Calculus Examples

$f (x) = x^{3} + 3 x^{2} - 5 x - 6$

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{3} + 3 x^{2} - 5 x - 6$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 6]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 6]$

Step 1.1.2

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

$3 x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 6]$

Step 1.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 1.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 6]$

Step 1.2.2

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

$3 x^{2} + 3 (2 x) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 6]$

Step 1.2.3

Multiply $2$ by $3$ .

$3 x^{2} + 6 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 6]$

Step 1.3

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 1.3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$3 x^{2} + 6 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 6]$

Step 1.3.2

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

$3 x^{2} + 6 x - 5 \cdot 1 + \frac{d}{d x} [- 6]$

Step 1.3.3

Multiply $- 5$ by $1$ .

$3 x^{2} + 6 x - 5 + \frac{d}{d x} [- 6]$

Step 1.4

Differentiate using the Constant Rule.

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Step 1.4.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$3 x^{2} + 6 x - 5 + 0$

Step 1.4.2

Add $3 x^{2} + 6 x - 5$ and $0$ .

$3 x^{2} + 6 x - 5$

Step 2

Set the first derivative equal to $0$ and solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 3$ , $b = 6$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (3 \cdot - 5)}}{2 \cdot 3}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Step 2.3.1.2

Multiply $- 4 \cdot 3 \cdot - 5$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 6 \pm \sqrt{36 - 12 \cdot - 5}}{2 \cdot 3}$

Step 2.3.1.2.2

Multiply $- 12$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 + 60}}{2 \cdot 3}$

Step 2.3.1.3

Add $36$ and $60$ .

$x = \frac{- 6 \pm \sqrt{96}}{2 \cdot 3}$

Step 2.3.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 2.3.1.4.1

Factor $16$ out of $96$ .

$x = \frac{- 6 \pm \sqrt{16 (6)}}{2 \cdot 3}$

Step 2.3.1.4.2

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

$x = \frac{- 6 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 3}$

Step 2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 4 \sqrt{6}}{2 \cdot 3}$

Step 2.3.2

Multiply $2$ by $3$ .

$x = \frac{- 6 \pm 4 \sqrt{6}}{6}$

Step 2.3.3

Simplify $\frac{- 6 \pm 4 \sqrt{6}}{6}$ .

$x = \frac{- 3 \pm 2 \sqrt{6}}{3}$

Step 2.4

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

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Step 2.4.1.2

Multiply $- 4 \cdot 3 \cdot - 5$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 6 \pm \sqrt{36 - 12 \cdot - 5}}{2 \cdot 3}$

Step 2.4.1.2.2

Multiply $- 12$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 + 60}}{2 \cdot 3}$

Step 2.4.1.3

Add $36$ and $60$ .

$x = \frac{- 6 \pm \sqrt{96}}{2 \cdot 3}$

Step 2.4.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 2.4.1.4.1

Factor $16$ out of $96$ .

$x = \frac{- 6 \pm \sqrt{16 (6)}}{2 \cdot 3}$

Step 2.4.1.4.2

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

$x = \frac{- 6 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 3}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 4 \sqrt{6}}{2 \cdot 3}$

Step 2.4.2

Multiply $2$ by $3$ .

$x = \frac{- 6 \pm 4 \sqrt{6}}{6}$

Step 2.4.3

Simplify $\frac{- 6 \pm 4 \sqrt{6}}{6}$ .

$x = \frac{- 3 \pm 2 \sqrt{6}}{3}$

Step 2.4.4

Change the $\pm$ to $+$ .

$x = \frac{- 3 + 2 \sqrt{6}}{3}$

Step 2.4.5

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + 2 \sqrt{6}}{3}$

Step 2.4.6

Factor $- 1$ out of $2 \sqrt{6}$ .

$x = \frac{- 1 \cdot 3 - (- 2 \sqrt{6})}{3}$

Step 2.4.7

Factor $- 1$ out of $- 1 (3) - (- 2 \sqrt{6})$ .

$x = \frac{- 1 (3 - 2 \sqrt{6})}{3}$

Step 2.4.8

Move the negative in front of the fraction.

$x = - \frac{3 - 2 \sqrt{6}}{3}$

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 3 \cdot - 5}}{2 \cdot 3}$

Step 2.5.1.2

Multiply $- 4 \cdot 3 \cdot - 5$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 6 \pm \sqrt{36 - 12 \cdot - 5}}{2 \cdot 3}$

Step 2.5.1.2.2

Multiply $- 12$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 + 60}}{2 \cdot 3}$

Step 2.5.1.3

Add $36$ and $60$ .

$x = \frac{- 6 \pm \sqrt{96}}{2 \cdot 3}$

Step 2.5.1.4

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 2.5.1.4.1

Factor $16$ out of $96$ .

$x = \frac{- 6 \pm \sqrt{16 (6)}}{2 \cdot 3}$

Step 2.5.1.4.2

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

$x = \frac{- 6 \pm \sqrt{4^{2} \cdot 6}}{2 \cdot 3}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 4 \sqrt{6}}{2 \cdot 3}$

Step 2.5.2

Multiply $2$ by $3$ .

$x = \frac{- 6 \pm 4 \sqrt{6}}{6}$

Step 2.5.3

Simplify $\frac{- 6 \pm 4 \sqrt{6}}{6}$ .

$x = \frac{- 3 \pm 2 \sqrt{6}}{3}$

Step 2.5.4

Change the $\pm$ to $-$ .

$x = \frac{- 3 - 2 \sqrt{6}}{3}$

Step 2.5.5

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - 2 \sqrt{6}}{3}$

Step 2.5.6

Factor $- 1$ out of $- 2 \sqrt{6}$ .

$x = \frac{- 1 \cdot 3 - (2 \sqrt{6})}{3}$

Step 2.5.7

Factor $- 1$ out of $- 1 (3) - (2 \sqrt{6})$ .

$x = \frac{- 1 (3 + 2 \sqrt{6})}{3}$

Step 2.5.8

Move the negative in front of the fraction.

