the numerator.

$6 \frac{- {(4 - 2 \sqrt{13})}^{3}}{729} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.4.2

Factor $3$ out of $6$ .

$3 (2) \frac{- {(4 - 2 \sqrt{13})}^{3}}{729} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.4.3

Factor $3$ out of $729$ .

$3 \cdot 2 \frac{- {(4 - 2 \sqrt{13})}^{3}}{3 \cdot 243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.4.4

Cancel the common factor.

$3 \cdot 2 \frac{- {(4 - 2 \sqrt{13})}^{3}}{3 \cdot 243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.4.5

Rewrite the expression.

$2 \frac{- {(4 - 2 \sqrt{13})}^{3}}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.5

Combine $2$ and $\frac{- {(4 - 2 \sqrt{13})}^{3}}{243}$ .

$\frac{2 (- {(4 - 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.6

Multiply $- 1$ by $2$ .

$\frac{- 2 {(4 - 2 \sqrt{13})}^{3}}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.7

Use the Binomial Theorem.

$\frac{- 2 (4^{3} + 3 \cdot 4^{2} (- 2 \sqrt{13}) + 3 \cdot 4 {(- 2 \sqrt{13})}^{2} + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8

Simplify each term.

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Step 1.4.1.2.1.8.1

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

$\frac{- 2 (64 + 3 \cdot 4^{2} (- 2 \sqrt{13}) + 3 \cdot 4 {(- 2 \sqrt{13})}^{2} + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.2

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

$\frac{- 2 (64 + 3 \cdot 16 (- 2 \sqrt{13}) + 3 \cdot 4 {(- 2 \sqrt{13})}^{2} + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.3

Multiply $3$ by $16$ .

$\frac{- 2 (64 + 48 (- 2 \sqrt{13}) + 3 \cdot 4 {(- 2 \sqrt{13})}^{2} + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.4

Multiply $- 2$ by $48$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 3 \cdot 4 {(- 2 \sqrt{13})}^{2} + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.5

Multiply $3$ by $4$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 12 {(- 2 \sqrt{13})}^{2} + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.6

Apply the product rule to $- 2 \sqrt{13}$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 12 ({(- 2)}^{2} {\sqrt{13}}^{2}) + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.7

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

$\frac{- 2 (64 - 96 \sqrt{13} + 12 (4 {\sqrt{13}}^{2}) + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.8

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.4.1.2.1.8.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 12 (4 {(13^{\frac{1}{2}})}^{2}) + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 12 (4 \cdot 13^{\frac{1}{2} \cdot 2}) + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.8.3

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

$\frac{- 2 (64 - 96 \sqrt{13} + 12 (4 \cdot 13^{\frac{2}{2}}) + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.8.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.8.8.4.1

Cancel the common factor.

$\frac{- 2 (64 - 96 \sqrt{13} + 12 (4 \cdot 13^{\frac{2}{2}}) + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.8.4.2

Rewrite the expression.

$\frac{- 2 (64 - 96 \sqrt{13} + 12 (4 \cdot 13^{1}) + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.8.5

Evaluate the exponent.

$\frac{- 2 (64 - 96 \sqrt{13} + 12 (4 \cdot 13) + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.9

Multiply $12 (4 \cdot 13)$ .

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Step 1.4.1.2.1.8.9.1

Multiply $4$ by $13$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 12 \cdot 52 + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.9.2

Multiply $12$ by $52$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 624 + {(- 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.10

Apply the product rule to $- 2 \sqrt{13}$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 624 + {(- 2)}^{3} {\sqrt{13}}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.11

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

$\frac{- 2 (64 - 96 \sqrt{13} + 624 - 8 {\sqrt{13}}^{3})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.12

Rewrite ${\sqrt{13}}^{3}$ as $\sqrt{13^{3}}$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 624 - 8 \sqrt{13^{3}})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.13

Raise $13$ to the power of $3$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 624 - 8 \sqrt{2197})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.14

Rewrite $2197$ as $13^{2} \cdot 13$ .

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Step 1.4.1.2.1.8.14.1

Factor $169$ out of $2197$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 624 - 8 \sqrt{169 (13)})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.14.2

Rewrite $169$ as $13^{2}$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 624 - 8 \sqrt{13^{2} \cdot 13})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.15

Pull terms out from under the radical.

$\frac{- 2 (64 - 96 \sqrt{13} + 624 - 8 (13 \sqrt{13}))}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.8.16

Multiply $13$ by $- 8$ .

$\frac{- 2 (64 - 96 \sqrt{13} + 624 - 104 \sqrt{13})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.9

Add $64$ and $624$ .

