Calculus Examples

$f (x) = x^{4} - 8 x^{3} + 6 x^{2} + 9 x - 5$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{4} - 8 x^{3} + 6 x^{2} + 9 x - 5$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 8 x^{3}] + \frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [- 5]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 8 x^{3}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 5)$

Step 1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 8 x^{3}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 5)$

Step 1.1.2

Evaluate $\frac{d}{d x} [- 8 x^{3}]$ .

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Step 1.1.2.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 4 x^{3} - 8 \frac{d}{d x} x^{3} + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 5)$

Step 1.1.2.2

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

$f' (x) = 4 x^{3} - 8 (3 x^{2}) + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 5)$

Step 1.1.2.3

Multiply $3$ by $- 8$ .

$f' (x) = 4 x^{3} - 24 x^{2} + \frac{d}{d x} (6 x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 5)$

Step 1.1.3

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 1.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 6 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 5)$

Step 1.1.3.2

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

$f' (x) = 4 x^{3} - 24 x^{2} + 6 (2 x) + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 5)$

Step 1.1.3.3

Multiply $2$ by $6$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 12 x + \frac{d}{d x} (9 x) + \frac{d}{d x} (- 5)$

Step 1.1.4

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 1.1.4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 12 x + 9 \frac{d}{d x} (x) + \frac{d}{d x} (- 5)$

Step 1.1.4.2

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

$f' (x) = 4 x^{3} - 24 x^{2} + 12 x + 9 \cdot 1 + \frac{d}{d x} (- 5)$

Step 1.1.4.3

Multiply $9$ by $1$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 12 x + 9 + \frac{d}{d x} (- 5)$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 12 x + 9 + 0$

Step 1.1.5.2

Add $4 x^{3} - 24 x^{2} + 12 x + 9$ and $0$ .

$f' (x) = 4 x^{3} - 24 x^{2} + 12 x + 9$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $4 x^{3} - 24 x^{2} + 12 x + 9$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 24 x^{2}] + \frac{d}{d x} [12 x] + \frac{d}{d x} [9]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (9)$

Step 1.2.2

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (9)$

Step 1.2.2.2

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

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (9)$

Step 1.2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 24 x^{2}) + \frac{d}{d x} (12 x) + \frac{d}{d x} (9)$

Step 1.2.3

Evaluate $\frac{d}{d x} [- 24 x^{2}]$ .

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Step 1.2.3.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x^{2}$ with respect to $x$ is $- 24 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 x^{2} - 24 \frac{d}{d x} x^{2} + \frac{d}{d x} (12 x) + \frac{d}{d x} (9)$

Step 1.2.3.2

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

$f'' (x) = 12 x^{2} - 24 (2 x) + \frac{d}{d x} (12 x) + \frac{d}{d x} (9)$

Step 1.2.3.3

Multiply $2$ by $- 24$ .

$f'' (x) = 12 x^{2} - 48 x + \frac{d}{d x} (12 x) + \frac{d}{d x} (9)$

Step 1.2.4

Evaluate $\frac{d}{d x} [12 x]$ .

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Step 1.2.4.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x$ with respect to $x$ is $12 \frac{d}{d x} [x]$ .

$f'' (x) = 12 x^{2} - 48 x + 12 \frac{d}{d x} (x) + \frac{d}{d x} (9)$

Step 1.2.4.2

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

$f'' (x) = 12 x^{2} - 48 x + 12 \cdot 1 + \frac{d}{d x} (9)$

Step 1.2.4.3

Multiply $12$ by $1$ .

$f'' (x) = 12 x^{2} - 48 x + 12 + \frac{d}{d x} (9)$

Step 1.2.5

Differentiate using the Constant Rule.

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Step 1.2.5.1

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$f'' (x) = 12 x^{2} - 48 x + 12 + 0$

Step 1.2.5.2

Add $12 x^{2} - 48 x + 12$ and $0$ .

$f'' (x) = 12 x^{2} - 48 x + 12$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 48 x + 12$ .

$12 x^{2} - 48 x + 12$

Step 2

Set the second derivative equal to $0$ then solve the equation $12 x^{2} - 48 x + 12 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$12 x^{2} - 48 x + 12 = 0$

Step 2.2

Factor $12$ out of $12 x^{2} - 48 x + 12$ .

