Calculus Examples

$y = x^{3} - 4 x^{2} + x + 2$

Step 1

Set $y$ as a function of $x$ .

$f (x) = x^{3} - 4 x^{2} + x + 2$

Step 2

Find the derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.1.2

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

$3 x^{2} + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.2

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

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Step 2.2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $- 4$ .

$3 x^{2} - 8 x + \frac{d}{d x} [x] + \frac{d}{d x} [2]$

Step 2.3

Differentiate.

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Step 2.3.1

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

$3 x^{2} - 8 x + 1 + \frac{d}{d x} [2]$

Step 2.3.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 x^{2} - 8 x + 1 + 0$

Step 2.3.3

Add $3 x^{2} - 8 x + 1$ and $0$ .

$3 x^{2} - 8 x + 1$

Step 3

Set the derivative equal to $0$ then solve the equation $3 x^{2} - 8 x + 1 = 0$ .

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Step 3.1

Use the quadratic formula to find the solutions.

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

Step 3.2

Substitute the values $a = 3$ , $b = - 8$ , and $c = 1$ into the quadratic formula and solve for $x$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

Step 3.3.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 3.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 1}}{2 \cdot 3}$

Step 3.3.1.2.2

Multiply $- 12$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 12}}{2 \cdot 3}$

Step 3.3.1.3

Subtract $12$ from $64$ .

$x = \frac{8 \pm \sqrt{52}}{2 \cdot 3}$

Step 3.3.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 3.3.1.4.1

Factor $4$ out of $52$ .

$x = \frac{8 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 3.3.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 3.3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 3.3.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 2 \sqrt{13}}{6}$

Step 3.3.3

Simplify $\frac{8 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{4 \pm \sqrt{13}}{3}$

Step 3.4

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

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

Step 3.4.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 1}}{2 \cdot 3}$

Step 3.4.1.2.2

Multiply $- 12$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 12}}{2 \cdot 3}$

Step 3.4.1.3

Subtract $12$ from $64$ .

$x = \frac{8 \pm \sqrt{52}}{2 \cdot 3}$

Step 3.4.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 3.4.1.4.1

Factor $4$ out of $52$ .

$x = \frac{8 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 3.4.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 3.4.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 3.4.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 2 \sqrt{13}}{6}$

Step 3.4.3

Simplify $\frac{8 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{4 \pm \sqrt{13}}{3}$

Step 3.4.4

Change the $\pm$ to $+$ .

$x = \frac{4 + \sqrt{13}}{3}$

Step 3.5

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

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

Step 3.5.1.2

Multiply $- 4 \cdot 3 \cdot 1$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 1}}{2 \cdot 3}$

Step 3.5.1.2.2

Multiply $- 12$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 12}}{2 \cdot 3}$

Step 3.5.1.3

Subtract $12$ from $64$ .

$x = \frac{8 \pm \sqrt{52}}{2 \cdot 3}$

Step 3.5.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 3.5.1.4.1

Factor $4$ out of $52$ .

$x = \frac{8 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 3.5.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 3.5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 3.5.2

Multiply $2$ by $3$ .

$x = \frac{8 \pm 2 \sqrt{13}}{6}$

Step 3.5.3

Simplify $\frac{8 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{4 \pm \sqrt{13}}{3}$

Step 3.5.4

Change the $\pm$ to $-$ .

$x = \frac{4 - \sqrt{13}}{3}$

Step 3.6

The final answer is the combination of both solutions.

$x = \frac{4 + \sqrt{13}}{3}, \frac{4 - \sqrt{13}}{3}$

Step 4

Solve the original function $f (x) = x^{3} - 4 x^{2} + x + 2$ at $x = \frac{4 + \sqrt{13}}{3}$ .

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Step 4.1

Replace the variable $x$ with $\frac{4 + \sqrt{13}}{3}$ in the expression.

$f (\frac{4 + \sqrt{13}}{3}) = {(\frac{4 + \sqrt{13}}{3})}^{3} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} + \frac{4 + \sqrt{13}}{3} + 2$

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

$f (\frac{4 + \sqrt{13}}{3}) = {(\frac{4 + \sqrt{13}}{3})}^{3} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} + \frac{4 + \sqrt{13}}{3} + 2$

Step 4.2.2

Find the common denominator.