$x = - \frac{3 + 2 \sqrt{6}}{3}$

Step 2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - 2 \sqrt{6}}{3}, - \frac{3 + 2 \sqrt{6}}{3}$

Step 3

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - \frac{3 + 2 \sqrt{6}}{3}) \cup (- \frac{3 + 2 \sqrt{6}}{3}, - \frac{3 - 2 \sqrt{6}}{3}) \cup (- \frac{3 - 2 \sqrt{6}}{3}, \infty)$

Step 4

Substitute any number, such as $- 5$ , from the interval $(- \infty, - \frac{3 + 2 \sqrt{6}}{3})$ in the first derivative $3 x^{2} + 6 x - 5$ to check if the result is negative or positive.

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Step 4.1

Replace the variable $x$ with $- 5$ in the expression.

$f' (- 5) = 3 {(- 5)}^{2} + 6 (- 5) - 5$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f' (- 5) = 3 \cdot 25 + 6 (- 5) - 5$

Step 4.2.1.2

Multiply $3$ by $25$ .

$f' (- 5) = 75 + 6 (- 5) - 5$

Step 4.2.1.3

Multiply $6$ by $- 5$ .

$f' (- 5) = 75 - 30 - 5$

Step 4.2.2

Simplify by subtracting numbers.

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Step 4.2.2.1

Subtract $30$ from $75$ .

$f' (- 5) = 45 - 5$

Step 4.2.2.2

Subtract $5$ from $45$ .

$f' (- 5) = 40$

Step 4.2.3

The final answer is $40$ .

$40$

Step 5

Substitute any number, such as $0$ , from the interval $(- \frac{3 + 2 \sqrt{6}}{3}, - \frac{3 - 2 \sqrt{6}}{3})$ in the first derivative $3 x^{2} + 6 x - 5$ to check if the result is negative or positive.

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Step 5.1

Replace the variable $x$ with $0$ in the expression.

$f' (0) = 3 {(0)}^{2} + 6 (0) - 5$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f' (0) = 3 \cdot 0 + 6 (0) - 5$

Step 5.2.1.2

Multiply $3$ by $0$ .

$f' (0) = 0 + 6 (0) - 5$

Step 5.2.1.3

Multiply $6$ by $0$ .

$f' (0) = 0 + 0 - 5$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 - 5$

Step 5.2.2.2

Subtract $5$ from $0$ .

$f' (0) = - 5$

Step 5.2.3

The final answer is $- 5$ .

$- 5$

Step 6

Substitute any number, such as $3$ , from the interval $(- \frac{3 - 2 \sqrt{6}}{3}, \infty)$ in the first derivative $3 x^{2} + 6 x - 5$ to check if the result is negative or positive.

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Step 6.1

Replace the variable $x$ with $3$ in the expression.

$f' (3) = 3 {(3)}^{2} + 6 (3) - 5$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 6.2.1.1.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 6.2.1.1.1.1

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

$f' (3) = 3 {(3)}^{2} + 6 (3) - 5$

Step 6.2.1.1.1.2

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

$f' (3) = 3^{1 + 2} + 6 (3) - 5$

Step 6.2.1.1.2

Add $1$ and $2$ .

$f' (3) = 3^{3} + 6 (3) - 5$

Step 6.2.1.2

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

$f' (3) = 27 + 6 (3) - 5$

Step 6.2.1.3

Multiply $6$ by $3$ .

$f' (3) = 27 + 18 - 5$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Add $27$ and $18$ .

$f' (3) = 45 - 5$

Step 6.2.2.2

Subtract $5$ from $45$ .

$f' (3) = 40$

Step 6.2.3

The final answer is $40$ .

$40$

Step 7

Since the first derivative changed signs from positive to negative around $x = - \frac{3 + 2 \sqrt{6}}{3}$ , then there is a turning point at $x = - \frac{3 + 2 \sqrt{6}}{3}$ .

Step 8

Find the y-coordinate of $- \frac{3 + 2 \sqrt{6}}{3}$ to find the turning point.

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Step 8.1

Find $f (- \frac{3 + 2 \sqrt{6}}{3})$ to find the y-coordinate of $- \frac{3 + 2 \sqrt{6}}{3}$ .

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Step 8.1.1

Replace the variable $x$ with $- \frac{3 + 2 \sqrt{6}}{3}$ in the expression.

$f (- \frac{3 + 2 \sqrt{6}}{3}) = {(- \frac{3 + 2 \sqrt{6}}{3})}^{3} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2

Simplify ${(- \frac{3 + 2 \sqrt{6}}{3})}^{3} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$ .

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Step 8.1.2.1

Remove parentheses.

${(- \frac{3 + 2 \sqrt{6}}{3})}^{3} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2

Simplify each term.

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Step 8.1.2.2.1

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

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Step 8.1.2.2.1.1

Apply the product rule to $- \frac{3 + 2 \sqrt{6}}{3}$ .

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

Step 8.1.2.2.1.2

Apply the product rule to $\frac{3 + 2 \sqrt{6}}{3}$ .

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

Step 8.1.2.2.2

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

$- \frac{{(3 + 2 \sqrt{6})}^{3}}{3^{3}} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.3

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

$- \frac{{(3 + 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.4

Use the Binomial Theorem.

$- \frac{3^{3} + 3 \cdot 3^{2} (2 \sqrt{6}) + 3 \cdot 3 {(2 \sqrt{6})}^{2} + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5

Simplify each term.

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Step 8.1.2.2.5.1

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

$- \frac{27 + 3 \cdot 3^{2} (2 \sqrt{6}) + 3 \cdot 3 {(2 \sqrt{6})}^{2} + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 8.1.2.2.5.2.1

Multiply $3$ by $3^{2}$ .

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Step 8.1.2.2.5.2.1.1

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

$- \frac{27 + 3^{1} \cdot 3^{2} (2 \sqrt{6}) + 3 \cdot 3 {(2 \sqrt{6})}^{2} + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.2.1.2

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

$- \frac{27 + 3^{1 + 2} (2 \sqrt{6}) + 3 \cdot 3 {(2 \sqrt{6})}^{2} + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.2.2

Add $1$ and $2$ .

$- \frac{27 + 3^{3} (2 \sqrt{6}) + 3 \cdot 3 {(2 \sqrt{6})}^{2} + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.3

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

$- \frac{27 + 27 (2 \sqrt{6}) + 3 \cdot 3 {(2 \sqrt{6})}^{2} + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.4

Multiply $2$ by $27$ .