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Step 2.2.1

Factor $12$ out of $12 x^{2}$ .

$12 (x^{2}) - 48 x + 12 = 0$

Step 2.2.2

Factor $12$ out of $- 48 x$ .

$12 (x^{2}) + 12 (- 4 x) + 12 = 0$

Step 2.2.3

Factor $12$ out of $12$ .

$12 (x^{2}) + 12 (- 4 x) + 12 (1) = 0$

Step 2.2.4

Factor $12$ out of $12 (x^{2}) + 12 (- 4 x)$ .

$12 (x^{2} - 4 x) + 12 (1) = 0$

Step 2.2.5

Factor $12$ out of $12 (x^{2} - 4 x) + 12 (1)$ .

$12 (x^{2} - 4 x + 1) = 0$

Step 2.3

Divide each term in $12 (x^{2} - 4 x + 1) = 0$ by $12$ and simplify.

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Step 2.3.1

Divide each term in $12 (x^{2} - 4 x + 1) = 0$ by $12$ .

$\frac{12 (x^{2} - 4 x + 1)}{12} = \frac{0}{12}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{12 (x^{2} - 4 x + 1)}{12} = \frac{0}{12}$

Step 2.3.2.1.2

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

$x^{2} - 4 x + 1 = \frac{0}{12}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $12$ .

$x^{2} - 4 x + 1 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

Substitute the values $a = 1$ , $b = - 4$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 2.6.1.3

Subtract $4$ from $16$ .

$x = \frac{4 \pm \sqrt{12}}{2 \cdot 1}$

Step 2.6.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.6.1.4.1

Factor $4$ out of $12$ .

$x = \frac{4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.6.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

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

Step 2.6.3

Simplify $\frac{4 \pm 2 \sqrt{3}}{2}$ .

$x = 2 \pm \sqrt{3}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 2.7.1.3

Subtract $4$ from $16$ .

$x = \frac{4 \pm \sqrt{12}}{2 \cdot 1}$

Step 2.7.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.7.1.4.1

Factor $4$ out of $12$ .

$x = \frac{4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.7.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

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

Step 2.7.3

Simplify $\frac{4 \pm 2 \sqrt{3}}{2}$ .

$x = 2 \pm \sqrt{3}$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{3}$

Step 2.8

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

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.8.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Step 2.8.1.3

Subtract $4$ from $16$ .

$x = \frac{4 \pm \sqrt{12}}{2 \cdot 1}$

Step 2.8.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.8.1.4.1

Factor $4$ out of $12$ .

$x = \frac{4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.8.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 2.8.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Step 2.8.2

Multiply $2$ by $1$ .

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

Step 2.8.3

Simplify $\frac{4 \pm 2 \sqrt{3}}{2}$ .

$x = 2 \pm \sqrt{3}$

Step 2.8.4

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{3}$

Step 2.9

The final answer is the combination of both solutions.

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

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $2 + \sqrt{3}$ in $f (x) = x^{4} - 8 x^{3} + 6 x^{2} + 9 x - 5$ to find the value of $y$ .

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Step 3.1.1

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

$f (2 + \sqrt{3}) = {(2 + \sqrt{3})}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Use the Binomial Theorem.

$f (2 + \sqrt{3}) = 2^{4} + 4 \cdot (2^{3} \sqrt{3}) + 6 \cdot (2^{2} {\sqrt{3}}^{2}) + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2

Simplify each term.

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Step 3.1.2.1.2.1

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

$f (2 + \sqrt{3}) = 16 + 4 \cdot (2^{3} \sqrt{3}) + 6 \cdot (2^{2} {\sqrt{3}}^{2}) + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.2

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

$f (2 + \sqrt{3}) = 16 + 4 \cdot (8 \sqrt{3}) + 6 \cdot (2^{2} {\sqrt{3}}^{2}) + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.3

Multiply $4$ by $8$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 6 \cdot (2^{2} {\sqrt{3}}^{2}) + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.4

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

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 6 \cdot (4 {\sqrt{3}}^{2}) + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.5

Multiply $6$ by $4$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 24 {\sqrt{3}}^{2} + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.2.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 24 {(3^{\frac{1}{2}})}^{2} + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.6.2

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

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 24 \cdot 3^{\frac{1}{2} \cdot 2} + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.6.3

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

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 24 \cdot 3^{\frac{2}{2}} + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.2.6.4.1

Cancel the common factor.