$\frac{- 2 (688 - 96 \sqrt{13} - 104 \sqrt{13})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.10

Subtract $104 \sqrt{13}$ from $- 96 \sqrt{13}$ .

$\frac{- 2 (688 - 200 \sqrt{13})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.11

Move the negative in front of the fraction.

$- \frac{(2) (688 - 200 \sqrt{13})}{243} + 8 {(- \frac{4 - 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.12

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

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Step 1.4.1.2.1.12.1

Apply the product rule to $- \frac{4 - 2 \sqrt{13}}{9}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 ({(- 1)}^{2} {(\frac{4 - 2 \sqrt{13}}{9})}^{2}) - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.12.2

Apply the product rule to $\frac{4 - 2 \sqrt{13}}{9}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 ({(- 1)}^{2} \frac{{(4 - 2 \sqrt{13})}^{2}}{9^{2}}) - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.13

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

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 (1 \frac{{(4 - 2 \sqrt{13})}^{2}}{9^{2}}) - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.14

Multiply $\frac{{(4 - 2 \sqrt{13})}^{2}}{9^{2}}$ by $1$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{{(4 - 2 \sqrt{13})}^{2}}{9^{2}} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.15

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

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{{(4 - 2 \sqrt{13})}^{2}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.16

Rewrite ${(4 - 2 \sqrt{13})}^{2}$ as $(4 - 2 \sqrt{13}) (4 - 2 \sqrt{13})$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{(4 - 2 \sqrt{13}) (4 - 2 \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.17

Expand $(4 - 2 \sqrt{13}) (4 - 2 \sqrt{13})$ using the FOIL Method.

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Step 1.4.1.2.1.17.1

Apply the distributive property.

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{4 (4 - 2 \sqrt{13}) - 2 \sqrt{13} (4 - 2 \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.17.2

Apply the distributive property.

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{4 \cdot 4 + 4 (- 2 \sqrt{13}) - 2 \sqrt{13} (4 - 2 \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.17.3

Apply the distributive property.

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{4 \cdot 4 + 4 (- 2 \sqrt{13}) - 2 \sqrt{13} \cdot 4 - 2 \sqrt{13} (- 2 \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18

Simplify and combine like terms.

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Step 1.4.1.2.1.18.1

Simplify each term.

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Step 1.4.1.2.1.18.1.1

Multiply $4$ by $4$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 + 4 (- 2 \sqrt{13}) - 2 \sqrt{13} \cdot 4 - 2 \sqrt{13} (- 2 \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.2

Multiply $- 2$ by $4$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 2 \sqrt{13} \cdot 4 - 2 \sqrt{13} (- 2 \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.3

Multiply $4$ by $- 2$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} - 2 \sqrt{13} (- 2 \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.4

Multiply $- 2 \sqrt{13} (- 2 \sqrt{13})$ .

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Step 1.4.1.2.1.18.1.4.1

Multiply $- 2$ by $- 2$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 \sqrt{13} \sqrt{13}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.4.2

Raise $\sqrt{13}$ to the power of $1$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 ({\sqrt{13}}^{1} \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.4.4

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

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 {\sqrt{13}}^{1 + 1}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.4.5

Add $1$ and $1$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 {\sqrt{13}}^{2}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.5

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.4.1.2.1.18.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 {(13^{\frac{1}{2}})}^{2}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 \cdot 13^{\frac{1}{2} \cdot 2}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.5.3

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

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 \cdot 13^{\frac{2}{2}}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.5.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.18.1.5.4.1

Cancel the common factor.

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 \cdot 13^{\frac{2}{2}}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.5.4.2

Rewrite the expression.

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 \cdot 13^{1}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.5.5

Evaluate the exponent.

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 4 \cdot 13}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.1.6

Multiply $4$ by $13$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{16 - 8 \sqrt{13} - 8 \sqrt{13} + 52}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.2

Add $16$ and $52$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{68 - 8 \sqrt{13} - 8 \sqrt{13}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.18.3

Subtract $8 \sqrt{13}$ from $- 8 \sqrt{13}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + 8 \frac{68 - 16 \sqrt{13}}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.19

Combine $8$ and $\frac{68 - 16 \sqrt{13}}{81}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13})}{81} - 8 (- \frac{4 - 2 \sqrt{13}}{9}) - 7$

Step 1.4.1.2.1.20

Multiply $- 8 (- \frac{4 - 2 \sqrt{13}}{9})$ .

