Calculus Examples

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

Step 1

Find the first derivative of the function.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $4$ by $- 1$ .

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

Step 1.3

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Multiply $2$ by $2$ .

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

Step 2

Find the second derivative of the function.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $- 4$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $2$ by $3$ .

$f'' (x) = - 12 x^{2} + 6 x + \frac{d}{d x} (4 x)$

Step 2.4

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

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Step 2.4.1

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

$f'' (x) = - 12 x^{2} + 6 x + 4 \frac{d}{d x} (x)$

Step 2.4.2

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

$f'' (x) = - 12 x^{2} + 6 x + 4 \cdot 1$

Step 2.4.3

Multiply $4$ by $1$ .

$f'' (x) = - 12 x^{2} + 6 x + 4$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$- 4 x^{3} + 3 x^{2} + 4 x = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

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

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

Step 4.1.2

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

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Step 4.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

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

Step 4.1.2.2

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

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

Step 4.1.2.3

Multiply $4$ by $- 1$ .

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

Step 4.1.3

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

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

Step 4.1.4

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

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Step 4.1.4.1

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

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

Step 4.1.4.2

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

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

Step 4.1.4.3

Multiply $2$ by $2$ .

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

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- 4 x^{3} + 3 x^{2} + 4 x$ .

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

Step 5

Set the first derivative equal to $0$ then solve the equation $- 4 x^{3} + 3 x^{2} + 4 x = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- 4 x^{3} + 3 x^{2} + 4 x = 0$

Step 5.2

Factor $- x$ out of $- 4 x^{3} + 3 x^{2} + 4 x$ .

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Step 5.2.1

Factor $- x$ out of $- 4 x^{3}$ .

$- x (4 x^{2}) + 3 x^{2} + 4 x = 0$

Step 5.2.2

Factor $- x$ out of $3 x^{2}$ .

$- x (4 x^{2}) - x (- 3 x) + 4 x = 0$

Step 5.2.3

Factor $- x$ out of $4 x$ .

$- x (4 x^{2}) - x (- 3 x) - x \cdot - 4 = 0$

Step 5.2.4

Factor $- x$ out of $- x (4 x^{2}) - x (- 3 x)$ .

$- x (4 x^{2} - 3 x) - x \cdot - 4 = 0$

Step 5.2.5

Factor $- x$ out of $- x (4 x^{2} - 3 x) - x (- 4)$ .

$- x (4 x^{2} - 3 x - 4) = 0$

Step 5.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$4 x^{2} - 3 x - 4 = 0$

Step 5.4

Set $x$ equal to $0$ .

$x = 0$

Step 5.5

Set $4 x^{2} - 3 x - 4$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $4 x^{2} - 3 x - 4$ equal to $0$ .

$4 x^{2} - 3 x - 4 = 0$

Step 5.5.2

Solve $4 x^{2} - 3 x - 4 = 0$ for $x$ .

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Step 5.5.2.1

Use the quadratic formula to find the solutions.

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

Step 5.5.2.2

Substitute the values $a = 4$ , $b = - 3$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 5.5.2.3

Simplify.

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Step 5.5.2.3.1

Simplify the numerator.

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Step 5.5.2.3.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 4}}{2 \cdot 4}$

Step 5.5.2.3.1.2

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

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Step 5.5.2.3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 4}}{2 \cdot 4}$

Step 5.5.2.3.1.2.2

Multiply $- 16$ by $- 4$ .

$x = \frac{3 \pm \sqrt{9 + 64}}{2 \cdot 4}$

Step 5.5.2.3.1.3

Add $9$ and $64$ .

$x = \frac{3 \pm \sqrt{73}}{2 \cdot 4}$

Step 5.5.2.3.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{73}}{8}$

Step 5.5.2.4

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

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Step 5.5.2.4.1

Simplify the numerator.

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Step 5.5.2.4.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 4}}{2 \cdot 4}$

Step 5.5.2.4.1.2

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

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Step 5.5.2.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 4}}{2 \cdot 4}$

Step 5.5.2.4.1.2.2

Multiply $- 16$ by $- 4$ .

$x = \frac{3 \pm \sqrt{9 + 64}}{2 \cdot 4}$

Step 5.5.2.4.1.3

Add $9$ and $64$ .

$x = \frac{3 \pm \sqrt{73}}{2 \cdot 4}$

Step 5.5.2.4.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{73}}{8}$

Step 5.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{73}}{8}$

Step 5.5.2.5

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

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Step 5.5.2.5.1

Simplify the numerator.

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Step 5.5.2.5.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 4}}{2 \cdot 4}$

Step 5.5.2.5.1.2

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

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Step 5.5.2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 4}}{2 \cdot 4}$

Step 5.5.2.5.1.2.2

Multiply $- 16$ by $- 4$ .

$x = \frac{3 \pm \sqrt{9 + 64}}{2 \cdot 4}$

Step 5.5.2.5.1.3

Add $9$ and $64$ .

$x = \frac{3 \pm \sqrt{73}}{2 \cdot 4}$

Step 5.5.2.5.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{73}}{8}$

Step 5.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{73}}{8}$

Step 5.5.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{73}}{8}, \frac{3 - \sqrt{73}}{8}$

Step 5.6

The final solution is all the values that make $- x (4 x^{2} - 3 x - 4) = 0$ true.

$x = 0, \frac{3 + \sqrt{73}}{8}, \frac{3 - \sqrt{73}}{8}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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

Step 7

Critical points to evaluate.

$x = 0, \frac{3 + \sqrt{73}}{8}, \frac{3 - \sqrt{73}}{8}$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 {(0)}^{2} + 6 (0) + 4$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

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

$- 12 \cdot 0 + 6 (0) + 4$

Step 9.1.2

Multiply $- 12$ by $0$ .

$0 + 6 (0) + 4$

Step 9.1.3

Multiply $6$ by $0$ .

$0 + 0 + 4$

Step 9.2

Simplify by adding numbers.

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Step 9.2.1

Add $0$ and $0$ .

$0 + 4$

Step 9.2.2

Add $0$ and $4$ .

$4$

Step 10

$x = 0$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 0$ is a local minimum

Step 11

Find the y-value when $x = 0$ .

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Step 11.1

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

$f (0) = - {(0)}^{4} + {(0)}^{3} + 2 {(0)}^{2}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

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

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

Step 11.2.1.2

Multiply $- 1$ by $0$ .

$f (0) = 0 + {(0)}^{3} + 2 {(0)}^{2}$

Step 11.2.1.3

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

$f (0) = 0 + 0 + 2 {(0)}^{2}$

Step 11.2.1.4

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

$f (0) = 0 + 0 + 2 \cdot 0$

Step 11.2.1.5

Multiply $2$ by $0$ .