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 24 \cdot 3^{\frac{2}{2}} + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.6.4.2

Rewrite the expression.

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 24 \cdot 3 + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.6.5

Evaluate the exponent.

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 24 \cdot 3 + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.7

Multiply $24$ by $3$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 4 \cdot (2 {\sqrt{3}}^{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.8

Multiply $4$ by $2$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 8 {\sqrt{3}}^{3} + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.9

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

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 8 \sqrt{3^{3}} + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.10

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

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 8 \sqrt{27} + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.11

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.1.2.1.2.11.1

Factor $9$ out of $27$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 8 \sqrt{9 (3)} + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.11.2

Rewrite $9$ as $3^{2}$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 8 \sqrt{3^{2} \cdot 3} + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.12

Pull terms out from under the radical.

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 8 (3 \sqrt{3}) + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.13

Multiply $3$ by $8$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + {\sqrt{3}}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.14

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 3.1.2.1.2.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + {(3^{\frac{1}{2}})}^{4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.14.2

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

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + 3^{\frac{1}{2} \cdot 4} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.14.3

Combine $\frac{1}{2}$ and $4$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + 3^{\frac{4}{2}} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.14.4

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

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Step 3.1.2.1.2.14.4.1

Factor $2$ out of $4$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + 3^{\frac{2 \cdot 2}{2}} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.14.4.2

Cancel the common factors.

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Step 3.1.2.1.2.14.4.2.1

Factor $2$ out of $2$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.14.4.2.2

Cancel the common factor.

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.14.4.2.3

Rewrite the expression.

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + 3^{\frac{2}{1}} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.14.4.2.4

Divide $2$ by $1$ .

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + 3^{2} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.2.15

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

$f (2 + \sqrt{3}) = 16 + 32 \sqrt{3} + 72 + 24 \sqrt{3} + 9 - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.3

Add $16$ and $72$ .

$f (2 + \sqrt{3}) = 88 + 32 \sqrt{3} + 24 \sqrt{3} + 9 - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.4

Add $88$ and $9$ .

$f (2 + \sqrt{3}) = 97 + 32 \sqrt{3} + 24 \sqrt{3} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.5

Add $32 \sqrt{3}$ and $24 \sqrt{3}$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 {(2 + \sqrt{3})}^{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.6

Use the Binomial Theorem.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (2^{3} + 3 \cdot (2^{2} \sqrt{3}) + 3 \cdot (2 {\sqrt{3}}^{2}) + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7

Simplify each term.

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Step 3.1.2.1.7.1

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

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 3 \cdot (2^{2} \sqrt{3}) + 3 \cdot (2 {\sqrt{3}}^{2}) + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.2

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

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 3 \cdot (4 \sqrt{3}) + 3 \cdot (2 {\sqrt{3}}^{2}) + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.3

Multiply $3$ by $4$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 3 \cdot (2 {\sqrt{3}}^{2}) + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.4

Multiply $3$ by $2$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 6 {\sqrt{3}}^{2} + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.1.2.1.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 6 {(3^{\frac{1}{2}})}^{2} + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.5.2

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

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 6 \cdot 3^{\frac{1}{2} \cdot 2} + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.5.3

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

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.5.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.7.5.4.1

Cancel the common factor.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.5.4.2

Rewrite the expression.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 6 \cdot 3 + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.5.5

Evaluate the exponent.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 6 \cdot 3 + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.6

Multiply $6$ by $3$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 18 + {\sqrt{3}}^{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.7

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

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 18 + \sqrt{3^{3}}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.8

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

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 18 + \sqrt{27}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.9

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.1.2.1.7.9.1

Factor $9$ out of $27$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 18 + \sqrt{9 (3)}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.9.2

Rewrite $9$ as $3^{2}$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 18 + \sqrt{3^{2} \cdot 3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.7.10

Pull terms out from under the radical.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (8 + 12 \sqrt{3} + 18 + 3 \sqrt{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.8

Add $8$ and $18$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (26 + 12 \sqrt{3} + 3 \sqrt{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.9

Add $12 \sqrt{3}$ and $3 \sqrt{3}$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 (26 + 15 \sqrt{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.10

Apply the distributive property.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 8 \cdot 26 - 8 (15 \sqrt{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.11