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Step 1.4.1.2.1.20.1

Multiply $- 1$ by $- 8$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13})}{81} + 8 \frac{4 - 2 \sqrt{13}}{9} - 7$

Step 1.4.1.2.1.20.2

Combine $8$ and $\frac{4 - 2 \sqrt{13}}{9}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13})}{81} + \frac{8 (4 - 2 \sqrt{13})}{9} - 7$

Step 1.4.1.2.2

Find the common denominator.

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Step 1.4.1.2.2.1

Multiply $\frac{8 (68 - 16 \sqrt{13})}{81}$ by $\frac{3}{3}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13})}{81} \cdot \frac{3}{3} + \frac{8 (4 - 2 \sqrt{13})}{9} - 7$

Step 1.4.1.2.2.2

Multiply $\frac{8 (68 - 16 \sqrt{13})}{81}$ by $\frac{3}{3}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 - 2 \sqrt{13})}{9} - 7$

Step 1.4.1.2.2.3

Multiply $\frac{8 (4 - 2 \sqrt{13})}{9}$ by $\frac{27}{27}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 - 2 \sqrt{13})}{9} \cdot \frac{27}{27} - 7$

Step 1.4.1.2.2.4

Multiply $\frac{8 (4 - 2 \sqrt{13})}{9}$ by $\frac{27}{27}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 - 2 \sqrt{13}) \cdot 27}{9 \cdot 27} - 7$

Step 1.4.1.2.2.5

Write $- 7$ as a fraction with denominator $1$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 - 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7}{1}$

Step 1.4.1.2.2.6

Multiply $\frac{- 7}{1}$ by $\frac{243}{243}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 - 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7}{1} \cdot \frac{243}{243}$

Step 1.4.1.2.2.7

Multiply $\frac{- 7}{1}$ by $\frac{243}{243}$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 - 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7 \cdot 243}{243}$

Step 1.4.1.2.2.8

Reorder the factors of $81 \cdot 3$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{3 \cdot 81} + \frac{8 (4 - 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7 \cdot 243}{243}$

Step 1.4.1.2.2.9

Multiply $3$ by $81$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{243} + \frac{8 (4 - 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7 \cdot 243}{243}$

Step 1.4.1.2.2.10

Multiply $9$ by $27$ .

$- \frac{2 (688 - 200 \sqrt{13})}{243} + \frac{8 (68 - 16 \sqrt{13}) \cdot 3}{243} + \frac{8 (4 - 2 \sqrt{13}) \cdot 27}{243} + \frac{- 7 \cdot 243}{243}$

Step 1.4.1.2.3

Combine the numerators over the common denominator.

$\frac{- 2 (688 - 200 \sqrt{13}) + 8 (68 - 16 \sqrt{13}) \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4

Simplify each term.

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Step 1.4.1.2.4.1

Apply the distributive property.

$\frac{- 2 \cdot 688 - 2 (- 200 \sqrt{13}) + 8 (68 - 16 \sqrt{13}) \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.2

Multiply $- 2$ by $688$ .

$\frac{- 1376 - 2 (- 200 \sqrt{13}) + 8 (68 - 16 \sqrt{13}) \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.3

Multiply $- 200$ by $- 2$ .

$\frac{- 1376 + 400 \sqrt{13} + 8 (68 - 16 \sqrt{13}) \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.4

Apply the distributive property.

$\frac{- 1376 + 400 \sqrt{13} + (8 \cdot 68 + 8 (- 16 \sqrt{13})) \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.5

Multiply $8$ by $68$ .

$\frac{- 1376 + 400 \sqrt{13} + (544 + 8 (- 16 \sqrt{13})) \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.6

Multiply $- 16$ by $8$ .

$\frac{- 1376 + 400 \sqrt{13} + (544 - 128 \sqrt{13}) \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.7

Apply the distributive property.

$\frac{- 1376 + 400 \sqrt{13} + 544 \cdot 3 - 128 \sqrt{13} \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.8

Multiply $544$ by $3$ .

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 128 \sqrt{13} \cdot 3 + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.9

Multiply $3$ by $- 128$ .

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 384 \sqrt{13} + 8 (4 - 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.10

Apply the distributive property.

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 384 \sqrt{13} + (8 \cdot 4 + 8 (- 2 \sqrt{13})) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.11

Multiply $8$ by $4$ .

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 384 \sqrt{13} + (32 + 8 (- 2 \sqrt{13})) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.12

Multiply $- 2$ by $8$ .

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 384 \sqrt{13} + (32 - 16 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.13

Apply the distributive property.