$f (0) = 0 + 0 + 0$

Step 11.2.2

Simplify by adding numbers.

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Step 11.2.2.1

Add $0$ and $0$ .

$f (0) = 0 + 0$

Step 11.2.2.2

Add $0$ and $0$ .

$f (0) = 0$

Step 11.2.3

The final answer is $0$ .

$y = 0$

Step 12

Evaluate the second derivative at $x = \frac{3 + \sqrt{73}}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 {(\frac{3 + \sqrt{73}}{8})}^{2} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13

Evaluate the second derivative.

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Step 13.1

Simplify each term.

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Step 13.1.1

Apply the product rule to $\frac{3 + \sqrt{73}}{8}$ .

$- 12 \frac{{(3 + \sqrt{73})}^{2}}{8^{2}} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.2

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

$- 12 \frac{{(3 + \sqrt{73})}^{2}}{64} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.3

Cancel the common factor of $4$ .

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Step 13.1.3.1

Factor $4$ out of $- 12$ .

$4 (- 3) \frac{{(3 + \sqrt{73})}^{2}}{64} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.3.2

Factor $4$ out of $64$ .

$4 \cdot - 3 \frac{{(3 + \sqrt{73})}^{2}}{4 \cdot 16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.3.3

Cancel the common factor.

$4 \cdot - 3 \frac{{(3 + \sqrt{73})}^{2}}{4 \cdot 16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.3.4

Rewrite the expression.

$- 3 \frac{{(3 + \sqrt{73})}^{2}}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.4

Combine $- 3$ and $\frac{{(3 + \sqrt{73})}^{2}}{16}$ .

$\frac{- 3 {(3 + \sqrt{73})}^{2}}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.5

Rewrite ${(3 + \sqrt{73})}^{2}$ as $(3 + \sqrt{73}) (3 + \sqrt{73})$ .

$\frac{- 3 ((3 + \sqrt{73}) (3 + \sqrt{73}))}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.6

Expand $(3 + \sqrt{73}) (3 + \sqrt{73})$ using the FOIL Method.

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Step 13.1.6.1

Apply the distributive property.

$\frac{- 3 (3 (3 + \sqrt{73}) + \sqrt{73} (3 + \sqrt{73}))}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.6.2

Apply the distributive property.

$\frac{- 3 (3 \cdot 3 + 3 \sqrt{73} + \sqrt{73} (3 + \sqrt{73}))}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.6.3

Apply the distributive property.

$\frac{- 3 (3 \cdot 3 + 3 \sqrt{73} + \sqrt{73} \cdot 3 + \sqrt{73} \sqrt{73})}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.7

Simplify and combine like terms.

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Step 13.1.7.1

Simplify each term.

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Step 13.1.7.1.1

Multiply $3$ by $3$ .

$\frac{- 3 (9 + 3 \sqrt{73} + \sqrt{73} \cdot 3 + \sqrt{73} \sqrt{73})}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.7.1.2

Move $3$ to the left of $\sqrt{73}$ .

$\frac{- 3 (9 + 3 \sqrt{73} + 3 \cdot \sqrt{73} + \sqrt{73} \sqrt{73})}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.7.1.3

Combine using the product rule for radicals.

$\frac{- 3 (9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{73 \cdot 73})}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.7.1.4

Multiply $73$ by $73$ .

$\frac{- 3 (9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{5329})}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.7.1.5

Rewrite $5329$ as $73^{2}$ .

$\frac{- 3 (9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{73^{2}})}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.7.1.6

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

$\frac{- 3 (9 + 3 \sqrt{73} + 3 \sqrt{73} + 73)}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.7.2

Add $9$ and $73$ .

$\frac{- 3 (82 + 3 \sqrt{73} + 3 \sqrt{73})}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.7.3

Add $3 \sqrt{73}$ and $3 \sqrt{73}$ .

$\frac{- 3 (82 + 6 \sqrt{73})}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.8

Cancel the common factor of $82 + 6 \sqrt{73}$ and $16$ .

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Step 13.1.8.1

Factor $2$ out of $- 3 (82 + 6 \sqrt{73})$ .

$\frac{2 (- 3 (41 + 3 \sqrt{73}))}{16} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.8.2

Cancel the common factors.

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Step 13.1.8.2.1

Factor $2$ out of $16$ .

$\frac{2 (- 3 (41 + 3 \sqrt{73}))}{2 (8)} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.8.2.2

Cancel the common factor.

$\frac{2 (- 3 (41 + 3 \sqrt{73}))}{2 \cdot 8} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.8.2.3

Rewrite the expression.

$\frac{- 3 (41 + 3 \sqrt{73})}{8} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.9

Move the negative in front of the fraction.

$- \frac{3 (41 + 3 \sqrt{73})}{8} + 6 (\frac{3 + \sqrt{73}}{8}) + 4$

Step 13.1.10

Cancel the common factor of $2$ .

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Step 13.1.10.1

Factor $2$ out of $6$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + 2 (3) \frac{3 + \sqrt{73}}{8} + 4$

Step 13.1.10.2

Factor $2$ out of $8$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + 2 \cdot 3 \frac{3 + \sqrt{73}}{2 \cdot 4} + 4$

Step 13.1.10.3

Cancel the common factor.

$- \frac{3 (41 + 3 \sqrt{73})}{8} + 2 \cdot 3 \frac{3 + \sqrt{73}}{2 \cdot 4} + 4$

Step 13.1.10.4

Rewrite the expression.

$- \frac{3 (41 + 3 \sqrt{73})}{8} + 3 \frac{3 + \sqrt{73}}{4} + 4$

Step 13.1.11

Combine $3$ and $\frac{3 + \sqrt{73}}{4}$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (3 + \sqrt{73})}{4} + 4$

Step 13.2

Find the common denominator.

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Step 13.2.1

Multiply $\frac{3 (3 + \sqrt{73})}{4}$ by $\frac{2}{2}$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (3 + \sqrt{73})}{4} \cdot \frac{2}{2} + 4$

Step 13.2.2

Multiply $\frac{3 (3 + \sqrt{73})}{4}$ by $\frac{2}{2}$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (3 + \sqrt{73}) \cdot 2}{4 \cdot 2} + 4$

Step 13.2.3

Write $4$ as a fraction with denominator $1$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (3 + \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{4}{1}$

Step 13.2.4

Multiply $\frac{4}{1}$ by $\frac{8}{8}$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (3 + \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{4}{1} \cdot \frac{8}{8}$

Step 13.2.5

Multiply $\frac{4}{1}$ by $\frac{8}{8}$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (3 + \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{4 \cdot 8}{8}$

Step 13.2.6

Reorder the factors of $4 \cdot 2$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (3 + \sqrt{73}) \cdot 2}{2 \cdot 4} + \frac{4 \cdot 8}{8}$

Step 13.2.7

Multiply $2$ by $4$ .