Multiply $- 8$ by $26$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 8 (15 \sqrt{3}) + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.12

Multiply $15$ by $- 8$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 {(2 + \sqrt{3})}^{2} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.13

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

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 ((2 + \sqrt{3}) (2 + \sqrt{3})) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.14

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

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Step 3.1.2.1.14.1

Apply the distributive property.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (2 (2 + \sqrt{3}) + \sqrt{3} (2 + \sqrt{3})) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.14.2

Apply the distributive property.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (2 \cdot 2 + 2 \sqrt{3} + \sqrt{3} (2 + \sqrt{3})) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.14.3

Apply the distributive property.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (2 \cdot 2 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.15

Simplify and combine like terms.

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Step 3.1.2.1.15.1

Simplify each term.

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Step 3.1.2.1.15.1.1

Multiply $2$ by $2$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (4 + 2 \sqrt{3} + \sqrt{3} \cdot 2 + \sqrt{3} \sqrt{3}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.15.1.2

Move $2$ to the left of $\sqrt{3}$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (4 + 2 \sqrt{3} + 2 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.15.1.3

Combine using the product rule for radicals.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3 \cdot 3}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.15.1.4

Multiply $3$ by $3$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{9}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.15.1.5

Rewrite $9$ as $3^{2}$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (4 + 2 \sqrt{3} + 2 \sqrt{3} + \sqrt{3^{2}}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.15.1.6

Pull terms out from under the radical, assuming positive real numbers.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (4 + 2 \sqrt{3} + 2 \sqrt{3} + 3) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.15.2

Add $4$ and $3$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (7 + 2 \sqrt{3} + 2 \sqrt{3}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.15.3

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 (7 + 4 \sqrt{3}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.16

Apply the distributive property.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 6 \cdot 7 + 6 (4 \sqrt{3}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.17

Multiply $6$ by $7$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 42 + 6 (4 \sqrt{3}) + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.18

Multiply $4$ by $6$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 42 + 24 \sqrt{3} + 9 (2 + \sqrt{3}) - 5$

Step 3.1.2.1.19

Apply the distributive property.

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 42 + 24 \sqrt{3} + 9 \cdot 2 + 9 \sqrt{3} - 5$

Step 3.1.2.1.20

Multiply $9$ by $2$ .

$f (2 + \sqrt{3}) = 97 + 56 \sqrt{3} - 208 - 120 \sqrt{3} + 42 + 24 \sqrt{3} + 18 + 9 \sqrt{3} - 5$

Step 3.1.2.2

Simplify by adding terms.

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Step 3.1.2.2.1

Subtract $208$ from $97$ .

$f (2 + \sqrt{3}) = - 111 + 56 \sqrt{3} - 120 \sqrt{3} + 42 + 24 \sqrt{3} + 18 + 9 \sqrt{3} - 5$

Step 3.1.2.2.2

Simplify by adding and subtracting.

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Step 3.1.2.2.2.1

Add $- 111$ and $42$ .

$f (2 + \sqrt{3}) = - 69 + 56 \sqrt{3} - 120 \sqrt{3} + 24 \sqrt{3} + 18 + 9 \sqrt{3} - 5$

Step 3.1.2.2.2.2

Add $- 69$ and $18$ .

$f (2 + \sqrt{3}) = - 51 + 56 \sqrt{3} - 120 \sqrt{3} + 24 \sqrt{3} + 9 \sqrt{3} - 5$

Step 3.1.2.2.2.3

Subtract $5$ from $- 51$ .

$f (2 + \sqrt{3}) = - 56 + 56 \sqrt{3} - 120 \sqrt{3} + 24 \sqrt{3} + 9 \sqrt{3}$

Step 3.1.2.2.3

Subtract $120 \sqrt{3}$ from $56 \sqrt{3}$ .

$f (2 + \sqrt{3}) = - 56 - 64 \sqrt{3} + 24 \sqrt{3} + 9 \sqrt{3}$

Step 3.1.2.2.4

Add $- 64 \sqrt{3}$ and $24 \sqrt{3}$ .

$f (2 + \sqrt{3}) = - 56 - 40 \sqrt{3} + 9 \sqrt{3}$

Step 3.1.2.2.5

Add $- 40 \sqrt{3}$ and $9 \sqrt{3}$ .