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 384 \sqrt{13} + 32 \cdot 27 - 16 \sqrt{13} \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.14

Multiply $32$ by $27$ .

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 384 \sqrt{13} + 864 - 16 \sqrt{13} \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.1.2.4.15

Multiply $27$ by $- 16$ .

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 384 \sqrt{13} + 864 - 432 \sqrt{13} - 7 \cdot 243}{243}$

Step 1.4.1.2.4.16

Multiply $- 7$ by $243$ .

$\frac{- 1376 + 400 \sqrt{13} + 1632 - 384 \sqrt{13} + 864 - 432 \sqrt{13} - 1701}{243}$

Step 1.4.1.2.5

Simplify terms.

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Step 1.4.1.2.5.1

Add $- 1376$ and $1632$ .

$\frac{256 + 400 \sqrt{13} - 384 \sqrt{13} + 864 - 432 \sqrt{13} - 1701}{243}$

Step 1.4.1.2.5.2

Simplify by adding and subtracting.

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Step 1.4.1.2.5.2.1

Add $256$ and $864$ .

$\frac{1120 + 400 \sqrt{13} - 384 \sqrt{13} - 432 \sqrt{13} - 1701}{243}$

Step 1.4.1.2.5.2.2

Subtract $1701$ from $1120$ .

$\frac{- 581 + 400 \sqrt{13} - 384 \sqrt{13} - 432 \sqrt{13}}{243}$

Step 1.4.1.2.5.3

Subtract $384 \sqrt{13}$ from $400 \sqrt{13}$ .

$\frac{- 581 + 16 \sqrt{13} - 432 \sqrt{13}}{243}$

Step 1.4.1.2.5.4

Subtract $432 \sqrt{13}$ from $16 \sqrt{13}$ .

$\frac{- 581 - 416 \sqrt{13}}{243}$

Step 1.4.1.2.5.5

Rewrite $- 581$ as $- 1 (581)$ .

$\frac{- 1 (581) - 416 \sqrt{13}}{243}$

Step 1.4.1.2.5.6

Factor $- 1$ out of $- 416 \sqrt{13}$ .

$\frac{- 1 (581) - (416 \sqrt{13})}{243}$

Step 1.4.1.2.5.7

Factor $- 1$ out of $- 1 (581) - (416 \sqrt{13})$ .

$\frac{- 1 (581 + 416 \sqrt{13})}{243}$

Step 1.4.1.2.5.8

Move the negative in front of the fraction.

$- \frac{581 + 416 \sqrt{13}}{243}$

Step 1.4.2

Evaluate at $x = - \frac{4 + 2 \sqrt{13}}{9}$ .

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Step 1.4.2.1

Substitute $- \frac{4 + 2 \sqrt{13}}{9}$ for $x$ .

$6 {(- \frac{4 + 2 \sqrt{13}}{9})}^{3} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

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

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Step 1.4.2.2.1.1.1

Apply the product rule to $- \frac{4 + 2 \sqrt{13}}{9}$ .

$6 ({(- 1)}^{3} {(\frac{4 + 2 \sqrt{13}}{9})}^{3}) + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.1.2

Apply the product rule to $\frac{4 + 2 \sqrt{13}}{9}$ .

$6 ({(- 1)}^{3} \frac{{(4 + 2 \sqrt{13})}^{3}}{9^{3}}) + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.2

Raise $- 1$ to the power of $3$ .

$6 (- \frac{{(4 + 2 \sqrt{13})}^{3}}{9^{3}}) + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.3

Raise $9$ to the power of $3$ .

$6 (- \frac{{(4 + 2 \sqrt{13})}^{3}}{729}) + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.4

Cancel the common factor of $3$ .

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Step 1.4.2.2.1.4.1

Move the leading negative in $- \frac{{(4 + 2 \sqrt{13})}^{3}}{729}$ into the numerator.

$6 \frac{- {(4 + 2 \sqrt{13})}^{3}}{729} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.4.2

Factor $3$ out of $6$ .

$3 (2) \frac{- {(4 + 2 \sqrt{13})}^{3}}{729} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.4.3

Factor $3$ out of $729$ .

$3 \cdot 2 \frac{- {(4 + 2 \sqrt{13})}^{3}}{3 \cdot 243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.4.4

Cancel the common factor.

$3 \cdot 2 \frac{- {(4 + 2 \sqrt{13})}^{3}}{3 \cdot 243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.4.5

Rewrite the expression.

$2 \frac{- {(4 + 2 \sqrt{13})}^{3}}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.5

Combine $2$ and $\frac{- {(4 + 2 \sqrt{13})}^{3}}{243}$ .