$- \frac{3 (41 + 3 \sqrt{73})}{8} + \frac{3 (3 + \sqrt{73}) \cdot 2}{8} + \frac{4 \cdot 8}{8}$

Step 13.3

Combine the numerators over the common denominator.

$\frac{- 3 (41 + 3 \sqrt{73}) + 3 (3 + \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 13.4

Simplify each term.

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Step 13.4.1

Apply the distributive property.

$\frac{- 3 \cdot 41 - 3 (3 \sqrt{73}) + 3 (3 + \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 13.4.2

Multiply $- 3$ by $41$ .

$\frac{- 123 - 3 (3 \sqrt{73}) + 3 (3 + \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 13.4.3

Multiply $3$ by $- 3$ .

$\frac{- 123 - 9 \sqrt{73} + 3 (3 + \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 13.4.4

Apply the distributive property.

$\frac{- 123 - 9 \sqrt{73} + (3 \cdot 3 + 3 \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 13.4.5

Multiply $3$ by $3$ .

$\frac{- 123 - 9 \sqrt{73} + (9 + 3 \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 13.4.6

Apply the distributive property.

$\frac{- 123 - 9 \sqrt{73} + 9 \cdot 2 + 3 \sqrt{73} \cdot 2 + 4 \cdot 8}{8}$

Step 13.4.7

Multiply $9$ by $2$ .

$\frac{- 123 - 9 \sqrt{73} + 18 + 3 \sqrt{73} \cdot 2 + 4 \cdot 8}{8}$

Step 13.4.8

Multiply $2$ by $3$ .

$\frac{- 123 - 9 \sqrt{73} + 18 + 6 \sqrt{73} + 4 \cdot 8}{8}$

Step 13.4.9

Multiply $4$ by $8$ .

$\frac{- 123 - 9 \sqrt{73} + 18 + 6 \sqrt{73} + 32}{8}$

Step 13.5

Simplify terms.

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Step 13.5.1

Add $- 123$ and $18$ .

$\frac{- 105 - 9 \sqrt{73} + 6 \sqrt{73} + 32}{8}$

Step 13.5.2

Add $- 105$ and $32$ .

$\frac{- 73 - 9 \sqrt{73} + 6 \sqrt{73}}{8}$

Step 13.5.3

Add $- 9 \sqrt{73}$ and $6 \sqrt{73}$ .

$\frac{- 73 - 3 \sqrt{73}}{8}$

Step 13.5.4

Rewrite $- 73$ as $- 1 (73)$ .

$\frac{- 1 (73) - 3 \sqrt{73}}{8}$

Step 13.5.5

Factor $- 1$ out of $- 3 \sqrt{73}$ .

$\frac{- 1 (73) - (3 \sqrt{73})}{8}$

Step 13.5.6

Factor $- 1$ out of $- 1 (73) - (3 \sqrt{73})$ .

$\frac{- 1 (73 + 3 \sqrt{73})}{8}$

Step 13.5.7

Move the negative in front of the fraction.

$- \frac{73 + 3 \sqrt{73}}{8}$

Step 14

$x = \frac{3 + \sqrt{73}}{8}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{3 + \sqrt{73}}{8}$ is a local maximum

Step 15

Find the y-value when $x = \frac{3 + \sqrt{73}}{8}$ .

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Step 15.1

Replace the variable $x$ with $\frac{3 + \sqrt{73}}{8}$ in the expression.

$f (\frac{3 + \sqrt{73}}{8}) = - {(\frac{3 + \sqrt{73}}{8})}^{4} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Apply the product rule to $\frac{3 + \sqrt{73}}{8}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{{(3 + \sqrt{73})}^{4}}{8^{4}} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.2

Raise $8$ to the power of $4$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{{(3 + \sqrt{73})}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.3

Use the Binomial Theorem.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{3^{4} + 4 \cdot (3^{3} \sqrt{73}) + 6 \cdot (3^{2} {\sqrt{73}}^{2}) + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4

Simplify each term.

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Step 15.2.1.4.1

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 4 \cdot (3^{3} \sqrt{73}) + 6 \cdot (3^{2} {\sqrt{73}}^{2}) + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.2

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 4 \cdot (27 \sqrt{73}) + 6 \cdot (3^{2} {\sqrt{73}}^{2}) + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.3

Multiply $4$ by $27$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 6 \cdot (3^{2} {\sqrt{73}}^{2}) + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.4

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 6 \cdot (9 {\sqrt{73}}^{2}) + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.5

Multiply $6$ by $9$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 54 {\sqrt{73}}^{2} + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.6

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 15.2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 54 {(73^{\frac{1}{2}})}^{2} + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.6.2

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 54 \cdot 73^{\frac{1}{2} \cdot 2} + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.6.3

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 54 \cdot 73^{\frac{2}{2}} + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.6.4

Cancel the common factor of $2$ .

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Step 15.2.1.4.6.4.1

Cancel the common factor.

Step 15.2.1.4.6.4.2

Rewrite the expression.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 54 \cdot 73 + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.6.5

Evaluate the exponent.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 54 \cdot 73 + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.7

Multiply $54$ by $73$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 4 \cdot (3 {\sqrt{73}}^{3}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.8

Multiply $4$ by $3$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 12 {\sqrt{73}}^{3} + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.9

Rewrite ${\sqrt{73}}^{3}$ as $\sqrt{73^{3}}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 12 \sqrt{73^{3}} + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.10

Raise $73$ to the power of $3$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 12 \sqrt{389017} + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.11

Rewrite $389017$ as $73^{2} \cdot 73$ .

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Step 15.2.1.4.11.1

Factor $5329$ out of $389017$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 12 \sqrt{5329 (73)} + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.11.2

Rewrite $5329$ as $73^{2}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 12 \sqrt{73^{2} \cdot 73} + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.12

Pull terms out from under the radical.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 12 (73 \sqrt{73}) + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.13

Multiply $73$ by $12$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + {\sqrt{73}}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.14

Rewrite ${\sqrt{73}}^{4}$ as $73^{2}$ .

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Step 15.2.1.4.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + {(73^{\frac{1}{2}})}^{4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.14.2

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + 73^{\frac{1}{2} \cdot 4}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.14.3

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + 73^{\frac{4}{2}}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.14.4

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

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Step 15.2.1.4.14.4.1

Factor $2$ out of $4$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + 73^{\frac{2 \cdot 2}{2}}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.14.4.2

Cancel the common factors.