$- \frac{27 + 54 \sqrt{6} + 3 \cdot 3 {(2 \sqrt{6})}^{2} + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.5

Multiply $3$ by $3$ .

$- \frac{27 + 54 \sqrt{6} + 9 {(2 \sqrt{6})}^{2} + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.6

Apply the product rule to $2 \sqrt{6}$ .

$- \frac{27 + 54 \sqrt{6} + 9 (2^{2} {\sqrt{6}}^{2}) + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.7

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

$- \frac{27 + 54 \sqrt{6} + 9 (4 {\sqrt{6}}^{2}) + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.8

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 8.1.2.2.5.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$- \frac{27 + 54 \sqrt{6} + 9 (4 {(6^{\frac{1}{2}})}^{2}) + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.8.2

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

$- \frac{27 + 54 \sqrt{6} + 9 (4 \cdot 6^{\frac{1}{2} \cdot 2}) + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.8.3

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

$- \frac{27 + 54 \sqrt{6} + 9 (4 \cdot 6^{\frac{2}{2}}) + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.8.4

Cancel the common factor of $2$ .

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Step 8.1.2.2.5.8.4.1

Cancel the common factor.

$- \frac{27 + 54 \sqrt{6} + 9 (4 \cdot 6^{\frac{2}{2}}) + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.8.4.2

Rewrite the expression.

$- \frac{27 + 54 \sqrt{6} + 9 (4 \cdot 6^{1}) + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.8.5

Evaluate the exponent.

$- \frac{27 + 54 \sqrt{6} + 9 (4 \cdot 6) + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.9

Multiply $9 (4 \cdot 6)$ .

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Step 8.1.2.2.5.9.1

Multiply $4$ by $6$ .

$- \frac{27 + 54 \sqrt{6} + 9 \cdot 24 + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.9.2

Multiply $9$ by $24$ .

$- \frac{27 + 54 \sqrt{6} + 216 + {(2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.10

Apply the product rule to $2 \sqrt{6}$ .

$- \frac{27 + 54 \sqrt{6} + 216 + 2^{3} {\sqrt{6}}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.11

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

$- \frac{27 + 54 \sqrt{6} + 216 + 8 {\sqrt{6}}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.12

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$- \frac{27 + 54 \sqrt{6} + 216 + 8 \sqrt{6^{3}}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.13

Raise $6$ to the power of $3$ .

$- \frac{27 + 54 \sqrt{6} + 216 + 8 \sqrt{216}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.14

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 8.1.2.2.5.14.1

Factor $36$ out of $216$ .

$- \frac{27 + 54 \sqrt{6} + 216 + 8 \sqrt{36 (6)}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.14.2

Rewrite $36$ as $6^{2}$ .

$- \frac{27 + 54 \sqrt{6} + 216 + 8 \sqrt{6^{2} \cdot 6}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.15

Pull terms out from under the radical.

$- \frac{27 + 54 \sqrt{6} + 216 + 8 (6 \sqrt{6})}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.5.16

Multiply $6$ by $8$ .

$- \frac{27 + 54 \sqrt{6} + 216 + 48 \sqrt{6}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.6

Add $27$ and $216$ .

$- \frac{243 + 54 \sqrt{6} + 48 \sqrt{6}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.7

Add $54 \sqrt{6}$ and $48 \sqrt{6}$ .

$- \frac{243 + 102 \sqrt{6}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.8

Cancel the common factor of $243 + 102 \sqrt{6}$ and $27$ .

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Step 8.1.2.2.8.1

Factor $3$ out of $243$ .

$- \frac{3 (81) + 102 \sqrt{6}}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.8.2

Factor $3$ out of $102 \sqrt{6}$ .

$- \frac{3 (81) + 3 (34 \sqrt{6})}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.8.3

Factor $3$ out of $3 (81) + 3 (34 \sqrt{6})$ .

$- \frac{3 (81 + 34 \sqrt{6})}{27} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.8.4

Cancel the common factors.

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Step 8.1.2.2.8.4.1

Factor $3$ out of $27$ .

$- \frac{3 (81 + 34 \sqrt{6})}{3 \cdot 9} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.8.4.2

Cancel the common factor.

$- \frac{3 (81 + 34 \sqrt{6})}{3 \cdot 9} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.8.4.3

Rewrite the expression.

$- \frac{81 + 34 \sqrt{6}}{9} + 3 {(- \frac{3 + 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.9

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

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Step 8.1.2.2.9.1

Apply the product rule to $- \frac{3 + 2 \sqrt{6}}{3}$ .

$- \frac{81 + 34 \sqrt{6}}{9} + 3 ({(- 1)}^{2} {(\frac{3 + 2 \sqrt{6}}{3})}^{2}) - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.9.2

Apply the product rule to $\frac{3 + 2 \sqrt{6}}{3}$ .

$- \frac{81 + 34 \sqrt{6}}{9} + 3 ({(- 1)}^{2} \frac{{(3 + 2 \sqrt{6})}^{2}}{3^{2}}) - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.10

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

$- \frac{81 + 34 \sqrt{6}}{9} + 3 (1 \frac{{(3 + 2 \sqrt{6})}^{2}}{3^{2}}) - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.11

Multiply $\frac{{(3 + 2 \sqrt{6})}^{2}}{3^{2}}$ by $1$ .

$- \frac{81 + 34 \sqrt{6}}{9} + 3 \frac{{(3 + 2 \sqrt{6})}^{2}}{3^{2}} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.12

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

$- \frac{81 + 34 \sqrt{6}}{9} + 3 \frac{{(3 + 2 \sqrt{6})}^{2}}{9} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.13

Cancel the common factor of $3$ .

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Step 8.1.2.2.13.1

Factor $3$ out of $9$ .

$- \frac{81 + 34 \sqrt{6}}{9} + 3 \frac{{(3 + 2 \sqrt{6})}^{2}}{3 (3)} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.13.2

Cancel the common factor.

$- \frac{81 + 34 \sqrt{6}}{9} + 3 \frac{{(3 + 2 \sqrt{6})}^{2}}{3 \cdot 3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.13.3

Rewrite the expression.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{{(3 + 2 \sqrt{6})}^{2}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.14

Rewrite ${(3 + 2 \sqrt{6})}^{2}$ as $(3 + 2 \sqrt{6}) (3 + 2 \sqrt{6})$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{(3 + 2 \sqrt{6}) (3 + 2 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.15

Expand $(3 + 2 \sqrt{6}) (3 + 2 \sqrt{6})$ using the FOIL Method.