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Step 4.2.2.1

Write ${(\frac{4 + \sqrt{13}}{3})}^{3}$ as a fraction with denominator $1$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3}}{1} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} + \frac{4 + \sqrt{13}}{3} + 2$

Step 4.2.2.2

Multiply $\frac{{(\frac{4 + \sqrt{13}}{3})}^{3}}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3}}{1} \cdot \frac{3}{3} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} + \frac{4 + \sqrt{13}}{3} + 2$

Step 4.2.2.3

Multiply $\frac{{(\frac{4 + \sqrt{13}}{3})}^{3}}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3} \cdot 3}{3} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} + \frac{4 + \sqrt{13}}{3} + 2$

Step 4.2.2.4

Write $- 4 {(\frac{4 + \sqrt{13}}{3})}^{2}$ as a fraction with denominator $1$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 + \sqrt{13}}{3})}^{2}}{1} + \frac{4 + \sqrt{13}}{3} + 2$

Step 4.2.2.5

Multiply $\frac{- 4 {(\frac{4 + \sqrt{13}}{3})}^{2}}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 + \sqrt{13}}{3})}^{2}}{1} \cdot \frac{3}{3} + \frac{4 + \sqrt{13}}{3} + 2$

Step 4.2.2.6

Multiply $\frac{- 4 {(\frac{4 + \sqrt{13}}{3})}^{2}}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3}{3} + \frac{4 + \sqrt{13}}{3} + 2$

Step 4.2.2.7

Write $2$ as a fraction with denominator $1$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3}{3} + \frac{4 + \sqrt{13}}{3} + \frac{2}{1}$

Step 4.2.2.8

Multiply $\frac{2}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3}{3} + \frac{4 + \sqrt{13}}{3} + \frac{2}{1} \cdot \frac{3}{3}$

Step 4.2.2.9

Multiply $\frac{2}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3}{3} + \frac{4 + \sqrt{13}}{3} + \frac{2 \cdot 3}{3}$

Step 4.2.3

Combine the numerators over the common denominator.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{{(\frac{4 + \sqrt{13}}{3})}^{3} \cdot 3 - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Apply the product rule to $\frac{4 + \sqrt{13}}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{{(4 + \sqrt{13})}^{3}}{3^{3}} \cdot 3 - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.2

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{{(4 + \sqrt{13})}^{3}}{27} \cdot 3 - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.3

Cancel the common factor of $3$ .

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Step 4.2.4.3.1

Factor $3$ out of $27$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{{(4 + \sqrt{13})}^{3}}{3 (9)} \cdot 3 - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.3.2

Cancel the common factor.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{{(4 + \sqrt{13})}^{3}}{3 \cdot 9} \cdot 3 - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.3.3

Rewrite the expression.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{{(4 + \sqrt{13})}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.4

Use the Binomial Theorem.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{4^{3} + 3 \cdot (4^{2} \sqrt{13}) + 3 \cdot (4 {\sqrt{13}}^{2}) + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5

Simplify each term.

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Step 4.2.4.5.1

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 3 \cdot (4^{2} \sqrt{13}) + 3 \cdot (4 {\sqrt{13}}^{2}) + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.2

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 3 \cdot (16 \sqrt{13}) + 3 \cdot (4 {\sqrt{13}}^{2}) + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.3

Multiply $3$ by $16$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 3 \cdot (4 {\sqrt{13}}^{2}) + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.4

Multiply $3$ by $4$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 12 {\sqrt{13}}^{2} + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.5

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

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Step 4.2.4.5.5.1

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 12 {(13^{\frac{1}{2}})}^{2} + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.5.2

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 12 \cdot 13^{\frac{1}{2} \cdot 2} + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.5.3

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 12 \cdot 13^{\frac{2}{2}} + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.5.4

Cancel the common factor of $2$ .