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Step 15.2.1.4.14.4.2.1

Factor $2$ out of $2$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + 73^{\frac{2 \cdot 2}{2 (1)}}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.14.4.2.2

Cancel the common factor.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + 73^{\frac{2 \cdot 2}{2 \cdot 1}}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.14.4.2.3

Rewrite the expression.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + 73^{\frac{2}{1}}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.14.4.2.4

Divide $2$ by $1$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + 73^{2}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.4.15

Raise $73$ to the power of $2$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{81 + 108 \sqrt{73} + 3942 + 876 \sqrt{73} + 5329}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.5

Add $81$ and $3942$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{4023 + 108 \sqrt{73} + 876 \sqrt{73} + 5329}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.6

Add $4023$ and $5329$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{9352 + 108 \sqrt{73} + 876 \sqrt{73}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.7

Add $108 \sqrt{73}$ and $876 \sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{9352 + 984 \sqrt{73}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.8

Cancel the common factor of $9352 + 984 \sqrt{73}$ and $4096$ .

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Step 15.2.1.8.1

Factor $8$ out of $9352$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{8 (1169) + 984 \sqrt{73}}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.8.2

Factor $8$ out of $984 \sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{8 (1169) + 8 (123 \sqrt{73})}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.8.3

Factor $8$ out of $8 (1169) + 8 (123 \sqrt{73})$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{8 (1169 + 123 \sqrt{73})}{4096} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.8.4

Cancel the common factors.

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Step 15.2.1.8.4.1

Factor $8$ out of $4096$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{8 (1169 + 123 \sqrt{73})}{8 \cdot 512} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.8.4.2

Cancel the common factor.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{8 (1169 + 123 \sqrt{73})}{8 \cdot 512} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.8.4.3

Rewrite the expression.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + {(\frac{3 + \sqrt{73}}{8})}^{3} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.9

Apply the product rule to $\frac{3 + \sqrt{73}}{8}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{{(3 + \sqrt{73})}^{3}}{8^{3}} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.10

Raise $8$ to the power of $3$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{{(3 + \sqrt{73})}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.11

Use the Binomial Theorem.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{3^{3} + 3 \cdot (3^{2} \sqrt{73}) + 3 \cdot (3 {\sqrt{73}}^{2}) + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12

Simplify each term.

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Step 15.2.1.12.1

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 3 \cdot (3^{2} \sqrt{73}) + 3 \cdot (3 {\sqrt{73}}^{2}) + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.2

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

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Step 15.2.1.12.2.1

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

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Step 15.2.1.12.2.1.1

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 3 \cdot (3^{2} \sqrt{73}) + 3 \cdot (3 {\sqrt{73}}^{2}) + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.2.1.2

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 3^{1 + 2} \sqrt{73} + 3 \cdot (3 {\sqrt{73}}^{2}) + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.2.2

Add $1$ and $2$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 3^{3} \sqrt{73} + 3 \cdot (3 {\sqrt{73}}^{2}) + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.3

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 3 \cdot (3 {\sqrt{73}}^{2}) + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.4

Multiply $3$ by $3$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 9 {\sqrt{73}}^{2} + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.5

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 15.2.1.12.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 9 {(73^{\frac{1}{2}})}^{2} + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.5.2

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 9 \cdot 73^{\frac{1}{2} \cdot 2} + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.5.3

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 9 \cdot 73^{\frac{2}{2}} + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.5.4

Cancel the common factor of $2$ .

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Step 15.2.1.12.5.4.1

Cancel the common factor.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 9 \cdot 73^{\frac{2}{2}} + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.5.4.2

Rewrite the expression.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 9 \cdot 73 + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.5.5

Evaluate the exponent.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 9 \cdot 73 + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.6

Multiply $9$ by $73$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 657 + {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.7

Rewrite ${\sqrt{73}}^{3}$ as $\sqrt{73^{3}}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 657 + \sqrt{73^{3}}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.8

Raise $73$ to the power of $3$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 657 + \sqrt{389017}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.9

Rewrite $389017$ as $73^{2} \cdot 73$ .

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Step 15.2.1.12.9.1

Factor $5329$ out of $389017$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 657 + \sqrt{5329 (73)}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.9.2

Rewrite $5329$ as $73^{2}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 657 + \sqrt{73^{2} \cdot 73}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.12.10

Pull terms out from under the radical.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{27 + 27 \sqrt{73} + 657 + 73 \sqrt{73}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.13

Add $27$ and $657$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{684 + 27 \sqrt{73} + 73 \sqrt{73}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.14

Add $27 \sqrt{73}$ and $73 \sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{684 + 100 \sqrt{73}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.15

Cancel the common factor of $684 + 100 \sqrt{73}$ and $512$ .

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Step 15.2.1.15.1

Factor $4$ out of $684$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{4 (171) + 100 \sqrt{73}}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.15.2

Factor $4$ out of $100 \sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{4 (171) + 4 (25 \sqrt{73})}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.15.3

Factor $4$ out of $4 (171) + 4 (25 \sqrt{73})$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{4 (171 + 25 \sqrt{73})}{512} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.15.4

Cancel the common factors.

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Step 15.2.1.15.4.1

Factor $4$ out of $512$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{4 (171 + 25 \sqrt{73})}{4 \cdot 128} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.15.4.2

Cancel the common factor.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{4 (171 + 25 \sqrt{73})}{4 \cdot 128} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.15.4.3

Rewrite the expression.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + 2 {(\frac{3 + \sqrt{73}}{8})}^{2}$

Step 15.2.1.16

Apply the product rule to $\frac{3 + \sqrt{73}}{8}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + 2 (\frac{{(3 + \sqrt{73})}^{2}}{8^{2}})$

Step 15.2.1.17

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + 2 (\frac{{(3 + \sqrt{73})}^{2}}{64})$

Step 15.2.1.18

Cancel the common factor of $2$ .

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Step 15.2.1.18.1

Factor $2$ out of $64$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + 2 (\frac{{(3 + \sqrt{73})}^{2}}{2 (32)})$

Step 15.2.1.18.2

Cancel the common factor.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + 2 (\frac{{(3 + \sqrt{73})}^{2}}{2 \cdot 32})$

Step 15.2.1.18.3

Rewrite the expression.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{{(3 + \sqrt{73})}^{2}}{32}$

Step 15.2.1.19

Rewrite ${(3 + \sqrt{73})}^{2}$ as $(3 + \sqrt{73}) (3 + \sqrt{73})$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{(3 + \sqrt{73}) (3 + \sqrt{73})}{32}$

Step 15.2.1.20

Expand $(3 + \sqrt{73}) (3 + \sqrt{73})$ using the FOIL Method.