$f (2 + \sqrt{3}) = - 56 - 31 \sqrt{3}$

Step 3.1.2.3

The final answer is $- 56 - 31 \sqrt{3}$ .

$- 56 - 31 \sqrt{3}$

Step 3.2

The point found by substituting $2 + \sqrt{3}$ in $f (x) = x^{4} - 8 x^{3} + 6 x^{2} + 9 x - 5$ is $(2 + \sqrt{3}, - 56 - 31 \sqrt{3})$ . This point can be an inflection point.

$(2 + \sqrt{3}, - 56 - 31 \sqrt{3})$

Step 3.3

Substitute $2 - \sqrt{3}$ in $f (x) = x^{4} - 8 x^{3} + 6 x^{2} + 9 x - 5$ to find the value of $y$ .

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Step 3.3.1

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

$f (2 - \sqrt{3}) = {(2 - \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Use the Binomial Theorem.

$f (2 - \sqrt{3}) = 2^{4} + 4 \cdot (2^{3} (- \sqrt{3})) + 6 \cdot (2^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2

Simplify each term.

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Step 3.3.2.1.2.1

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

$f (2 - \sqrt{3}) = 16 + 4 \cdot (2^{3} (- \sqrt{3})) + 6 \cdot (2^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.2

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

$f (2 - \sqrt{3}) = 16 + 4 \cdot (8 (- \sqrt{3})) + 6 \cdot (2^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.3

Multiply $4$ by $8$ .

$f (2 - \sqrt{3}) = 16 + 32 (- \sqrt{3}) + 6 \cdot (2^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.4

Multiply $- 1$ by $32$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 6 \cdot (2^{2} {(- \sqrt{3})}^{2}) + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.5

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 6 \cdot (4 {(- \sqrt{3})}^{2}) + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.6

Multiply $6$ by $4$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 {(- \sqrt{3})}^{2} + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.7

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 ({(- 1)}^{2} {\sqrt{3}}^{2}) + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.8

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 (1 {\sqrt{3}}^{2}) + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.9

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 {\sqrt{3}}^{2} + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.10

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.2.1.2.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 {(3^{\frac{1}{2}})}^{2} + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.10.2

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 \cdot 3^{\frac{1}{2} \cdot 2} + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.10.3

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 \cdot 3^{\frac{2}{2}} + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.10.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.10.4.1

Cancel the common factor.

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 \cdot 3^{\frac{2}{2}} + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.10.4.2

Rewrite the expression.

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 \cdot 3 + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.10.5

Evaluate the exponent.

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 24 \cdot 3 + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.11

Multiply $24$ by $3$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 4 \cdot (2 {(- \sqrt{3})}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.12

Multiply $4$ by $2$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 {(- \sqrt{3})}^{3} + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.13

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 ({(- 1)}^{3} {\sqrt{3}}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.14

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 (- {\sqrt{3}}^{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.15

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 (- \sqrt{3^{3}}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.16

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 (- \sqrt{27}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.17

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.3.2.1.2.17.1

Factor $9$ out of $27$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 (- \sqrt{9 (3)}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.17.2

Rewrite $9$ as $3^{2}$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 (- \sqrt{3^{2} \cdot 3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.18

Pull terms out from under the radical.

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 (- (3 \sqrt{3})) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.19

Multiply $3$ by $- 1$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 + 8 (- 3 \sqrt{3}) + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.20

Multiply $- 3$ by $8$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + {(- \sqrt{3})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.21

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + {(- 1)}^{4} {\sqrt{3}}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.22

Raise $- 1$ to the power of $4$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 1 {\sqrt{3}}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.23

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + {\sqrt{3}}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.24

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 3.3.2.1.2.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + {(3^{\frac{1}{2}})}^{4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.24.2

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 3^{\frac{1}{2} \cdot 4} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.24.3

Combine $\frac{1}{2}$ and $4$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 3^{\frac{4}{2}} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.24.4

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

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Step 3.3.2.1.2.24.4.1

Factor $2$ out of $4$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 3^{\frac{2 \cdot 2}{2}} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.24.4.2

Cancel the common factors.

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Step 3.3.2.1.2.24.4.2.1

Factor $2$ out of $2$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 (1)}} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.24.4.2.2

Cancel the common factor.