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Step 4.2.4.5.5.4.1

Cancel the common factor.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 12 \cdot 13^{\frac{2}{2}} + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.5.4.2

Rewrite the expression.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 12 \cdot 13 + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.5.5

Evaluate the exponent.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 12 \cdot 13 + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.6

Multiply $12$ by $13$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 156 + {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.7

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 156 + \sqrt{13^{3}}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.8

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 156 + \sqrt{2197}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.9

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

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Step 4.2.4.5.9.1

Factor $169$ out of $2197$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 156 + \sqrt{169 (13)}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.9.2

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 156 + \sqrt{13^{2} \cdot 13}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.5.10

Pull terms out from under the radical.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{64 + 48 \sqrt{13} + 156 + 13 \sqrt{13}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.6

Add $64$ and $156$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 48 \sqrt{13} + 13 \sqrt{13}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.7

Add $48 \sqrt{13}$ and $13 \sqrt{13}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 {(\frac{4 + \sqrt{13}}{3})}^{2} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.8

Apply the product rule to $\frac{4 + \sqrt{13}}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{{(4 + \sqrt{13})}^{2}}{3^{2}} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.9

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{{(4 + \sqrt{13})}^{2}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.10

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{(4 + \sqrt{13}) (4 + \sqrt{13})}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.11

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

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Step 4.2.4.11.1

Apply the distributive property.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{4 (4 + \sqrt{13}) + \sqrt{13} (4 + \sqrt{13})}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.11.2

Apply the distributive property.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{4 \cdot 4 + 4 \sqrt{13} + \sqrt{13} (4 + \sqrt{13})}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.11.3

Apply the distributive property.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{4 \cdot 4 + 4 \sqrt{13} + \sqrt{13} \cdot 4 + \sqrt{13} \sqrt{13}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.12

Simplify and combine like terms.

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Step 4.2.4.12.1

Simplify each term.

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Step 4.2.4.12.1.1

Multiply $4$ by $4$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{16 + 4 \sqrt{13} + \sqrt{13} \cdot 4 + \sqrt{13} \sqrt{13}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.12.1.2

Move $4$ to the left of $\sqrt{13}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{16 + 4 \sqrt{13} + 4 \cdot \sqrt{13} + \sqrt{13} \sqrt{13}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.12.1.3

Combine using the product rule for radicals.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{16 + 4 \sqrt{13} + 4 \sqrt{13} + \sqrt{13 \cdot 13}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.12.1.4

Multiply $13$ by $13$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{16 + 4 \sqrt{13} + 4 \sqrt{13} + \sqrt{169}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.12.1.5

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{16 + 4 \sqrt{13} + 4 \sqrt{13} + \sqrt{13^{2}}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.12.1.6

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

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{16 + 4 \sqrt{13} + 4 \sqrt{13} + 13}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.12.2

Add $16$ and $13$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{29 + 4 \sqrt{13} + 4 \sqrt{13}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.12.3

Add $4 \sqrt{13}$ and $4 \sqrt{13}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - 4 \frac{29 + 8 \sqrt{13}}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.13

Combine $- 4$ and $\frac{29 + 8 \sqrt{13}}{9}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} + \frac{- 4 (29 + 8 \sqrt{13})}{9} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.14

Cancel the common factor of $3$ .

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Step 4.2.4.14.1

Factor $3$ out of $9$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} + \frac{- 4 (29 + 8 \sqrt{13})}{3 (3)} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.14.2

Cancel the common factor.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} + \frac{- 4 (29 + 8 \sqrt{13})}{3 \cdot 3} \cdot 3 + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.14.3

Rewrite the expression.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} + \frac{- 4 (29 + 8 \sqrt{13})}{3} + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.15

Move the negative in front of the fraction.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - \frac{(4) (29 + 8 \sqrt{13})}{3} + 4 + \sqrt{13} + 2 \cdot 3}{3}$

Step 4.2.4.16

Multiply $2$ by $3$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - \frac{4 (29 + 8 \sqrt{13})}{3} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.5

To write $- \frac{4 (29 + 8 \sqrt{13})}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - \frac{4 (29 + 8 \sqrt{13})}{3} \cdot \frac{3}{3} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.6

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.6.1

Multiply $\frac{4 (29 + 8 \sqrt{13})}{3}$ by $\frac{3}{3}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - \frac{4 (29 + 8 \sqrt{13}) \cdot 3}{3 \cdot 3} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.6.2

Multiply $3$ by $3$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13}}{9} - \frac{4 (29 + 8 \sqrt{13}) \cdot 3}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.7

Combine the numerators over the common denominator.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13} - 4 (29 + 8 \sqrt{13}) \cdot 3}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.8

Simplify the numerator.