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Step 15.2.1.20.1

Apply the distributive property.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{3 (3 + \sqrt{73}) + \sqrt{73} (3 + \sqrt{73})}{32}$

Step 15.2.1.20.2

Apply the distributive property.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{3 \cdot 3 + 3 \sqrt{73} + \sqrt{73} (3 + \sqrt{73})}{32}$

Step 15.2.1.20.3

Apply the distributive property.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{3 \cdot 3 + 3 \sqrt{73} + \sqrt{73} \cdot 3 + \sqrt{73} \sqrt{73}}{32}$

Step 15.2.1.21

Simplify and combine like terms.

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Step 15.2.1.21.1

Simplify each term.

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Step 15.2.1.21.1.1

Multiply $3$ by $3$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{9 + 3 \sqrt{73} + \sqrt{73} \cdot 3 + \sqrt{73} \sqrt{73}}{32}$

Step 15.2.1.21.1.2

Move $3$ to the left of $\sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{9 + 3 \sqrt{73} + 3 \cdot \sqrt{73} + \sqrt{73} \sqrt{73}}{32}$

Step 15.2.1.21.1.3

Combine using the product rule for radicals.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{73 \cdot 73}}{32}$

Step 15.2.1.21.1.4

Multiply $73$ by $73$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{5329}}{32}$

Step 15.2.1.21.1.5

Rewrite $5329$ as $73^{2}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{9 + 3 \sqrt{73} + 3 \sqrt{73} + \sqrt{73^{2}}}{32}$

Step 15.2.1.21.1.6

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

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{9 + 3 \sqrt{73} + 3 \sqrt{73} + 73}{32}$

Step 15.2.1.21.2

Add $9$ and $73$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{82 + 3 \sqrt{73} + 3 \sqrt{73}}{32}$

Step 15.2.1.21.3

Add $3 \sqrt{73}$ and $3 \sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{82 + 6 \sqrt{73}}{32}$

Step 15.2.1.22

Cancel the common factor of $82 + 6 \sqrt{73}$ and $32$ .

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Step 15.2.1.22.1

Factor $2$ out of $82$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{2 (41) + 6 \sqrt{73}}{32}$

Step 15.2.1.22.2

Factor $2$ out of $6 \sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{2 (41) + 2 (3 \sqrt{73})}{32}$

Step 15.2.1.22.3

Factor $2$ out of $2 (41) + 2 (3 \sqrt{73})$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{2 (41 + 3 \sqrt{73})}{32}$

Step 15.2.1.22.4

Cancel the common factors.

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Step 15.2.1.22.4.1

Factor $2$ out of $32$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{2 (41 + 3 \sqrt{73})}{2 \cdot 16}$

Step 15.2.1.22.4.2

Cancel the common factor.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{2 (41 + 3 \sqrt{73})}{2 \cdot 16}$

Step 15.2.1.22.4.3

Rewrite the expression.

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} + \frac{41 + 3 \sqrt{73}}{16}$

Step 15.2.2

Find the common denominator.

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Step 15.2.2.1

Multiply $\frac{171 + 25 \sqrt{73}}{128}$ by $\frac{4}{4}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{171 + 25 \sqrt{73}}{128} \cdot \frac{4}{4} + \frac{41 + 3 \sqrt{73}}{16}$

Step 15.2.2.2

Multiply $\frac{171 + 25 \sqrt{73}}{128}$ by $\frac{4}{4}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{(171 + 25 \sqrt{73}) \cdot 4}{128 \cdot 4} + \frac{41 + 3 \sqrt{73}}{16}$

Step 15.2.2.3

Multiply $\frac{41 + 3 \sqrt{73}}{16}$ by $\frac{32}{32}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{(171 + 25 \sqrt{73}) \cdot 4}{128 \cdot 4} + \frac{41 + 3 \sqrt{73}}{16} \cdot \frac{32}{32}$

Step 15.2.2.4

Multiply $\frac{41 + 3 \sqrt{73}}{16}$ by $\frac{32}{32}$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{(171 + 25 \sqrt{73}) \cdot 4}{128 \cdot 4} + \frac{(41 + 3 \sqrt{73}) \cdot 32}{16 \cdot 32}$

Step 15.2.2.5

Reorder the factors of $128 \cdot 4$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{(171 + 25 \sqrt{73}) \cdot 4}{4 \cdot 128} + \frac{(41 + 3 \sqrt{73}) \cdot 32}{16 \cdot 32}$

Step 15.2.2.6

Multiply $4$ by $128$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{(171 + 25 \sqrt{73}) \cdot 4}{512} + \frac{(41 + 3 \sqrt{73}) \cdot 32}{16 \cdot 32}$

Step 15.2.2.7

Multiply $16$ by $32$ .

$f (\frac{3 + \sqrt{73}}{8}) = - \frac{1169 + 123 \sqrt{73}}{512} + \frac{(171 + 25 \sqrt{73}) \cdot 4}{512} + \frac{(41 + 3 \sqrt{73}) \cdot 32}{512}$

Step 15.2.3

Combine the numerators over the common denominator.

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- (1169 + 123 \sqrt{73}) + (171 + 25 \sqrt{73}) \cdot 4 + (41 + 3 \sqrt{73}) \cdot 32}{512}$

Step 15.2.4

Simplify each term.

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Step 15.2.4.1

Apply the distributive property.

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1 \cdot 1169 - (123 \sqrt{73}) + (171 + 25 \sqrt{73}) \cdot 4 + (41 + 3 \sqrt{73}) \cdot 32}{512}$

Step 15.2.4.2

Multiply $- 1$ by $1169$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1169 - (123 \sqrt{73}) + (171 + 25 \sqrt{73}) \cdot 4 + (41 + 3 \sqrt{73}) \cdot 32}{512}$

Step 15.2.4.3

Multiply $123$ by $- 1$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1169 - 123 \sqrt{73} + (171 + 25 \sqrt{73}) \cdot 4 + (41 + 3 \sqrt{73}) \cdot 32}{512}$

Step 15.2.4.4

Apply the distributive property.

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1169 - 123 \sqrt{73} + 171 \cdot 4 + 25 \sqrt{73} \cdot 4 + (41 + 3 \sqrt{73}) \cdot 32}{512}$

Step 15.2.4.5

Multiply $171$ by $4$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1169 - 123 \sqrt{73} + 684 + 25 \sqrt{73} \cdot 4 + (41 + 3 \sqrt{73}) \cdot 32}{512}$

Step 15.2.4.6

Multiply $4$ by $25$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1169 - 123 \sqrt{73} + 684 + 100 \sqrt{73} + (41 + 3 \sqrt{73}) \cdot 32}{512}$

Step 15.2.4.7

Apply the distributive property.