$\frac{2 (- {(4 + 2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.6

Multiply $- 1$ by $2$ .

$\frac{- 2 {(4 + 2 \sqrt{13})}^{3}}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.7

Use the Binomial Theorem.

$\frac{- 2 (4^{3} + 3 \cdot 4^{2} (2 \sqrt{13}) + 3 \cdot 4 {(2 \sqrt{13})}^{2} + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8

Simplify each term.

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Step 1.4.2.2.1.8.1

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

$\frac{- 2 (64 + 3 \cdot 4^{2} (2 \sqrt{13}) + 3 \cdot 4 {(2 \sqrt{13})}^{2} + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.2

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

$\frac{- 2 (64 + 3 \cdot 16 (2 \sqrt{13}) + 3 \cdot 4 {(2 \sqrt{13})}^{2} + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.3

Multiply $3$ by $16$ .

$\frac{- 2 (64 + 48 (2 \sqrt{13}) + 3 \cdot 4 {(2 \sqrt{13})}^{2} + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.4

Multiply $2$ by $48$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 3 \cdot 4 {(2 \sqrt{13})}^{2} + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.5

Multiply $3$ by $4$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 12 {(2 \sqrt{13})}^{2} + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.6

Apply the product rule to $2 \sqrt{13}$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 12 (2^{2} {\sqrt{13}}^{2}) + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.7

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

$\frac{- 2 (64 + 96 \sqrt{13} + 12 (4 {\sqrt{13}}^{2}) + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.8

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.4.2.2.1.8.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 12 (4 {(13^{\frac{1}{2}})}^{2}) + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 12 (4 \cdot 13^{\frac{1}{2} \cdot 2}) + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.8.3

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

$\frac{- 2 (64 + 96 \sqrt{13} + 12 (4 \cdot 13^{\frac{2}{2}}) + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.8.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.8.8.4.1

Cancel the common factor.

$\frac{- 2 (64 + 96 \sqrt{13} + 12 (4 \cdot 13^{\frac{2}{2}}) + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.8.4.2

Rewrite the expression.

$\frac{- 2 (64 + 96 \sqrt{13} + 12 (4 \cdot 13^{1}) + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.8.5

Evaluate the exponent.

$\frac{- 2 (64 + 96 \sqrt{13} + 12 (4 \cdot 13) + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.9

Multiply $12 (4 \cdot 13)$ .

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Step 1.4.2.2.1.8.9.1

Multiply $4$ by $13$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 12 \cdot 52 + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.9.2

Multiply $12$ by $52$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + {(2 \sqrt{13})}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.10

Apply the product rule to $2 \sqrt{13}$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + 2^{3} {\sqrt{13}}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.11

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

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + 8 {\sqrt{13}}^{3})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.12

Rewrite ${\sqrt{13}}^{3}$ as $\sqrt{13^{3}}$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + 8 \sqrt{13^{3}})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.13

Raise $13$ to the power of $3$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + 8 \sqrt{2197})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.14

Rewrite $2197$ as $13^{2} \cdot 13$ .

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Step 1.4.2.2.1.8.14.1

Factor $169$ out of $2197$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + 8 \sqrt{169 (13)})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.14.2

Rewrite $169$ as $13^{2}$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + 8 \sqrt{13^{2} \cdot 13})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.15

Pull terms out from under the radical.

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + 8 (13 \sqrt{13}))}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.8.16

Multiply $13$ by $8$ .

$\frac{- 2 (64 + 96 \sqrt{13} + 624 + 104 \sqrt{13})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.9

Add $64$ and $624$ .

$\frac{- 2 (688 + 96 \sqrt{13} + 104 \sqrt{13})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.10

Add $96 \sqrt{13}$ and $104 \sqrt{13}$ .

$\frac{- 2 (688 + 200 \sqrt{13})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.11

Move the negative in front of the fraction.