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Step 4.2.8.1

Apply the distributive property.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13} + (- 4 \cdot 29 - 4 (8 \sqrt{13})) \cdot 3}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.8.2

Multiply $- 4$ by $29$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13} + (- 116 - 4 (8 \sqrt{13})) \cdot 3}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.8.3

Multiply $8$ by $- 4$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13} + (- 116 - 32 \sqrt{13}) \cdot 3}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.8.4

Apply the distributive property.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13} - 116 \cdot 3 - 32 \sqrt{13} \cdot 3}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.8.5

Multiply $- 116$ by $3$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13} - 348 - 32 \sqrt{13} \cdot 3}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.8.6

Multiply $3$ by $- 32$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{220 + 61 \sqrt{13} - 348 - 96 \sqrt{13}}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.8.7

Subtract $348$ from $220$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 128 + 61 \sqrt{13} - 96 \sqrt{13}}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.8.8

Subtract $96 \sqrt{13}$ from $61 \sqrt{13}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 128 - 35 \sqrt{13}}{9} + 4 + \sqrt{13} + 6}{3}$

Step 4.2.9

To write $4$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 128 - 35 \sqrt{13}}{9} + 4 \cdot \frac{9}{9} + \sqrt{13} + 6}{3}$

Step 4.2.10

Combine $4$ and $\frac{9}{9}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 128 - 35 \sqrt{13}}{9} + \frac{4 \cdot 9}{9} + \sqrt{13} + 6}{3}$

Step 4.2.11

Simplify the expression.

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Step 4.2.11.1

Combine the numerators over the common denominator.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 128 - 35 \sqrt{13} + 4 \cdot 9}{9} + \sqrt{13} + 6}{3}$

Step 4.2.11.2

Multiply $4$ by $9$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 128 - 35 \sqrt{13} + 36}{9} + \sqrt{13} + 6}{3}$

Step 4.2.11.3

Add $- 128$ and $36$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 35 \sqrt{13}}{9} + \sqrt{13} + 6}{3}$

Step 4.2.12

To write $\sqrt{13}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 35 \sqrt{13}}{9} + \sqrt{13} \cdot \frac{9}{9} + 6}{3}$

Step 4.2.13

Combine $\sqrt{13}$ and $\frac{9}{9}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 35 \sqrt{13}}{9} + \frac{\sqrt{13} \cdot 9}{9} + 6}{3}$

Step 4.2.14

Simplify the expression.

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Step 4.2.14.1

Combine the numerators over the common denominator.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 35 \sqrt{13} + \sqrt{13} \cdot 9}{9} + 6}{3}$

Step 4.2.14.2

Reorder the factors of $\sqrt{13} \cdot 9$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 35 \sqrt{13} + 9 \sqrt{13}}{9} + 6}{3}$

Step 4.2.15

Add $- 35 \sqrt{13}$ and $9 \sqrt{13}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 26 \sqrt{13}}{9} + 6}{3}$

Step 4.2.16

To write $6$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 26 \sqrt{13}}{9} + 6 \cdot \frac{9}{9}}{3}$

Step 4.2.17

Combine fractions.

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Step 4.2.17.1

Combine $6$ and $\frac{9}{9}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 26 \sqrt{13}}{9} + \frac{6 \cdot 9}{9}}{3}$

Step 4.2.17.2

Combine the numerators over the common denominator.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 26 \sqrt{13} + 6 \cdot 9}{9}}{3}$

Step 4.2.18

Simplify the numerator.

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Step 4.2.18.1

Multiply $6$ by $9$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 92 - 26 \sqrt{13} + 54}{9}}{3}$

Step 4.2.18.2

Add $- 92$ and $54$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 38 - 26 \sqrt{13}}{9}}{3}$

Step 4.2.19

Simplify with factoring out.

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Step 4.2.19.1

Rewrite $- 38$ as $- 1 (38)$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 1 \cdot 38 - 26 \sqrt{13}}{9}}{3}$

Step 4.2.19.2

Factor $- 1$ out of $- 26 \sqrt{13}$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 1 \cdot 38 - (26 \sqrt{13})}{9}}{3}$

Step 4.2.19.3

Factor $- 1$ out of $- 1 (38) - (26 \sqrt{13})$ .

$f (\frac{4 + \sqrt{13}}{3}) = \frac{\frac{- 1 (38 + 26 \sqrt{13})}{9}}{3}$

Step 4.2.19.4

Move the negative in front of the fraction.