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1169 - 123 \sqrt{73} + 684 + 100 \sqrt{73} + 41 \cdot 32 + 3 \sqrt{73} \cdot 32}{512}$

Step 15.2.4.8

Multiply $41$ by $32$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1169 - 123 \sqrt{73} + 684 + 100 \sqrt{73} + 1312 + 3 \sqrt{73} \cdot 32}{512}$

Step 15.2.4.9

Multiply $32$ by $3$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 1169 - 123 \sqrt{73} + 684 + 100 \sqrt{73} + 1312 + 96 \sqrt{73}}{512}$

Step 15.2.5

Simplify by adding terms.

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Step 15.2.5.1

Add $- 1169$ and $684$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{- 485 - 123 \sqrt{73} + 100 \sqrt{73} + 1312 + 96 \sqrt{73}}{512}$

Step 15.2.5.2

Add $- 485$ and $1312$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{827 - 123 \sqrt{73} + 100 \sqrt{73} + 96 \sqrt{73}}{512}$

Step 15.2.5.3

Add $- 123 \sqrt{73}$ and $100 \sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{827 - 23 \sqrt{73} + 96 \sqrt{73}}{512}$

Step 15.2.5.4

Add $- 23 \sqrt{73}$ and $96 \sqrt{73}$ .

$f (\frac{3 + \sqrt{73}}{8}) = \frac{827 + 73 \sqrt{73}}{512}$

Step 15.2.6

The final answer is $\frac{827 + 73 \sqrt{73}}{512}$ .

$y = \frac{827 + 73 \sqrt{73}}{512}$

Step 16

Evaluate the second derivative at $x = \frac{3 - \sqrt{73}}{8}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- 12 {(\frac{3 - \sqrt{73}}{8})}^{2} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17

Evaluate the second derivative.

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Step 17.1

Simplify each term.

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Step 17.1.1

Apply the product rule to $\frac{3 - \sqrt{73}}{8}$ .

$- 12 \frac{{(3 - \sqrt{73})}^{2}}{8^{2}} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.2

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

$- 12 \frac{{(3 - \sqrt{73})}^{2}}{64} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.3

Cancel the common factor of $4$ .

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Step 17.1.3.1

Factor $4$ out of $- 12$ .

$4 (- 3) \frac{{(3 - \sqrt{73})}^{2}}{64} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.3.2

Factor $4$ out of $64$ .

$4 \cdot - 3 \frac{{(3 - \sqrt{73})}^{2}}{4 \cdot 16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.3.3

Cancel the common factor.

$4 \cdot - 3 \frac{{(3 - \sqrt{73})}^{2}}{4 \cdot 16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.3.4

Rewrite the expression.

$- 3 \frac{{(3 - \sqrt{73})}^{2}}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.4

Combine $- 3$ and $\frac{{(3 - \sqrt{73})}^{2}}{16}$ .

$\frac{- 3 {(3 - \sqrt{73})}^{2}}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.5

Rewrite ${(3 - \sqrt{73})}^{2}$ as $(3 - \sqrt{73}) (3 - \sqrt{73})$ .

$\frac{- 3 ((3 - \sqrt{73}) (3 - \sqrt{73}))}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.6

Expand $(3 - \sqrt{73}) (3 - \sqrt{73})$ using the FOIL Method.

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Step 17.1.6.1

Apply the distributive property.

$\frac{- 3 (3 (3 - \sqrt{73}) - \sqrt{73} (3 - \sqrt{73}))}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.6.2

Apply the distributive property.

$\frac{- 3 (3 \cdot 3 + 3 (- \sqrt{73}) - \sqrt{73} (3 - \sqrt{73}))}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.6.3

Apply the distributive property.

$\frac{- 3 (3 \cdot 3 + 3 (- \sqrt{73}) - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73}))}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7

Simplify and combine like terms.

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Step 17.1.7.1

Simplify each term.

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Step 17.1.7.1.1

Multiply $3$ by $3$ .

$\frac{- 3 (9 + 3 (- \sqrt{73}) - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73}))}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.2

Multiply $- 1$ by $3$ .

$\frac{- 3 (9 - 3 \sqrt{73} - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73}))}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.3

Multiply $3$ by $- 1$ .

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} - \sqrt{73} (- \sqrt{73}))}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.4

Multiply $- \sqrt{73} (- \sqrt{73})$ .

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Step 17.1.7.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 1 \sqrt{73} \sqrt{73})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.4.2

Multiply $\sqrt{73}$ by $1$ .

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + \sqrt{73} \sqrt{73})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.4.3

Raise $\sqrt{73}$ to the power of $1$ .

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{1} \sqrt{73})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.4.4

Raise $\sqrt{73}$ to the power of $1$ .

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{1} {\sqrt{73}}^{1})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.4.5

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

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{1 + 1})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.4.6

Add $1$ and $1$ .

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{2})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.5

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 17.1.7.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + {(73^{\frac{1}{2}})}^{2})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.5.2

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

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{1}{2} \cdot 2})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.5.3

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

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{2}{2}})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.5.4

Cancel the common factor of $2$ .

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Step 17.1.7.1.5.4.1

Cancel the common factor.

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{2}{2}})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.5.4.2

Rewrite the expression.

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{1})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.1.5.5

Evaluate the exponent.

$\frac{- 3 (9 - 3 \sqrt{73} - 3 \sqrt{73} + 73)}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.2

Add $9$ and $73$ .

$\frac{- 3 (82 - 3 \sqrt{73} - 3 \sqrt{73})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.7.3

Subtract $3 \sqrt{73}$ from $- 3 \sqrt{73}$ .

$\frac{- 3 (82 - 6 \sqrt{73})}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.8

Cancel the common factor of $82 - 6 \sqrt{73}$ and $16$ .

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Step 17.1.8.1

Factor $2$ out of $- 3 (82 - 6 \sqrt{73})$ .

$\frac{2 (- 3 (41 - 3 \sqrt{73}))}{16} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.8.2

Cancel the common factors.

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Step 17.1.8.2.1

Factor $2$ out of $16$ .

$\frac{2 (- 3 (41 - 3 \sqrt{73}))}{2 (8)} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.8.2.2

Cancel the common factor.

$\frac{2 (- 3 (41 - 3 \sqrt{73}))}{2 \cdot 8} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.8.2.3

Rewrite the expression.

$\frac{- 3 (41 - 3 \sqrt{73})}{8} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.9

Move the negative in front of the fraction.

$- \frac{3 (41 - 3 \sqrt{73})}{8} + 6 (\frac{3 - \sqrt{73}}{8}) + 4$

Step 17.1.10

Cancel the common factor of $2$ .