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Step 8.1.2.2.15.1

Apply the distributive property.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{3 (3 + 2 \sqrt{6}) + 2 \sqrt{6} (3 + 2 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.15.2

Apply the distributive property.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{3 \cdot 3 + 3 (2 \sqrt{6}) + 2 \sqrt{6} (3 + 2 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.15.3

Apply the distributive property.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{3 \cdot 3 + 3 (2 \sqrt{6}) + 2 \sqrt{6} \cdot 3 + 2 \sqrt{6} (2 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16

Simplify and combine like terms.

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Step 8.1.2.2.16.1

Simplify each term.

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Step 8.1.2.2.16.1.1

Multiply $3$ by $3$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 3 (2 \sqrt{6}) + 2 \sqrt{6} \cdot 3 + 2 \sqrt{6} (2 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.2

Multiply $2$ by $3$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 2 \sqrt{6} \cdot 3 + 2 \sqrt{6} (2 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.3

Multiply $3$ by $2$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 2 \sqrt{6} (2 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.4

Multiply $2 \sqrt{6} (2 \sqrt{6})$ .

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Step 8.1.2.2.16.1.4.1

Multiply $2$ by $2$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 \sqrt{6} \sqrt{6}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 ({\sqrt{6}}^{1} \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.4.4

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

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 {\sqrt{6}}^{1 + 1}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.4.5

Add $1$ and $1$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 {\sqrt{6}}^{2}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 8.1.2.2.16.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 {(6^{\frac{1}{2}})}^{2}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.5.2

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

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 \cdot 6^{\frac{1}{2} \cdot 2}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.5.3

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

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 \cdot 6^{\frac{2}{2}}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.5.4

Cancel the common factor of $2$ .

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Step 8.1.2.2.16.1.5.4.1

Cancel the common factor.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 \cdot 6^{\frac{2}{2}}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.5.4.2

Rewrite the expression.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 \cdot 6^{1}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.5.5

Evaluate the exponent.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 4 \cdot 6}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.1.6

Multiply $4$ by $6$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{9 + 6 \sqrt{6} + 6 \sqrt{6} + 24}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.2

Add $9$ and $24$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{33 + 6 \sqrt{6} + 6 \sqrt{6}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.16.3

Add $6 \sqrt{6}$ and $6 \sqrt{6}$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{33 + 12 \sqrt{6}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.17

Cancel the common factor of $33 + 12 \sqrt{6}$ and $3$ .

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Step 8.1.2.2.17.1

Factor $3$ out of $33$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{3 \cdot 11 + 12 \sqrt{6}}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.17.2

Factor $3$ out of $12 \sqrt{6}$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{3 \cdot 11 + 3 (4 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.17.3

Factor $3$ out of $3 (11) + 3 (4 \sqrt{6})$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{3 (11 + 4 \sqrt{6})}{3} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.17.4

Cancel the common factors.

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Step 8.1.2.2.17.4.1

Factor $3$ out of $3$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{3 (11 + 4 \sqrt{6})}{3 (1)} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.17.4.2

Cancel the common factor.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{3 (11 + 4 \sqrt{6})}{3 \cdot 1} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.17.4.3

Rewrite the expression.

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{11 + 4 \sqrt{6}}{1} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.17.4.4

Divide $11 + 4 \sqrt{6}$ by $1$ .

$- \frac{81 + 34 \sqrt{6}}{9} + 11 + 4 \sqrt{6} - 5 (- \frac{3 + 2 \sqrt{6}}{3}) - 6$

Step 8.1.2.2.18

Multiply $- 5 (- \frac{3 + 2 \sqrt{6}}{3})$ .

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Step 8.1.2.2.18.1

Multiply $- 1$ by $- 5$ .

$- \frac{81 + 34 \sqrt{6}}{9} + 11 + 4 \sqrt{6} + 5 \frac{3 + 2 \sqrt{6}}{3} - 6$

Step 8.1.2.2.18.2

Combine $5$ and $\frac{3 + 2 \sqrt{6}}{3}$ .

$- \frac{81 + 34 \sqrt{6}}{9} + 11 + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.3

To write $11$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{81 + 34 \sqrt{6}}{9} + 11 \cdot \frac{9}{9} + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.4

Combine fractions.

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Step 8.1.2.4.1

Combine $11$ and $\frac{9}{9}$ .

$- \frac{81 + 34 \sqrt{6}}{9} + \frac{11 \cdot 9}{9} + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.4.2

Simplify the expression.

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Step 8.1.2.4.2.1

Combine the numerators over the common denominator.

$\frac{- (81 + 34 \sqrt{6}) + 11 \cdot 9}{9} + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.4.2.2

Multiply $11$ by $9$ .

$\frac{- (81 + 34 \sqrt{6}) + 99}{9} + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.5

Simplify the numerator.

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Step 8.1.2.5.1

Apply the distributive property.

$\frac{- 1 \cdot 81 - (34 \sqrt{6}) + 99}{9} + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.5.2

Multiply $- 1$ by $81$ .

$\frac{- 81 - (34 \sqrt{6}) + 99}{9} + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.5.3

Multiply $34$ by $- 1$ .

$\frac{- 81 - 34 \sqrt{6} + 99}{9} + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.5.4

Add $- 81$ and $99$ .

$\frac{18 - 34 \sqrt{6}}{9} + 4 \sqrt{6} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.6

Find the common denominator.