$f (\frac{4 + \sqrt{13}}{3}) = \frac{- \frac{38 + 26 \sqrt{13}}{9}}{3}$

Step 4.2.20

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{4 + \sqrt{13}}{3}) = - \frac{38 + 26 \sqrt{13}}{9} \cdot \frac{1}{3}$

Step 4.2.21

Multiply $- \frac{38 + 26 \sqrt{13}}{9} \cdot \frac{1}{3}$ .

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Step 4.2.21.1

Multiply $\frac{1}{3}$ by $\frac{38 + 26 \sqrt{13}}{9}$ .

$f (\frac{4 + \sqrt{13}}{3}) = - \frac{38 + 26 \sqrt{13}}{3 \cdot 9}$

Step 4.2.21.2

Multiply $3$ by $9$ .

$f (\frac{4 + \sqrt{13}}{3}) = - \frac{38 + 26 \sqrt{13}}{27}$

Step 4.2.22

The final answer is $- \frac{38 + 26 \sqrt{13}}{27}$ .

$- \frac{38 + 26 \sqrt{13}}{27}$

Step 5

Solve the original function $f (x) = x^{3} - 4 x^{2} + x + 2$ at $x = \frac{4 - \sqrt{13}}{3}$ .

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Step 5.1

Replace the variable $x$ with $\frac{4 - \sqrt{13}}{3}$ in the expression.

$f (\frac{4 - \sqrt{13}}{3}) = {(\frac{4 - \sqrt{13}}{3})}^{3} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} + \frac{4 - \sqrt{13}}{3} + 2$

Step 5.2

Simplify the result.

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Step 5.2.1

Remove parentheses.

$f (\frac{4 - \sqrt{13}}{3}) = {(\frac{4 - \sqrt{13}}{3})}^{3} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} + \frac{4 - \sqrt{13}}{3} + 2$

Step 5.2.2

Find the common denominator.

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Step 5.2.2.1

Write ${(\frac{4 - \sqrt{13}}{3})}^{3}$ as a fraction with denominator $1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3}}{1} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} + \frac{4 - \sqrt{13}}{3} + 2$

Step 5.2.2.2

Multiply $\frac{{(\frac{4 - \sqrt{13}}{3})}^{3}}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3}}{1} \cdot \frac{3}{3} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} + \frac{4 - \sqrt{13}}{3} + 2$

Step 5.2.2.3

Multiply $\frac{{(\frac{4 - \sqrt{13}}{3})}^{3}}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3} \cdot 3}{3} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} + \frac{4 - \sqrt{13}}{3} + 2$

Step 5.2.2.4

Write $- 4 {(\frac{4 - \sqrt{13}}{3})}^{2}$ as a fraction with denominator $1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 - \sqrt{13}}{3})}^{2}}{1} + \frac{4 - \sqrt{13}}{3} + 2$

Step 5.2.2.5

Multiply $\frac{- 4 {(\frac{4 - \sqrt{13}}{3})}^{2}}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 - \sqrt{13}}{3})}^{2}}{1} \cdot \frac{3}{3} + \frac{4 - \sqrt{13}}{3} + 2$

Step 5.2.2.6

Multiply $\frac{- 4 {(\frac{4 - \sqrt{13}}{3})}^{2}}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3}{3} + \frac{4 - \sqrt{13}}{3} + 2$

Step 5.2.2.7

Write $2$ as a fraction with denominator $1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3}{3} + \frac{4 - \sqrt{13}}{3} + \frac{2}{1}$

Step 5.2.2.8

Multiply $\frac{2}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3}{3} + \frac{4 - \sqrt{13}}{3} + \frac{2}{1} \cdot \frac{3}{3}$

Step 5.2.2.9

Multiply $\frac{2}{1}$ by $\frac{3}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3} \cdot 3}{3} + \frac{- 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3}{3} + \frac{4 - \sqrt{13}}{3} + \frac{2 \cdot 3}{3}$

Step 5.2.3

Combine the numerators over the common denominator.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{{(\frac{4 - \sqrt{13}}{3})}^{3} \cdot 3 - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the product rule to $\frac{4 - \sqrt{13}}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{{(4 - \sqrt{13})}^{3}}{3^{3}} \cdot 3 - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.2

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{{(4 - \sqrt{13})}^{3}}{27} \cdot 3 - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.3

Cancel the common factor of $3$ .