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 3^{\frac{2 \cdot 2}{2 \cdot 1}} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.24.4.2.3

Rewrite the expression.

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 3^{\frac{2}{1}} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.24.4.2.4

Divide $2$ by $1$ .

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 3^{2} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.2.25

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

$f (2 - \sqrt{3}) = 16 - 32 \sqrt{3} + 72 - 24 \sqrt{3} + 9 - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.3

Add $16$ and $72$ .

$f (2 - \sqrt{3}) = 88 - 32 \sqrt{3} - 24 \sqrt{3} + 9 - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.4

Add $88$ and $9$ .

$f (2 - \sqrt{3}) = 97 - 32 \sqrt{3} - 24 \sqrt{3} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.5

Subtract $24 \sqrt{3}$ from $- 32 \sqrt{3}$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 {(2 - \sqrt{3})}^{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.6

Use the Binomial Theorem.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (2^{3} + 3 \cdot (2^{2} (- \sqrt{3})) + 3 \cdot (2 {(- \sqrt{3})}^{2}) + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7

Simplify each term.

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Step 3.3.2.1.7.1

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 + 3 \cdot (2^{2} (- \sqrt{3})) + 3 \cdot (2 {(- \sqrt{3})}^{2}) + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.2

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 + 3 \cdot (4 (- \sqrt{3})) + 3 \cdot (2 {(- \sqrt{3})}^{2}) + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.3

Multiply $3$ by $4$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 + 12 (- \sqrt{3}) + 3 \cdot (2 {(- \sqrt{3})}^{2}) + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.4

Multiply $- 1$ by $12$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 3 \cdot (2 {(- \sqrt{3})}^{2}) + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.5

Multiply $3$ by $2$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 {(- \sqrt{3})}^{2} + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.6

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 ({(- 1)}^{2} {\sqrt{3}}^{2}) + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.7

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 (1 {\sqrt{3}}^{2}) + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.8

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 {\sqrt{3}}^{2} + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.9

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.2.1.7.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 {(3^{\frac{1}{2}})}^{2} + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.9.2

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 \cdot 3^{\frac{1}{2} \cdot 2} + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.9.3

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.9.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.7.9.4.1

Cancel the common factor.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 \cdot 3^{\frac{2}{2}} + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.9.4.2

Rewrite the expression.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 \cdot 3 + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.9.5

Evaluate the exponent.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 6 \cdot 3 + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.10

Multiply $6$ by $3$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 + {(- \sqrt{3})}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.11

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 + {(- 1)}^{3} {\sqrt{3}}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.12

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 - {\sqrt{3}}^{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.13

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 - \sqrt{3^{3}}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.14

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 - \sqrt{27}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.15

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 3.3.2.1.7.15.1

Factor $9$ out of $27$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 - \sqrt{9 (3)}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.15.2

Rewrite $9$ as $3^{2}$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 - \sqrt{3^{2} \cdot 3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.16

Pull terms out from under the radical.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 - (3 \sqrt{3})) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.7.17

Multiply $3$ by $- 1$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (8 - 12 \sqrt{3} + 18 - 3 \sqrt{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.8

Add $8$ and $18$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (26 - 12 \sqrt{3} - 3 \sqrt{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.9

Subtract $3 \sqrt{3}$ from $- 12 \sqrt{3}$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 (26 - 15 \sqrt{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.10

Apply the distributive property.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 8 \cdot 26 - 8 (- 15 \sqrt{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.11

Multiply $- 8$ by $26$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 - 8 (- 15 \sqrt{3}) + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.12

Multiply $- 15$ by $- 8$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 {(2 - \sqrt{3})}^{2} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.13

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 ((2 - \sqrt{3}) (2 - \sqrt{3})) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.14

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

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Step 3.3.2.1.14.1

Apply the distributive property.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (2 (2 - \sqrt{3}) - \sqrt{3} (2 - \sqrt{3})) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.14.2

Apply the distributive property.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (2 \cdot 2 + 2 (- \sqrt{3}) - \sqrt{3} (2 - \sqrt{3})) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.14.3

Apply the distributive property.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (2 \cdot 2 + 2 (- \sqrt{3}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3})) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15

Simplify and combine like terms.

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Step 3.3.2.1.15.1

Simplify each term.