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Step 17.1.10.1

Factor $2$ out of $6$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + 2 (3) \frac{3 - \sqrt{73}}{8} + 4$

Step 17.1.10.2

Factor $2$ out of $8$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + 2 \cdot 3 \frac{3 - \sqrt{73}}{2 \cdot 4} + 4$

Step 17.1.10.3

Cancel the common factor.

$- \frac{3 (41 - 3 \sqrt{73})}{8} + 2 \cdot 3 \frac{3 - \sqrt{73}}{2 \cdot 4} + 4$

Step 17.1.10.4

Rewrite the expression.

$- \frac{3 (41 - 3 \sqrt{73})}{8} + 3 \frac{3 - \sqrt{73}}{4} + 4$

Step 17.1.11

Combine $3$ and $\frac{3 - \sqrt{73}}{4}$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (3 - \sqrt{73})}{4} + 4$

Step 17.2

Find the common denominator.

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Step 17.2.1

Multiply $\frac{3 (3 - \sqrt{73})}{4}$ by $\frac{2}{2}$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (3 - \sqrt{73})}{4} \cdot \frac{2}{2} + 4$

Step 17.2.2

Multiply $\frac{3 (3 - \sqrt{73})}{4}$ by $\frac{2}{2}$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (3 - \sqrt{73}) \cdot 2}{4 \cdot 2} + 4$

Step 17.2.3

Write $4$ as a fraction with denominator $1$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (3 - \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{4}{1}$

Step 17.2.4

Multiply $\frac{4}{1}$ by $\frac{8}{8}$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (3 - \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{4}{1} \cdot \frac{8}{8}$

Step 17.2.5

Multiply $\frac{4}{1}$ by $\frac{8}{8}$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (3 - \sqrt{73}) \cdot 2}{4 \cdot 2} + \frac{4 \cdot 8}{8}$

Step 17.2.6

Reorder the factors of $4 \cdot 2$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (3 - \sqrt{73}) \cdot 2}{2 \cdot 4} + \frac{4 \cdot 8}{8}$

Step 17.2.7

Multiply $2$ by $4$ .

$- \frac{3 (41 - 3 \sqrt{73})}{8} + \frac{3 (3 - \sqrt{73}) \cdot 2}{8} + \frac{4 \cdot 8}{8}$

Step 17.3

Combine the numerators over the common denominator.

$\frac{- 3 (41 - 3 \sqrt{73}) + 3 (3 - \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 17.4

Simplify each term.

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Step 17.4.1

Apply the distributive property.

$\frac{- 3 \cdot 41 - 3 (- 3 \sqrt{73}) + 3 (3 - \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 17.4.2

Multiply $- 3$ by $41$ .

$\frac{- 123 - 3 (- 3 \sqrt{73}) + 3 (3 - \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 17.4.3

Multiply $- 3$ by $- 3$ .

$\frac{- 123 + 9 \sqrt{73} + 3 (3 - \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 17.4.4

Apply the distributive property.

$\frac{- 123 + 9 \sqrt{73} + (3 \cdot 3 + 3 (- \sqrt{73})) \cdot 2 + 4 \cdot 8}{8}$

Step 17.4.5

Multiply $3$ by $3$ .

$\frac{- 123 + 9 \sqrt{73} + (9 + 3 (- \sqrt{73})) \cdot 2 + 4 \cdot 8}{8}$

Step 17.4.6

Multiply $- 1$ by $3$ .

$\frac{- 123 + 9 \sqrt{73} + (9 - 3 \sqrt{73}) \cdot 2 + 4 \cdot 8}{8}$

Step 17.4.7

Apply the distributive property.

$\frac{- 123 + 9 \sqrt{73} + 9 \cdot 2 - 3 \sqrt{73} \cdot 2 + 4 \cdot 8}{8}$

Step 17.4.8

Multiply $9$ by $2$ .

$\frac{- 123 + 9 \sqrt{73} + 18 - 3 \sqrt{73} \cdot 2 + 4 \cdot 8}{8}$

Step 17.4.9

Multiply $2$ by $- 3$ .

$\frac{- 123 + 9 \sqrt{73} + 18 - 6 \sqrt{73} + 4 \cdot 8}{8}$

Step 17.4.10

Multiply $4$ by $8$ .

$\frac{- 123 + 9 \sqrt{73} + 18 - 6 \sqrt{73} + 32}{8}$

Step 17.5

Simplify terms.

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Step 17.5.1

Add $- 123$ and $18$ .

$\frac{- 105 + 9 \sqrt{73} - 6 \sqrt{73} + 32}{8}$

Step 17.5.2

Add $- 105$ and $32$ .

$\frac{- 73 + 9 \sqrt{73} - 6 \sqrt{73}}{8}$

Step 17.5.3

Subtract $6 \sqrt{73}$ from $9 \sqrt{73}$ .

$\frac{- 73 + 3 \sqrt{73}}{8}$

Step 17.5.4

Rewrite $- 73$ as $- 1 (73)$ .

$\frac{- 1 (73) + 3 \sqrt{73}}{8}$

Step 17.5.5

Factor $- 1$ out of $3 \sqrt{73}$ .

$\frac{- 1 (73) - (- 3 \sqrt{73})}{8}$

Step 17.5.6

Factor $- 1$ out of $- 1 (73) - (- 3 \sqrt{73})$ .

$\frac{- 1 (73 - 3 \sqrt{73})}{8}$

Step 17.5.7

Move the negative in front of the fraction.

$- \frac{73 - 3 \sqrt{73}}{8}$

Step 18

$x = \frac{3 - \sqrt{73}}{8}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{3 - \sqrt{73}}{8}$ is a local maximum

Step 19

Find the y-value when $x = \frac{3 - \sqrt{73}}{8}$ .

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Step 19.1

Replace the variable $x$ with $\frac{3 - \sqrt{73}}{8}$ in the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - {(\frac{3 - \sqrt{73}}{8})}^{4} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2

Simplify the result.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Apply the product rule to $\frac{3 - \sqrt{73}}{8}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{{(3 - \sqrt{73})}^{4}}{8^{4}} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.2

Raise $8$ to the power of $4$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{{(3 - \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.3

Use the Binomial Theorem.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{3^{4} + 4 \cdot (3^{3} (- \sqrt{73})) + 6 \cdot (3^{2} {(- \sqrt{73})}^{2}) + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4

Simplify each term.