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Step 5.2.4.3.1

Factor $3$ out of $27$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{{(4 - \sqrt{13})}^{3}}{3 (9)} \cdot 3 - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.3.2

Cancel the common factor.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{{(4 - \sqrt{13})}^{3}}{3 \cdot 9} \cdot 3 - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.3.3

Rewrite the expression.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{{(4 - \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.4

Use the Binomial Theorem.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{4^{3} + 3 \cdot (4^{2} (- \sqrt{13})) + 3 \cdot (4 {(- \sqrt{13})}^{2}) + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5

Simplify each term.

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Step 5.2.4.5.1

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 + 3 \cdot (4^{2} (- \sqrt{13})) + 3 \cdot (4 {(- \sqrt{13})}^{2}) + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.2

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 + 3 \cdot (16 (- \sqrt{13})) + 3 \cdot (4 {(- \sqrt{13})}^{2}) + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.3

Multiply $3$ by $16$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 + 48 (- \sqrt{13}) + 3 \cdot (4 {(- \sqrt{13})}^{2}) + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.4

Multiply $- 1$ by $48$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 3 \cdot (4 {(- \sqrt{13})}^{2}) + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.5

Multiply $3$ by $4$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 {(- \sqrt{13})}^{2} + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.6

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 ({(- 1)}^{2} {\sqrt{13}}^{2}) + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.7

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 (1 {\sqrt{13}}^{2}) + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.8

Multiply ${\sqrt{13}}^{2}$ by $1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 {\sqrt{13}}^{2} + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.9

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

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Step 5.2.4.5.9.1

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 {(13^{\frac{1}{2}})}^{2} + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.9.2

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 \cdot 13^{\frac{1}{2} \cdot 2} + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.9.3

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 \cdot 13^{\frac{2}{2}} + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.9.4

Cancel the common factor of $2$ .

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Step 5.2.4.5.9.4.1

Cancel the common factor.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 \cdot 13^{\frac{2}{2}} + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.9.4.2

Rewrite the expression.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 \cdot 13 + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.9.5

Evaluate the exponent.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 12 \cdot 13 + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.10

Multiply $12$ by $13$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 + {(- \sqrt{13})}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.11

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 + {(- 1)}^{3} {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.12

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 - {\sqrt{13}}^{3}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.13

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 - \sqrt{13^{3}}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.14

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 - \sqrt{2197}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.15

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

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Step 5.2.4.5.15.1

Factor $169$ out of $2197$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 - \sqrt{169 (13)}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.15.2

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 - \sqrt{13^{2} \cdot 13}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.16

Pull terms out from under the radical.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 - (13 \sqrt{13})}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.5.17

Multiply $13$ by $- 1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{64 - 48 \sqrt{13} + 156 - 13 \sqrt{13}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.6

Add $64$ and $156$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 48 \sqrt{13} - 13 \sqrt{13}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.7

Subtract $13 \sqrt{13}$ from $- 48 \sqrt{13}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 {(\frac{4 - \sqrt{13}}{3})}^{2} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.8

Apply the product rule to $\frac{4 - \sqrt{13}}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{{(4 - \sqrt{13})}^{2}}{3^{2}} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.9

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{{(4 - \sqrt{13})}^{2}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.10

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{(4 - \sqrt{13}) (4 - \sqrt{13})}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.11

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

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Step 5.2.4.11.1

Apply the distributive property.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{4 (4 - \sqrt{13}) - \sqrt{13} (4 - \sqrt{13})}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.11.2

Apply the distributive property.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{4 \cdot 4 + 4 (- \sqrt{13}) - \sqrt{13} (4 - \sqrt{13})}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.11.3

Apply the distributive property.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{4 \cdot 4 + 4 (- \sqrt{13}) - \sqrt{13} \cdot 4 - \sqrt{13} (- \sqrt{13})}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12

Simplify and combine like terms.

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Step 5.2.4.12.1

Simplify each term.