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Step 8.1.2.6.1

Write $4 \sqrt{6}$ as a fraction with denominator $1$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6}}{1} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.6.2

Multiply $\frac{4 \sqrt{6}}{1}$ by $\frac{9}{9}$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6}}{1} \cdot \frac{9}{9} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.6.3

Multiply $\frac{4 \sqrt{6}}{1}$ by $\frac{9}{9}$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 + 2 \sqrt{6})}{3} - 6$

Step 8.1.2.6.4

Multiply $\frac{5 (3 + 2 \sqrt{6})}{3}$ by $\frac{3}{3}$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 + 2 \sqrt{6})}{3} \cdot \frac{3}{3} - 6$

Step 8.1.2.6.5

Multiply $\frac{5 (3 + 2 \sqrt{6})}{3}$ by $\frac{3}{3}$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 + 2 \sqrt{6}) \cdot 3}{3 \cdot 3} - 6$

Step 8.1.2.6.6

Write $- 6$ as a fraction with denominator $1$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 + 2 \sqrt{6}) \cdot 3}{3 \cdot 3} + \frac{- 6}{1}$

Step 8.1.2.6.7

Multiply $\frac{- 6}{1}$ by $\frac{9}{9}$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 + 2 \sqrt{6}) \cdot 3}{3 \cdot 3} + \frac{- 6}{1} \cdot \frac{9}{9}$

Step 8.1.2.6.8

Multiply $\frac{- 6}{1}$ by $\frac{9}{9}$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 + 2 \sqrt{6}) \cdot 3}{3 \cdot 3} + \frac{- 6 \cdot 9}{9}$

Step 8.1.2.6.9

Multiply $3$ by $3$ .

$\frac{18 - 34 \sqrt{6}}{9} + \frac{4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 + 2 \sqrt{6}) \cdot 3}{9} + \frac{- 6 \cdot 9}{9}$

Step 8.1.2.7

Combine the numerators over the common denominator.

$\frac{18 - 34 \sqrt{6} + 4 \sqrt{6} \cdot 9 + 5 (3 + 2 \sqrt{6}) \cdot 3 - 6 \cdot 9}{9}$

Step 8.1.2.8

Simplify each term.

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Step 8.1.2.8.1

Multiply $9$ by $4$ .

$\frac{18 - 34 \sqrt{6} + 36 \sqrt{6} + 5 (3 + 2 \sqrt{6}) \cdot 3 - 6 \cdot 9}{9}$

Step 8.1.2.8.2

Apply the distributive property.

$\frac{18 - 34 \sqrt{6} + 36 \sqrt{6} + (5 \cdot 3 + 5 (2 \sqrt{6})) \cdot 3 - 6 \cdot 9}{9}$

Step 8.1.2.8.3

Multiply $5$ by $3$ .

$\frac{18 - 34 \sqrt{6} + 36 \sqrt{6} + (15 + 5 (2 \sqrt{6})) \cdot 3 - 6 \cdot 9}{9}$

Step 8.1.2.8.4

Multiply $2$ by $5$ .

$\frac{18 - 34 \sqrt{6} + 36 \sqrt{6} + (15 + 10 \sqrt{6}) \cdot 3 - 6 \cdot 9}{9}$

Step 8.1.2.8.5

Apply the distributive property.

$\frac{18 - 34 \sqrt{6} + 36 \sqrt{6} + 15 \cdot 3 + 10 \sqrt{6} \cdot 3 - 6 \cdot 9}{9}$

Step 8.1.2.8.6

Multiply $15$ by $3$ .

$\frac{18 - 34 \sqrt{6} + 36 \sqrt{6} + 45 + 10 \sqrt{6} \cdot 3 - 6 \cdot 9}{9}$

Step 8.1.2.8.7

Multiply $3$ by $10$ .

$\frac{18 - 34 \sqrt{6} + 36 \sqrt{6} + 45 + 30 \sqrt{6} - 6 \cdot 9}{9}$

Step 8.1.2.8.8

Multiply $- 6$ by $9$ .

$\frac{18 - 34 \sqrt{6} + 36 \sqrt{6} + 45 + 30 \sqrt{6} - 54}{9}$

Step 8.1.2.9

Simplify by adding terms.

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Step 8.1.2.9.1

Add $18$ and $45$ .

$\frac{63 - 34 \sqrt{6} + 36 \sqrt{6} + 30 \sqrt{6} - 54}{9}$

Step 8.1.2.9.2

Subtract $54$ from $63$ .

$\frac{9 - 34 \sqrt{6} + 36 \sqrt{6} + 30 \sqrt{6}}{9}$

Step 8.1.2.9.3

Add $- 34 \sqrt{6}$ and $36 \sqrt{6}$ .

$\frac{9 + 2 \sqrt{6} + 30 \sqrt{6}}{9}$

Step 8.1.2.9.4

Add $2 \sqrt{6}$ and $30 \sqrt{6}$ .

$\frac{9 + 32 \sqrt{6}}{9}$

Step 8.2

Write the $x$ and $y$ coordinates in point form.

$(- \frac{3 + 2 \sqrt{6}}{3}, \frac{9 + 32 \sqrt{6}}{9})$

Step 9

Since the first derivative changed signs from negative to positive around $x = - \frac{3 - 2 \sqrt{6}}{3}$ , then there is a turning point at $x = - \frac{3 - 2 \sqrt{6}}{3}$ .

Step 10

Find the y-coordinate of $- \frac{3 - 2 \sqrt{6}}{3}$ to find the turning point.

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Step 10.1

Find $f (- \frac{3 - 2 \sqrt{6}}{3})$ to find the y-coordinate of $- \frac{3 - 2 \sqrt{6}}{3}$ .

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Step 10.1.1

Replace the variable $x$ with $- \frac{3 - 2 \sqrt{6}}{3}$ in the expression.

$f (- \frac{3 - 2 \sqrt{6}}{3}) = {(- \frac{3 - 2 \sqrt{6}}{3})}^{3} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2

Simplify ${(- \frac{3 - 2 \sqrt{6}}{3})}^{3} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$ .

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Step 10.1.2.1

Remove parentheses.

${(- \frac{3 - 2 \sqrt{6}}{3})}^{3} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2

Simplify each term.

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Step 10.1.2.2.1

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

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Step 10.1.2.2.1.1

Apply the product rule to $- \frac{3 - 2 \sqrt{6}}{3}$ .

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

Step 10.1.2.2.1.2

Apply the product rule to $\frac{3 - 2 \sqrt{6}}{3}$ .

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

Step 10.1.2.2.2

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

$- \frac{{(3 - 2 \sqrt{6})}^{3}}{3^{3}} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.3

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

$- \frac{{(3 - 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.4

Use the Binomial Theorem.

$- \frac{3^{3} + 3 \cdot 3^{2} (- 2 \sqrt{6}) + 3 \cdot 3 {(- 2 \sqrt{6})}^{2} + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5

Simplify each term.