$- \frac{(2) (688 + 200 \sqrt{13})}{243} + 8 {(- \frac{4 + 2 \sqrt{13}}{9})}^{2} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.12

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

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Step 1.4.2.2.1.12.1

Apply the product rule to $- \frac{4 + 2 \sqrt{13}}{9}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 ({(- 1)}^{2} {(\frac{4 + 2 \sqrt{13}}{9})}^{2}) - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.12.2

Apply the product rule to $\frac{4 + 2 \sqrt{13}}{9}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 ({(- 1)}^{2} \frac{{(4 + 2 \sqrt{13})}^{2}}{9^{2}}) - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.13

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

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 (1 \frac{{(4 + 2 \sqrt{13})}^{2}}{9^{2}}) - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.14

Multiply $\frac{{(4 + 2 \sqrt{13})}^{2}}{9^{2}}$ by $1$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{{(4 + 2 \sqrt{13})}^{2}}{9^{2}} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.15

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

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{{(4 + 2 \sqrt{13})}^{2}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.16

Rewrite ${(4 + 2 \sqrt{13})}^{2}$ as $(4 + 2 \sqrt{13}) (4 + 2 \sqrt{13})$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{(4 + 2 \sqrt{13}) (4 + 2 \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.17

Expand $(4 + 2 \sqrt{13}) (4 + 2 \sqrt{13})$ using the FOIL Method.

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Step 1.4.2.2.1.17.1

Apply the distributive property.

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{4 (4 + 2 \sqrt{13}) + 2 \sqrt{13} (4 + 2 \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.17.2

Apply the distributive property.

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{4 \cdot 4 + 4 (2 \sqrt{13}) + 2 \sqrt{13} (4 + 2 \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.17.3

Apply the distributive property.

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{4 \cdot 4 + 4 (2 \sqrt{13}) + 2 \sqrt{13} \cdot 4 + 2 \sqrt{13} (2 \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18

Simplify and combine like terms.

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Step 1.4.2.2.1.18.1

Simplify each term.

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Step 1.4.2.2.1.18.1.1

Multiply $4$ by $4$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 4 (2 \sqrt{13}) + 2 \sqrt{13} \cdot 4 + 2 \sqrt{13} (2 \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.2

Multiply $2$ by $4$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 2 \sqrt{13} \cdot 4 + 2 \sqrt{13} (2 \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.3

Multiply $4$ by $2$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 2 \sqrt{13} (2 \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.4

Multiply $2 \sqrt{13} (2 \sqrt{13})$ .

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Step 1.4.2.2.1.18.1.4.1

Multiply $2$ by $2$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 \sqrt{13} \sqrt{13}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.4.2

Raise $\sqrt{13}$ to the power of $1$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 ({\sqrt{13}}^{1} \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.4.4

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

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 {\sqrt{13}}^{1 + 1}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.4.5

Add $1$ and $1$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 {\sqrt{13}}^{2}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.5

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 1.4.2.2.1.18.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 {(13^{\frac{1}{2}})}^{2}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 \cdot 13^{\frac{1}{2} \cdot 2}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.5.3

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

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 \cdot 13^{\frac{2}{2}}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.5.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.18.1.5.4.1

Cancel the common factor.

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 \cdot 13^{\frac{2}{2}}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.5.4.2

Rewrite the expression.

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 \cdot 13^{1}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.5.5

Evaluate the exponent.

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 4 \cdot 13}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.1.6

Multiply $4$ by $13$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{16 + 8 \sqrt{13} + 8 \sqrt{13} + 52}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.2

Add $16$ and $52$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{68 + 8 \sqrt{13} + 8 \sqrt{13}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.18.3

Add $8 \sqrt{13}$ and $8 \sqrt{13}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + 8 \frac{68 + 16 \sqrt{13}}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.19

Combine $8$ and $\frac{68 + 16 \sqrt{13}}{81}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13})}{81} - 8 (- \frac{4 + 2 \sqrt{13}}{9}) - 7$

Step 1.4.2.2.1.20

Multiply $- 8 (- \frac{4 + 2 \sqrt{13}}{9})$ .

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Step 1.4.2.2.1.20.1

Multiply $- 1$ by $- 8$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13})}{81} + 8 \frac{4 + 2 \sqrt{13}}{9} - 7$

Step 1.4.2.2.1.20.2

Combine $8$ and $\frac{4 + 2 \sqrt{13}}{9}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13})}{81} + \frac{8 (4 + 2 \sqrt{13})}{9} - 7$

Step 1.4.2.2.2

Find the common denominator.