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Step 5.2.4.12.1.1

Multiply $4$ by $4$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 + 4 (- \sqrt{13}) - \sqrt{13} \cdot 4 - \sqrt{13} (- \sqrt{13})}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.2

Multiply $- 1$ by $4$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - \sqrt{13} \cdot 4 - \sqrt{13} (- \sqrt{13})}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.3

Multiply $4$ by $- 1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} - \sqrt{13} (- \sqrt{13})}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.4

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

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Step 5.2.4.12.1.4.1

Multiply $- 1$ by $- 1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + 1 \sqrt{13} \sqrt{13}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.4.2

Multiply $\sqrt{13}$ by $1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + \sqrt{13} \sqrt{13}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.4.3

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + \sqrt{13} \sqrt{13}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.4.4

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + \sqrt{13} \sqrt{13}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.4.5

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + {\sqrt{13}}^{1 + 1}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.4.6

Add $1$ and $1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + {\sqrt{13}}^{2}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.5

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

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Step 5.2.4.12.1.5.1

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + {(13^{\frac{1}{2}})}^{2}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.5.2

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + 13^{\frac{1}{2} \cdot 2}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.5.3

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + 13^{\frac{2}{2}}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.4.12.1.5.4.1

Cancel the common factor.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + 13^{\frac{2}{2}}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.5.4.2

Rewrite the expression.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + 13}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.1.5.5

Evaluate the exponent.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{16 - 4 \sqrt{13} - 4 \sqrt{13} + 13}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.2

Add $16$ and $13$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{29 - 4 \sqrt{13} - 4 \sqrt{13}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.12.3

Subtract $4 \sqrt{13}$ from $- 4 \sqrt{13}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - 4 \frac{29 - 8 \sqrt{13}}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.13

Combine $- 4$ and $\frac{29 - 8 \sqrt{13}}{9}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} + \frac{- 4 (29 - 8 \sqrt{13})}{9} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.14

Cancel the common factor of $3$ .

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Step 5.2.4.14.1

Factor $3$ out of $9$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} + \frac{- 4 (29 - 8 \sqrt{13})}{3 (3)} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.14.2

Cancel the common factor.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} + \frac{- 4 (29 - 8 \sqrt{13})}{3 \cdot 3} \cdot 3 + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.14.3

Rewrite the expression.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} + \frac{- 4 (29 - 8 \sqrt{13})}{3} + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.15

Move the negative in front of the fraction.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - \frac{(4) (29 - 8 \sqrt{13})}{3} + 4 - \sqrt{13} + 2 \cdot 3}{3}$

Step 5.2.4.16

Multiply $2$ by $3$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - \frac{4 (29 - 8 \sqrt{13})}{3} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.5

To write $- \frac{4 (29 - 8 \sqrt{13})}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - \frac{4 (29 - 8 \sqrt{13})}{3} \cdot \frac{3}{3} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.6

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.6.1

Multiply $\frac{4 (29 - 8 \sqrt{13})}{3}$ by $\frac{3}{3}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - \frac{4 (29 - 8 \sqrt{13}) \cdot 3}{3 \cdot 3} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.6.2

Multiply $3$ by $3$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13}}{9} - \frac{4 (29 - 8 \sqrt{13}) \cdot 3}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.7

Combine the numerators over the common denominator.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13} - 4 (29 - 8 \sqrt{13}) \cdot 3}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.8

Simplify the numerator.

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Step 5.2.8.1

Apply the distributive property.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13} + (- 4 \cdot 29 - 4 (- 8 \sqrt{13})) \cdot 3}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.8.2

Multiply $- 4$ by $29$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13} + (- 116 - 4 (- 8 \sqrt{13})) \cdot 3}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.8.3

Multiply $- 8$ by $- 4$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13} + (- 116 + 32 \sqrt{13}) \cdot 3}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.8.4

Apply the distributive property.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13} - 116 \cdot 3 + 32 \sqrt{13} \cdot 3}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.8.5

Multiply $- 116$ by $3$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13} - 348 + 32 \sqrt{13} \cdot 3}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.8.6

Multiply $3$ by $32$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{220 - 61 \sqrt{13} - 348 + 96 \sqrt{13}}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.8.7

Subtract $348$ from $220$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 128 - 61 \sqrt{13} + 96 \sqrt{13}}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.8.8

Add $- 61 \sqrt{13}$ and $96 \sqrt{13}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 128 + 35 \sqrt{13}}{9} + 4 - \sqrt{13} + 6}{3}$

Step 5.2.9

To write $4$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 128 + 35 \sqrt{13}}{9} + 4 \cdot \frac{9}{9} - \sqrt{13} + 6}{3}$

Step 5.2.10

Combine $4$ and $\frac{9}{9}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 128 + 35 \sqrt{13}}{9} + \frac{4 \cdot 9}{9} - \sqrt{13} + 6}{3}$

Step 5.2.11

Simplify the expression.