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Step 10.1.2.2.5.1

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

$- \frac{27 + 3 \cdot 3^{2} (- 2 \sqrt{6}) + 3 \cdot 3 {(- 2 \sqrt{6})}^{2} + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.2

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 10.1.2.2.5.2.1

Multiply $3$ by $3^{2}$ .

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Step 10.1.2.2.5.2.1.1

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

$- \frac{27 + 3^{1} \cdot 3^{2} (- 2 \sqrt{6}) + 3 \cdot 3 {(- 2 \sqrt{6})}^{2} + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.2.1.2

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

$- \frac{27 + 3^{1 + 2} (- 2 \sqrt{6}) + 3 \cdot 3 {(- 2 \sqrt{6})}^{2} + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.2.2

Add $1$ and $2$ .

$- \frac{27 + 3^{3} (- 2 \sqrt{6}) + 3 \cdot 3 {(- 2 \sqrt{6})}^{2} + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.3

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

$- \frac{27 + 27 (- 2 \sqrt{6}) + 3 \cdot 3 {(- 2 \sqrt{6})}^{2} + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.4

Multiply $- 2$ by $27$ .

$- \frac{27 - 54 \sqrt{6} + 3 \cdot 3 {(- 2 \sqrt{6})}^{2} + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.5

Multiply $3$ by $3$ .

$- \frac{27 - 54 \sqrt{6} + 9 {(- 2 \sqrt{6})}^{2} + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.6

Apply the product rule to $- 2 \sqrt{6}$ .

$- \frac{27 - 54 \sqrt{6} + 9 ({(- 2)}^{2} {\sqrt{6}}^{2}) + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.7

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

$- \frac{27 - 54 \sqrt{6} + 9 (4 {\sqrt{6}}^{2}) + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.8

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 10.1.2.2.5.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$- \frac{27 - 54 \sqrt{6} + 9 (4 {(6^{\frac{1}{2}})}^{2}) + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.8.2

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

$- \frac{27 - 54 \sqrt{6} + 9 (4 \cdot 6^{\frac{1}{2} \cdot 2}) + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.8.3

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

$- \frac{27 - 54 \sqrt{6} + 9 (4 \cdot 6^{\frac{2}{2}}) + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.8.4

Cancel the common factor of $2$ .

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Step 10.1.2.2.5.8.4.1

Cancel the common factor.

$- \frac{27 - 54 \sqrt{6} + 9 (4 \cdot 6^{\frac{2}{2}}) + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.8.4.2

Rewrite the expression.

$- \frac{27 - 54 \sqrt{6} + 9 (4 \cdot 6^{1}) + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.8.5

Evaluate the exponent.

$- \frac{27 - 54 \sqrt{6} + 9 (4 \cdot 6) + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.9

Multiply $9 (4 \cdot 6)$ .

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Step 10.1.2.2.5.9.1

Multiply $4$ by $6$ .

$- \frac{27 - 54 \sqrt{6} + 9 \cdot 24 + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.9.2

Multiply $9$ by $24$ .

$- \frac{27 - 54 \sqrt{6} + 216 + {(- 2 \sqrt{6})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.10

Apply the product rule to $- 2 \sqrt{6}$ .

$- \frac{27 - 54 \sqrt{6} + 216 + {(- 2)}^{3} {\sqrt{6}}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.11

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

$- \frac{27 - 54 \sqrt{6} + 216 - 8 {\sqrt{6}}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.12

Rewrite ${\sqrt{6}}^{3}$ as $\sqrt{6^{3}}$ .

$- \frac{27 - 54 \sqrt{6} + 216 - 8 \sqrt{6^{3}}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.13

Raise $6$ to the power of $3$ .

$- \frac{27 - 54 \sqrt{6} + 216 - 8 \sqrt{216}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.14

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 10.1.2.2.5.14.1

Factor $36$ out of $216$ .

$- \frac{27 - 54 \sqrt{6} + 216 - 8 \sqrt{36 (6)}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.14.2

Rewrite $36$ as $6^{2}$ .

$- \frac{27 - 54 \sqrt{6} + 216 - 8 \sqrt{6^{2} \cdot 6}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.15

Pull terms out from under the radical.

$- \frac{27 - 54 \sqrt{6} + 216 - 8 (6 \sqrt{6})}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.5.16

Multiply $6$ by $- 8$ .

$- \frac{27 - 54 \sqrt{6} + 216 - 48 \sqrt{6}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.6

Add $27$ and $216$ .

$- \frac{243 - 54 \sqrt{6} - 48 \sqrt{6}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.7

Subtract $48 \sqrt{6}$ from $- 54 \sqrt{6}$ .

$- \frac{243 - 102 \sqrt{6}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.8

Cancel the common factor of $243 - 102 \sqrt{6}$ and $27$ .

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Step 10.1.2.2.8.1

Factor $3$ out of $243$ .

$- \frac{3 (81) - 102 \sqrt{6}}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.8.2

Factor $3$ out of $- 102 \sqrt{6}$ .

$- \frac{3 (81) + 3 (- 34 \sqrt{6})}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.8.3

Factor $3$ out of $3 (81) + 3 (- 34 \sqrt{6})$ .

$- \frac{3 (81 - 34 \sqrt{6})}{27} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.8.4

Cancel the common factors.

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Step 10.1.2.2.8.4.1

Factor $3$ out of $27$ .

$- \frac{3 (81 - 34 \sqrt{6})}{3 \cdot 9} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.8.4.2

Cancel the common factor.

$- \frac{3 (81 - 34 \sqrt{6})}{3 \cdot 9} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.8.4.3

Rewrite the expression.

$- \frac{81 - 34 \sqrt{6}}{9} + 3 {(- \frac{3 - 2 \sqrt{6}}{3})}^{2} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.9

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

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Step 10.1.2.2.9.1

Apply the product rule to $- \frac{3 - 2 \sqrt{6}}{3}$ .

$- \frac{81 - 34 \sqrt{6}}{9} + 3 ({(- 1)}^{2} {(\frac{3 - 2 \sqrt{6}}{3})}^{2}) - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.9.2

Apply the product rule to $\frac{3 - 2 \sqrt{6}}{3}$ .