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Step 3.3.2.1.15.1.1

Multiply $2$ by $2$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 + 2 (- \sqrt{3}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3})) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.2

Multiply $- 1$ by $2$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3})) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.3

Multiply $2$ by $- 1$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{3} (- \sqrt{3})) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 3.3.2.1.15.1.4.1

Multiply $- 1$ by $- 1$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{3} \sqrt{3}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.4.2

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{3} \sqrt{3}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{3} \sqrt{3}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.4.4

Raise $\sqrt{3}$ to the power of $1$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{3} \sqrt{3}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.4.5

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1 + 1}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.4.6

Add $1$ and $1$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{2}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.2.1.15.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.5.2

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.5.3

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

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{2}{2}}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.5.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.15.1.5.4.1

Cancel the common factor.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{2}{2}}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.5.4.2

Rewrite the expression.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.1.5.5

Evaluate the exponent.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.2

Add $4$ and $3$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (7 - 2 \sqrt{3} - 2 \sqrt{3}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.15.3

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 (7 - 4 \sqrt{3}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.16

Apply the distributive property.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 6 \cdot 7 + 6 (- 4 \sqrt{3}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.17

Multiply $6$ by $7$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 42 + 6 (- 4 \sqrt{3}) + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.18

Multiply $- 4$ by $6$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 42 - 24 \sqrt{3} + 9 (2 - \sqrt{3}) - 5$

Step 3.3.2.1.19

Apply the distributive property.

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 42 - 24 \sqrt{3} + 9 \cdot 2 + 9 (- \sqrt{3}) - 5$

Step 3.3.2.1.20

Multiply $9$ by $2$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 42 - 24 \sqrt{3} + 18 + 9 (- \sqrt{3}) - 5$

Step 3.3.2.1.21

Multiply $- 1$ by $9$ .

$f (2 - \sqrt{3}) = 97 - 56 \sqrt{3} - 208 + 120 \sqrt{3} + 42 - 24 \sqrt{3} + 18 - 9 \sqrt{3} - 5$

Step 3.3.2.2

Simplify by adding terms.

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Step 3.3.2.2.1

Subtract $208$ from $97$ .

$f (2 - \sqrt{3}) = - 111 - 56 \sqrt{3} + 120 \sqrt{3} + 42 - 24 \sqrt{3} + 18 - 9 \sqrt{3} - 5$

Step 3.3.2.2.2

Simplify by adding and subtracting.

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Step 3.3.2.2.2.1

Add $- 111$ and $42$ .

$f (2 - \sqrt{3}) = - 69 - 56 \sqrt{3} + 120 \sqrt{3} - 24 \sqrt{3} + 18 - 9 \sqrt{3} - 5$

Step 3.3.2.2.2.2

Add $- 69$ and $18$ .

$f (2 - \sqrt{3}) = - 51 - 56 \sqrt{3} + 120 \sqrt{3} - 24 \sqrt{3} - 9 \sqrt{3} - 5$

Step 3.3.2.2.2.3

Subtract $5$ from $- 51$ .

$f (2 - \sqrt{3}) = - 56 - 56 \sqrt{3} + 120 \sqrt{3} - 24 \sqrt{3} - 9 \sqrt{3}$

Step 3.3.2.2.3

Add $- 56 \sqrt{3}$ and $120 \sqrt{3}$ .

$f (2 - \sqrt{3}) = - 56 + 64 \sqrt{3} - 24 \sqrt{3} - 9 \sqrt{3}$

Step 3.3.2.2.4

Subtract $24 \sqrt{3}$ from $64 \sqrt{3}$ .

$f (2 - \sqrt{3}) = - 56 + 40 \sqrt{3} - 9 \sqrt{3}$

Step 3.3.2.2.5

Subtract $9 \sqrt{3}$ from $40 \sqrt{3}$ .

$f (2 - \sqrt{3}) = - 56 + 31 \sqrt{3}$

Step 3.3.2.3

The final answer is $- 56 + 31 \sqrt{3}$ .

$- 56 + 31 \sqrt{3}$

Step 3.4

The point found by substituting $2 - \sqrt{3}$ in $f (x) = x^{4} - 8 x^{3} + 6 x^{2} + 9 x - 5$ is $(2 - \sqrt{3}, - 56 + 31 \sqrt{3})$ . This point can be an inflection point.