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Step 5.2.11.1

Combine the numerators over the common denominator.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 128 + 35 \sqrt{13} + 4 \cdot 9}{9} - \sqrt{13} + 6}{3}$

Step 5.2.11.2

Multiply $4$ by $9$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 128 + 35 \sqrt{13} + 36}{9} - \sqrt{13} + 6}{3}$

Step 5.2.11.3

Add $- 128$ and $36$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 35 \sqrt{13}}{9} - \sqrt{13} + 6}{3}$

Step 5.2.12

To write $- \sqrt{13}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 35 \sqrt{13}}{9} - \sqrt{13} \cdot \frac{9}{9} + 6}{3}$

Step 5.2.13

Combine fractions.

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Step 5.2.13.1

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

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 35 \sqrt{13}}{9} + \frac{- \sqrt{13} \cdot 9}{9} + 6}{3}$

Step 5.2.13.2

Combine the numerators over the common denominator.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 35 \sqrt{13} - \sqrt{13} \cdot 9}{9} + 6}{3}$

Step 5.2.14

Simplify the numerator.

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Step 5.2.14.1

Multiply $9$ by $- 1$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 35 \sqrt{13} - 9 \sqrt{13}}{9} + 6}{3}$

Step 5.2.14.2

Subtract $9 \sqrt{13}$ from $35 \sqrt{13}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 26 \sqrt{13}}{9} + 6}{3}$

Step 5.2.15

To write $6$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 26 \sqrt{13}}{9} + 6 \cdot \frac{9}{9}}{3}$

Step 5.2.16

Combine fractions.

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Step 5.2.16.1

Combine $6$ and $\frac{9}{9}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 26 \sqrt{13}}{9} + \frac{6 \cdot 9}{9}}{3}$

Step 5.2.16.2

Combine the numerators over the common denominator.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 26 \sqrt{13} + 6 \cdot 9}{9}}{3}$

Step 5.2.17

Simplify the numerator.

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Step 5.2.17.1

Multiply $6$ by $9$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 92 + 26 \sqrt{13} + 54}{9}}{3}$

Step 5.2.17.2

Add $- 92$ and $54$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 38 + 26 \sqrt{13}}{9}}{3}$

Step 5.2.18

Simplify with factoring out.

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Step 5.2.18.1

Rewrite $- 38$ as $- 1 (38)$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 1 \cdot 38 + 26 \sqrt{13}}{9}}{3}$

Step 5.2.18.2

Factor $- 1$ out of $26 \sqrt{13}$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 1 \cdot 38 - (- 26 \sqrt{13})}{9}}{3}$

Step 5.2.18.3

Factor $- 1$ out of $- 1 (38) - (- 26 \sqrt{13})$ .

$f (\frac{4 - \sqrt{13}}{3}) = \frac{\frac{- 1 (38 - 26 \sqrt{13})}{9}}{3}$

Step 5.2.18.4

Move the negative in front of the fraction.

$f (\frac{4 - \sqrt{13}}{3}) = \frac{- \frac{38 - 26 \sqrt{13}}{9}}{3}$

Step 5.2.19

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{4 - \sqrt{13}}{3}) = - \frac{38 - 26 \sqrt{13}}{9} \cdot \frac{1}{3}$

Step 5.2.20

Multiply $- \frac{38 - 26 \sqrt{13}}{9} \cdot \frac{1}{3}$ .

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Step 5.2.20.1

Multiply $\frac{1}{3}$ by $\frac{38 - 26 \sqrt{13}}{9}$ .

$f (\frac{4 - \sqrt{13}}{3}) = - \frac{38 - 26 \sqrt{13}}{3 \cdot 9}$

Step 5.2.20.2

Multiply $3$ by $9$ .

$f (\frac{4 - \sqrt{13}}{3}) = - \frac{38 - 26 \sqrt{13}}{27}$

Step 5.2.21

The final answer is $- \frac{38 - 26 \sqrt{13}}{27}$ .

$- \frac{38 - 26 \sqrt{13}}{27}$

Step 6

The horizontal tangent lines on function $f (x) = x^{3} - 4 x^{2} + x + 2$ are $y = - \frac{38 + 26 \sqrt{13}}{27}, y = - \frac{38 - 26 \sqrt{13}}{27}$ .

$y = - \frac{38 + 26 \sqrt{13}}{27}, y = - \frac{38 - 26 \sqrt{13}}{27}$

Step 7