$- \frac{81 - 34 \sqrt{6}}{9} + 3 ({(- 1)}^{2} \frac{{(3 - 2 \sqrt{6})}^{2}}{3^{2}}) - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.10

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

$- \frac{81 - 34 \sqrt{6}}{9} + 3 (1 \frac{{(3 - 2 \sqrt{6})}^{2}}{3^{2}}) - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.11

Multiply $\frac{{(3 - 2 \sqrt{6})}^{2}}{3^{2}}$ by $1$ .

$- \frac{81 - 34 \sqrt{6}}{9} + 3 \frac{{(3 - 2 \sqrt{6})}^{2}}{3^{2}} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.12

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

$- \frac{81 - 34 \sqrt{6}}{9} + 3 \frac{{(3 - 2 \sqrt{6})}^{2}}{9} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.13

Cancel the common factor of $3$ .

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Step 10.1.2.2.13.1

Factor $3$ out of $9$ .

$- \frac{81 - 34 \sqrt{6}}{9} + 3 \frac{{(3 - 2 \sqrt{6})}^{2}}{3 (3)} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.13.2

Cancel the common factor.

$- \frac{81 - 34 \sqrt{6}}{9} + 3 \frac{{(3 - 2 \sqrt{6})}^{2}}{3 \cdot 3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.13.3

Rewrite the expression.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{{(3 - 2 \sqrt{6})}^{2}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.14

Rewrite ${(3 - 2 \sqrt{6})}^{2}$ as $(3 - 2 \sqrt{6}) (3 - 2 \sqrt{6})$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{(3 - 2 \sqrt{6}) (3 - 2 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.15

Expand $(3 - 2 \sqrt{6}) (3 - 2 \sqrt{6})$ using the FOIL Method.

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Step 10.1.2.2.15.1

Apply the distributive property.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{3 (3 - 2 \sqrt{6}) - 2 \sqrt{6} (3 - 2 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.15.2

Apply the distributive property.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{3 \cdot 3 + 3 (- 2 \sqrt{6}) - 2 \sqrt{6} (3 - 2 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.15.3

Apply the distributive property.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{3 \cdot 3 + 3 (- 2 \sqrt{6}) - 2 \sqrt{6} \cdot 3 - 2 \sqrt{6} (- 2 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16

Simplify and combine like terms.

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Step 10.1.2.2.16.1

Simplify each term.

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Step 10.1.2.2.16.1.1

Multiply $3$ by $3$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 + 3 (- 2 \sqrt{6}) - 2 \sqrt{6} \cdot 3 - 2 \sqrt{6} (- 2 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.2

Multiply $- 2$ by $3$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 2 \sqrt{6} \cdot 3 - 2 \sqrt{6} (- 2 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.3

Multiply $3$ by $- 2$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} - 2 \sqrt{6} (- 2 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.4

Multiply $- 2 \sqrt{6} (- 2 \sqrt{6})$ .

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Step 10.1.2.2.16.1.4.1

Multiply $- 2$ by $- 2$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \sqrt{6} \sqrt{6}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.4.2

Raise $\sqrt{6}$ to the power of $1$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 ({\sqrt{6}}^{1} \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.4.4

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

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 {\sqrt{6}}^{1 + 1}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.4.5

Add $1$ and $1$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 {\sqrt{6}}^{2}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.5

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 10.1.2.2.16.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 {(6^{\frac{1}{2}})}^{2}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.5.2

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

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 6^{\frac{1}{2} \cdot 2}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.5.3

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

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 6^{\frac{2}{2}}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.5.4

Cancel the common factor of $2$ .

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Step 10.1.2.2.16.1.5.4.1

Cancel the common factor.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 6^{\frac{2}{2}}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.5.4.2

Rewrite the expression.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 6^{1}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.5.5

Evaluate the exponent.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 6}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.1.6

Multiply $4$ by $6$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{9 - 6 \sqrt{6} - 6 \sqrt{6} + 24}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.2

Add $9$ and $24$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{33 - 6 \sqrt{6} - 6 \sqrt{6}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.16.3

Subtract $6 \sqrt{6}$ from $- 6 \sqrt{6}$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{33 - 12 \sqrt{6}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.17

Cancel the common factor of $33 - 12 \sqrt{6}$ and $3$ .

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Step 10.1.2.2.17.1

Factor $3$ out of $33$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{3 \cdot 11 - 12 \sqrt{6}}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.17.2

Factor $3$ out of $- 12 \sqrt{6}$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{3 \cdot 11 + 3 (- 4 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.17.3

Factor $3$ out of $3 (11) + 3 (- 4 \sqrt{6})$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{3 (11 - 4 \sqrt{6})}{3} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.17.4

Cancel the common factors.

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Step 10.1.2.2.17.4.1

Factor $3$ out of $3$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{3 (11 - 4 \sqrt{6})}{3 (1)} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.17.4.2

Cancel the common factor.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{3 (11 - 4 \sqrt{6})}{3 \cdot 1} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.17.4.3

Rewrite the expression.

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{11 - 4 \sqrt{6}}{1} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.17.4.4

Divide $11 - 4 \sqrt{6}$ by $1$ .

$- \frac{81 - 34 \sqrt{6}}{9} + 11 - 4 \sqrt{6} - 5 (- \frac{3 - 2 \sqrt{6}}{3}) - 6$

Step 10.1.2.2.18

Multiply $- 5 (- \frac{3 - 2 \sqrt{6}}{3})$ .

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Step 10.1.2.2.18.1

Multiply $- 1$ by $- 5$ .

$- \frac{81 - 34 \sqrt{6}}{9} + 11 - 4 \sqrt{6} + 5 \frac{3 - 2 \sqrt{6}}{3} - 6$

Step 10.1.2.2.18.2

Combine $5$ and $\frac{3 - 2 \sqrt{6}}{3}$ .

$- \frac{81 - 34 \sqrt{6}}{9} + 11 - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.3

To write $11$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{81 - 34 \sqrt{6}}{9} + 11 \cdot \frac{9}{9} - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.4

Combine fractions.

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Step 10.1.2.4.1

Combine $11$ and $\frac{9}{9}$ .

$- \frac{81 - 34 \sqrt{6}}{9} + \frac{11 \cdot 9}{9} - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.4.2

Simplify the expression.

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Step 10.1.2.4.2.1

Combine the numerators over the common denominator.

$\frac{- (81 - 34 \sqrt{6}) + 11 \cdot 9}{9} - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.4.2.2

Multiply $11$ by $9$ .