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Step 1.4.2.2.2.1

Multiply $\frac{8 (68 + 16 \sqrt{13})}{81}$ by $\frac{3}{3}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13})}{81} \cdot \frac{3}{3} + \frac{8 (4 + 2 \sqrt{13})}{9} - 7$

Step 1.4.2.2.2.2

Multiply $\frac{8 (68 + 16 \sqrt{13})}{81}$ by $\frac{3}{3}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 + 2 \sqrt{13})}{9} - 7$

Step 1.4.2.2.2.3

Multiply $\frac{8 (4 + 2 \sqrt{13})}{9}$ by $\frac{27}{27}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 + 2 \sqrt{13})}{9} \cdot \frac{27}{27} - 7$

Step 1.4.2.2.2.4

Multiply $\frac{8 (4 + 2 \sqrt{13})}{9}$ by $\frac{27}{27}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 + 2 \sqrt{13}) \cdot 27}{9 \cdot 27} - 7$

Step 1.4.2.2.2.5

Write $- 7$ as a fraction with denominator $1$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 + 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7}{1}$

Step 1.4.2.2.2.6

Multiply $\frac{- 7}{1}$ by $\frac{243}{243}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 + 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7}{1} \cdot \frac{243}{243}$

Step 1.4.2.2.2.7

Multiply $\frac{- 7}{1}$ by $\frac{243}{243}$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{81 \cdot 3} + \frac{8 (4 + 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7 \cdot 243}{243}$

Step 1.4.2.2.2.8

Reorder the factors of $81 \cdot 3$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{3 \cdot 81} + \frac{8 (4 + 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7 \cdot 243}{243}$

Step 1.4.2.2.2.9

Multiply $3$ by $81$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{243} + \frac{8 (4 + 2 \sqrt{13}) \cdot 27}{9 \cdot 27} + \frac{- 7 \cdot 243}{243}$

Step 1.4.2.2.2.10

Multiply $9$ by $27$ .

$- \frac{2 (688 + 200 \sqrt{13})}{243} + \frac{8 (68 + 16 \sqrt{13}) \cdot 3}{243} + \frac{8 (4 + 2 \sqrt{13}) \cdot 27}{243} + \frac{- 7 \cdot 243}{243}$

Step 1.4.2.2.3

Combine the numerators over the common denominator.

$\frac{- 2 (688 + 200 \sqrt{13}) + 8 (68 + 16 \sqrt{13}) \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4

Simplify each term.

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Step 1.4.2.2.4.1

Apply the distributive property.

$\frac{- 2 \cdot 688 - 2 (200 \sqrt{13}) + 8 (68 + 16 \sqrt{13}) \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.2

Multiply $- 2$ by $688$ .

$\frac{- 1376 - 2 (200 \sqrt{13}) + 8 (68 + 16 \sqrt{13}) \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.3

Multiply $200$ by $- 2$ .

$\frac{- 1376 - 400 \sqrt{13} + 8 (68 + 16 \sqrt{13}) \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.4

Apply the distributive property.

$\frac{- 1376 - 400 \sqrt{13} + (8 \cdot 68 + 8 (16 \sqrt{13})) \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.5

Multiply $8$ by $68$ .

$\frac{- 1376 - 400 \sqrt{13} + (544 + 8 (16 \sqrt{13})) \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.6

Multiply $16$ by $8$ .

$\frac{- 1376 - 400 \sqrt{13} + (544 + 128 \sqrt{13}) \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.7

Apply the distributive property.

$\frac{- 1376 - 400 \sqrt{13} + 544 \cdot 3 + 128 \sqrt{13} \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.8

Multiply $544$ by $3$ .

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 128 \sqrt{13} \cdot 3 + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.9

Multiply $3$ by $128$ .

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 384 \sqrt{13} + 8 (4 + 2 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.10

Apply the distributive property.

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 384 \sqrt{13} + (8 \cdot 4 + 8 (2 \sqrt{13})) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.11

Multiply $8$ by $4$ .

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 384 \sqrt{13} + (32 + 8 (2 \sqrt{13})) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.12

Multiply $2$ by $8$ .

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 384 \sqrt{13} + (32 + 16 \sqrt{13}) \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.13

Apply the distributive property.

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 384 \sqrt{13} + 32 \cdot 27 + 16 \sqrt{13} \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.14

Multiply $32$ by $27$ .

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 384 \sqrt{13} + 864 + 16 \sqrt{13} \cdot 27 - 7 \cdot 243}{243}$

Step 1.4.2.2.4.15

Multiply $27$ by $16$ .

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 384 \sqrt{13} + 864 + 432 \sqrt{13} - 7 \cdot 243}{243}$

Step 1.4.2.2.4.16

Multiply $- 7$ by $243$ .

$\frac{- 1376 - 400 \sqrt{13} + 1632 + 384 \sqrt{13} + 864 + 432 \sqrt{13} - 1701}{243}$

Step 1.4.2.2.5

Simplify terms.

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Step 1.4.2.2.5.1

Add $- 1376$ and $1632$ .

$\frac{256 - 400 \sqrt{13} + 384 \sqrt{13} + 864 + 432 \sqrt{13} - 1701}{243}$

Step 1.4.2.2.5.2

Simplify by adding and subtracting.