$(2 - \sqrt{3}, - 56 + 31 \sqrt{3})$

Step 3.5

Determine the points that could be inflection points.

$(2 + \sqrt{3}, - 56 - 31 \sqrt{3}), (2 - \sqrt{3}, - 56 + 31 \sqrt{3})$

Step 4

Split $(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

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

Step 5

Substitute a value from the interval $(- \infty, 2 - \sqrt{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $0.16794919$ in the expression.

$f'' (0.16794919) = 12 {(0.16794919)}^{2} - 48 \cdot 0.16794919 + 12$

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

Raise $0.16794919$ to the power of $2$ .

$f'' (0.16794919) = 12 \cdot 0.02820693 - 48 \cdot 0.16794919 + 12$

Step 5.2.1.2

Multiply $12$ by $0.02820693$ .

$f'' (0.16794919) = 0.33848317 - 48 \cdot 0.16794919 + 12$

Step 5.2.1.3

Multiply $- 48$ by $0.16794919$ .

$f'' (0.16794919) = 0.33848317 - 8.06156123 + 12$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $8.06156123$ from $0.33848317$ .

$f'' (0.16794919) = - 7.72307806 + 12$

Step 5.2.2.2

Add $- 7.72307806$ and $12$ .

$f'' (0.16794919) = 4.27692193$

Step 5.2.3

The final answer is $4.27692193$ .

$4.27692193$

Step 5.3

At $0.16794919$ , the second derivative is $4.27692193$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, 2 - \sqrt{3})$ .

Increasing on $(- \infty, 2 - \sqrt{3})$ since $f'' (x) > 0$

Step 6

Substitute a value from the interval $(2 - \sqrt{3}, 2 + \sqrt{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $2$ in the expression.

$f'' (2) = 12 {(2)}^{2} - 48 \cdot 2 + 12$

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

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

$f'' (2) = 12 \cdot 4 - 48 \cdot 2 + 12$

Step 6.2.1.2

Multiply $12$ by $4$ .

$f'' (2) = 48 - 48 \cdot 2 + 12$

Step 6.2.1.3

Multiply $- 48$ by $2$ .

$f'' (2) = 48 - 96 + 12$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $96$ from $48$ .

$f'' (2) = - 48 + 12$

Step 6.2.2.2

Add $- 48$ and $12$ .

$f'' (2) = - 36$

Step 6.2.3

The final answer is $- 36$ .

$- 36$

Step 6.3

At $2$ , the second derivative is $- 36$ . Since this is negative, the second derivative is decreasing on the interval $(2 - \sqrt{3}, 2 + \sqrt{3})$

Decreasing on $(2 - \sqrt{3}, 2 + \sqrt{3})$ since $f'' (x) < 0$

Step 7

Substitute a value from the interval $(2 + \sqrt{3}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3.8320508$ in the expression.

$f'' (3.8320508) = 12 {(3.8320508)}^{2} - 48 \cdot 3.8320508 + 12$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $3.8320508$ to the power of $2$ .

$f'' (3.8320508) = 12 \cdot 14.68461339 - 48 \cdot 3.8320508 + 12$

Step 7.2.1.2

Multiply $12$ by $14.68461339$ .

$f'' (3.8320508) = 176.2153607 - 48 \cdot 3.8320508 + 12$

Step 7.2.1.3

Multiply $- 48$ by $3.8320508$ .

$f'' (3.8320508) = 176.2153607 - 183.93843876 + 12$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Subtract $183.93843876$ from $176.2153607$ .

$f'' (3.8320508) = - 7.72307806 + 12$

Step 7.2.2.2

Add $- 7.72307806$ and $12$ .

$f'' (3.8320508) = 4.27692193$

Step 7.2.3

The final answer is $4.27692193$ .

$4.27692193$

Step 7.3

At $3.8320508$ , the second derivative is $4.27692193$ . Since this is positive, the second derivative is increasing on the interval $(2 + \sqrt{3}, \infty)$ .

Increasing on $(2 + \sqrt{3}, \infty)$ since $f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are $(2 - \sqrt{3}, - 56 + 31 \sqrt{3}), (2 + \sqrt{3}, - 56 - 31 \sqrt{3})$ .

$(2 - \sqrt{3}, - 56 + 31 \sqrt{3}), (2 + \sqrt{3}, - 56 - 31 \sqrt{3})$

Step 9