$\frac{- (81 - 34 \sqrt{6}) + 99}{9} - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.5

Simplify the numerator.

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Step 10.1.2.5.1

Apply the distributive property.

$\frac{- 1 \cdot 81 - (- 34 \sqrt{6}) + 99}{9} - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.5.2

Multiply $- 1$ by $81$ .

$\frac{- 81 - (- 34 \sqrt{6}) + 99}{9} - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.5.3

Multiply $- 34$ by $- 1$ .

$\frac{- 81 + 34 \sqrt{6} + 99}{9} - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.5.4

Add $- 81$ and $99$ .

$\frac{18 + 34 \sqrt{6}}{9} - 4 \sqrt{6} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.6

Find the common denominator.

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Step 10.1.2.6.1

Write $- 4 \sqrt{6}$ as a fraction with denominator $1$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6}}{1} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.6.2

Multiply $\frac{- 4 \sqrt{6}}{1}$ by $\frac{9}{9}$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6}}{1} \cdot \frac{9}{9} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.6.3

Multiply $\frac{- 4 \sqrt{6}}{1}$ by $\frac{9}{9}$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 - 2 \sqrt{6})}{3} - 6$

Step 10.1.2.6.4

Multiply $\frac{5 (3 - 2 \sqrt{6})}{3}$ by $\frac{3}{3}$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 - 2 \sqrt{6})}{3} \cdot \frac{3}{3} - 6$

Step 10.1.2.6.5

Multiply $\frac{5 (3 - 2 \sqrt{6})}{3}$ by $\frac{3}{3}$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 - 2 \sqrt{6}) \cdot 3}{3 \cdot 3} - 6$

Step 10.1.2.6.6

Write $- 6$ as a fraction with denominator $1$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 - 2 \sqrt{6}) \cdot 3}{3 \cdot 3} + \frac{- 6}{1}$

Step 10.1.2.6.7

Multiply $\frac{- 6}{1}$ by $\frac{9}{9}$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 - 2 \sqrt{6}) \cdot 3}{3 \cdot 3} + \frac{- 6}{1} \cdot \frac{9}{9}$

Step 10.1.2.6.8

Multiply $\frac{- 6}{1}$ by $\frac{9}{9}$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 - 2 \sqrt{6}) \cdot 3}{3 \cdot 3} + \frac{- 6 \cdot 9}{9}$

Step 10.1.2.6.9

Multiply $3$ by $3$ .

$\frac{18 + 34 \sqrt{6}}{9} + \frac{- 4 \sqrt{6} \cdot 9}{9} + \frac{5 (3 - 2 \sqrt{6}) \cdot 3}{9} + \frac{- 6 \cdot 9}{9}$

Step 10.1.2.7

Combine the numerators over the common denominator.

$\frac{18 + 34 \sqrt{6} - 4 \sqrt{6} \cdot 9 + 5 (3 - 2 \sqrt{6}) \cdot 3 - 6 \cdot 9}{9}$

Step 10.1.2.8

Simplify each term.

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Step 10.1.2.8.1

Multiply $9$ by $- 4$ .

$\frac{18 + 34 \sqrt{6} - 36 \sqrt{6} + 5 (3 - 2 \sqrt{6}) \cdot 3 - 6 \cdot 9}{9}$

Step 10.1.2.8.2

Apply the distributive property.

$\frac{18 + 34 \sqrt{6} - 36 \sqrt{6} + (5 \cdot 3 + 5 (- 2 \sqrt{6})) \cdot 3 - 6 \cdot 9}{9}$

Step 10.1.2.8.3

Multiply $5$ by $3$ .

$\frac{18 + 34 \sqrt{6} - 36 \sqrt{6} + (15 + 5 (- 2 \sqrt{6})) \cdot 3 - 6 \cdot 9}{9}$

Step 10.1.2.8.4

Multiply $- 2$ by $5$ .

$\frac{18 + 34 \sqrt{6} - 36 \sqrt{6} + (15 - 10 \sqrt{6}) \cdot 3 - 6 \cdot 9}{9}$

Step 10.1.2.8.5

Apply the distributive property.

$\frac{18 + 34 \sqrt{6} - 36 \sqrt{6} + 15 \cdot 3 - 10 \sqrt{6} \cdot 3 - 6 \cdot 9}{9}$

Step 10.1.2.8.6

Multiply $15$ by $3$ .

$\frac{18 + 34 \sqrt{6} - 36 \sqrt{6} + 45 - 10 \sqrt{6} \cdot 3 - 6 \cdot 9}{9}$

Step 10.1.2.8.7

Multiply $3$ by $- 10$ .

$\frac{18 + 34 \sqrt{6} - 36 \sqrt{6} + 45 - 30 \sqrt{6} - 6 \cdot 9}{9}$

Step 10.1.2.8.8

Multiply $- 6$ by $9$ .

$\frac{18 + 34 \sqrt{6} - 36 \sqrt{6} + 45 - 30 \sqrt{6} - 54}{9}$

Step 10.1.2.9

Simplify by adding terms.

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Step 10.1.2.9.1

Add $18$ and $45$ .

$\frac{63 + 34 \sqrt{6} - 36 \sqrt{6} - 30 \sqrt{6} - 54}{9}$

Step 10.1.2.9.2

Subtract $54$ from $63$ .

$\frac{9 + 34 \sqrt{6} - 36 \sqrt{6} - 30 \sqrt{6}}{9}$

Step 10.1.2.9.3

Subtract $36 \sqrt{6}$ from $34 \sqrt{6}$ .

$\frac{9 - 2 \sqrt{6} - 30 \sqrt{6}}{9}$

Step 10.1.2.9.4

Subtract $30 \sqrt{6}$ from $- 2 \sqrt{6}$ .

$\frac{9 - 32 \sqrt{6}}{9}$

Step 10.2

Write the $x$ and $y$ coordinates in point form.

$(- \frac{3 - 2 \sqrt{6}}{3}, \frac{9 - 32 \sqrt{6}}{9})$

Step 11

These are the turning points.

$(- \frac{3 + 2 \sqrt{6}}{3}, \frac{9 + 32 \sqrt{6}}{9})$

$(- \frac{3 - 2 \sqrt{6}}{3}, \frac{9 - 32 \sqrt{6}}{9})$

Step 12

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