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Step 1.4.2.2.5.2.1

Add $256$ and $864$ .

$\frac{1120 - 400 \sqrt{13} + 384 \sqrt{13} + 432 \sqrt{13} - 1701}{243}$

Step 1.4.2.2.5.2.2

Subtract $1701$ from $1120$ .

$\frac{- 581 - 400 \sqrt{13} + 384 \sqrt{13} + 432 \sqrt{13}}{243}$

Step 1.4.2.2.5.3

Add $- 400 \sqrt{13}$ and $384 \sqrt{13}$ .

$\frac{- 581 - 16 \sqrt{13} + 432 \sqrt{13}}{243}$

Step 1.4.2.2.5.4

Add $- 16 \sqrt{13}$ and $432 \sqrt{13}$ .

$\frac{- 581 + 416 \sqrt{13}}{243}$

Step 1.4.2.2.5.5

Rewrite $- 581$ as $- 1 (581)$ .

$\frac{- 1 (581) + 416 \sqrt{13}}{243}$

Step 1.4.2.2.5.6

Factor $- 1$ out of $416 \sqrt{13}$ .

$\frac{- 1 (581) - (- 416 \sqrt{13})}{243}$

Step 1.4.2.2.5.7

Factor $- 1$ out of $- 1 (581) - (- 416 \sqrt{13})$ .

$\frac{- 1 (581 - 416 \sqrt{13})}{243}$

Step 1.4.2.2.5.8

Move the negative in front of the fraction.

$- \frac{581 - 416 \sqrt{13}}{243}$

Step 1.4.3

List all of the points.

$(- \frac{4 - 2 \sqrt{13}}{9}, - \frac{581 + 416 \sqrt{13}}{243}), (- \frac{4 + 2 \sqrt{13}}{9}, - \frac{581 - 416 \sqrt{13}}{243})$

Step 2

Exclude the points that are not on the interval.

Step 3

Evaluate at the included endpoints.

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Step 3.1

Evaluate at $x = 1$ .

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Step 3.1.1

Substitute $1$ for $x$ .

$6 {(1)}^{3} + 8 {(1)}^{2} - 8 \cdot 1 - 7$

Step 3.1.2

Simplify.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

One to any power is one.

$6 \cdot 1 + 8 {(1)}^{2} - 8 \cdot 1 - 7$

Step 3.1.2.1.2

Multiply $6$ by $1$ .

$6 + 8 {(1)}^{2} - 8 \cdot 1 - 7$

Step 3.1.2.1.3

One to any power is one.

$6 + 8 \cdot 1 - 8 \cdot 1 - 7$

Step 3.1.2.1.4

Multiply $8$ by $1$ .

$6 + 8 - 8 \cdot 1 - 7$

Step 3.1.2.1.5

Multiply $- 8$ by $1$ .

$6 + 8 - 8 - 7$

Step 3.1.2.2

Simplify by adding and subtracting.

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Step 3.1.2.2.1

Add $6$ and $8$ .

$14 - 8 - 7$

Step 3.1.2.2.2

Subtract $8$ from $14$ .

$6 - 7$

Step 3.1.2.2.3

Subtract $7$ from $6$ .

$- 1$

Step 3.2

Evaluate at $x = 2$ .

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Step 3.2.1

Substitute $2$ for $x$ .

$6 {(2)}^{3} + 8 {(2)}^{2} - 8 \cdot 2 - 7$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

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

$6 \cdot 8 + 8 {(2)}^{2} - 8 \cdot 2 - 7$

Step 3.2.2.1.2

Multiply $6$ by $8$ .

$48 + 8 {(2)}^{2} - 8 \cdot 2 - 7$

Step 3.2.2.1.3

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

$48 + 8 \cdot 4 - 8 \cdot 2 - 7$

Step 3.2.2.1.4

Multiply $8$ by $4$ .

$48 + 32 - 8 \cdot 2 - 7$

Step 3.2.2.1.5

Multiply $- 8$ by $2$ .

$48 + 32 - 16 - 7$

Step 3.2.2.2

Simplify by adding and subtracting.

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Step 3.2.2.2.1

Add $48$ and $32$ .

$80 - 16 - 7$

Step 3.2.2.2.2

Subtract $16$ from $80$ .

$64 - 7$

Step 3.2.2.2.3

Subtract $7$ from $64$ .

$57$

Step 3.3

List all of the points.

$(1, - 1), (2, 57)$

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(2, 57)$

Absolute Minimum: $(1, - 1)$

Step 5