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Step 19.2.1.4.1

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 + 4 \cdot (3^{3} (- \sqrt{73})) + 6 \cdot (3^{2} {(- \sqrt{73})}^{2}) + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.2

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 + 4 \cdot (27 (- \sqrt{73})) + 6 \cdot (3^{2} {(- \sqrt{73})}^{2}) + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.3

Multiply $4$ by $27$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 + 108 (- \sqrt{73}) + 6 \cdot (3^{2} {(- \sqrt{73})}^{2}) + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.4

Multiply $- 1$ by $108$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 6 \cdot (3^{2} {(- \sqrt{73})}^{2}) + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.5

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 6 \cdot (9 {(- \sqrt{73})}^{2}) + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.6

Multiply $6$ by $9$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 54 {(- \sqrt{73})}^{2} + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.7

Apply the product rule to $- \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 54 ({(- 1)}^{2} {\sqrt{73}}^{2}) + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.8

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 54 (1 {\sqrt{73}}^{2}) + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.9

Multiply ${\sqrt{73}}^{2}$ by $1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 54 {\sqrt{73}}^{2} + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.10

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 19.2.1.4.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 54 {(73^{\frac{1}{2}})}^{2} + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.10.2

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 54 \cdot 73^{\frac{1}{2} \cdot 2} + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.10.3

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 54 \cdot 73^{\frac{2}{2}} + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.10.4

Cancel the common factor of $2$ .

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Step 19.2.1.4.10.4.1

Cancel the common factor.

Step 19.2.1.4.10.4.2

Rewrite the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 54 \cdot 73 + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.10.5

Evaluate the exponent.

Step 19.2.1.4.11

Multiply $54$ by $73$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 4 \cdot (3 {(- \sqrt{73})}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.12

Multiply $4$ by $3$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 {(- \sqrt{73})}^{3} + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.13

Apply the product rule to $- \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 ({(- 1)}^{3} {\sqrt{73}}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.14

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 (- {\sqrt{73}}^{3}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.15

Rewrite ${\sqrt{73}}^{3}$ as $\sqrt{73^{3}}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 (- \sqrt{73^{3}}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.16

Raise $73$ to the power of $3$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 (- \sqrt{389017}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.17

Rewrite $389017$ as $73^{2} \cdot 73$ .

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Step 19.2.1.4.17.1

Factor $5329$ out of $389017$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 (- \sqrt{5329 (73)}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.17.2

Rewrite $5329$ as $73^{2}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 (- \sqrt{73^{2} \cdot 73}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.18

Pull terms out from under the radical.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 (- (73 \sqrt{73})) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.19

Multiply $73$ by $- 1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 + 12 (- 73 \sqrt{73}) + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.20

Multiply $- 73$ by $12$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + {(- \sqrt{73})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.21

Apply the product rule to $- \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + {(- 1)}^{4} {\sqrt{73}}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.22

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 1 {\sqrt{73}}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.23

Multiply ${\sqrt{73}}^{4}$ by $1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + {\sqrt{73}}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.24

Rewrite ${\sqrt{73}}^{4}$ as $73^{2}$ .

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Step 19.2.1.4.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + {(73^{\frac{1}{2}})}^{4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.24.2

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 73^{\frac{1}{2} \cdot 4}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.24.3

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 73^{\frac{4}{2}}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.24.4

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

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Step 19.2.1.4.24.4.1

Factor $2$ out of $4$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 73^{\frac{2 \cdot 2}{2}}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.24.4.2

Cancel the common factors.

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Step 19.2.1.4.24.4.2.1

Factor $2$ out of $2$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 73^{\frac{2 \cdot 2}{2 (1)}}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.24.4.2.2

Cancel the common factor.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 73^{\frac{2 \cdot 2}{2 \cdot 1}}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.24.4.2.3

Rewrite the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 73^{\frac{2}{1}}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.24.4.2.4

Divide $2$ by $1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 73^{2}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.4.25

Raise $73$ to the power of $2$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{81 - 108 \sqrt{73} + 3942 - 876 \sqrt{73} + 5329}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.5

Add $81$ and $3942$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{4023 - 108 \sqrt{73} - 876 \sqrt{73} + 5329}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.6

Add $4023$ and $5329$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{9352 - 108 \sqrt{73} - 876 \sqrt{73}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.7

Subtract $876 \sqrt{73}$ from $- 108 \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{9352 - 984 \sqrt{73}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.8

Cancel the common factor of $9352 - 984 \sqrt{73}$ and $4096$ .

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Step 19.2.1.8.1

Factor $8$ out of $9352$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{8 (1169) - 984 \sqrt{73}}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.8.2

Factor $8$ out of $- 984 \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{8 (1169) + 8 (- 123 \sqrt{73})}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.8.3

Factor $8$ out of $8 (1169) + 8 (- 123 \sqrt{73})$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{8 (1169 - 123 \sqrt{73})}{4096} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.8.4

Cancel the common factors.

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Step 19.2.1.8.4.1

Factor $8$ out of $4096$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{8 (1169 - 123 \sqrt{73})}{8 \cdot 512} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.8.4.2

Cancel the common factor.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{8 (1169 - 123 \sqrt{73})}{8 \cdot 512} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.8.4.3

Rewrite the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + {(\frac{3 - \sqrt{73}}{8})}^{3} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.9

Apply the product rule to $\frac{3 - \sqrt{73}}{8}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{{(3 - \sqrt{73})}^{3}}{8^{3}} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.10

Raise $8$ to the power of $3$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{{(3 - \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.11

Use the Binomial Theorem.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{3^{3} + 3 \cdot (3^{2} (- \sqrt{73})) + 3 \cdot (3 {(- \sqrt{73})}^{2}) + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12

Simplify each term.

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Step 19.2.1.12.1

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 + 3 \cdot (3^{2} (- \sqrt{73})) + 3 \cdot (3 {(- \sqrt{73})}^{2}) + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.2

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

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Step 19.2.1.12.2.1

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

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Step 19.2.1.12.2.1.1

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

Step 19.2.1.12.2.1.2

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 + 3^{1 + 2} (- \sqrt{73}) + 3 \cdot (3 {(- \sqrt{73})}^{2}) + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.2.2

Add $1$ and $2$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 + 3^{3} (- \sqrt{73}) + 3 \cdot (3 {(- \sqrt{73})}^{2}) + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.3

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 + 27 (- \sqrt{73}) + 3 \cdot (3 {(- \sqrt{73})}^{2}) + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.4

Multiply $- 1$ by $27$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 3 \cdot (3 {(- \sqrt{73})}^{2}) + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.5

Multiply $3$ by $3$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 {(- \sqrt{73})}^{2} + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.6

Apply the product rule to $- \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 ({(- 1)}^{2} {\sqrt{73}}^{2}) + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.7

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 (1 {\sqrt{73}}^{2}) + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.8

Multiply ${\sqrt{73}}^{2}$ by $1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 {\sqrt{73}}^{2} + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.9

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 19.2.1.12.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 {(73^{\frac{1}{2}})}^{2} + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.9.2

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 \cdot 73^{\frac{1}{2} \cdot 2} + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.9.3

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 \cdot 73^{\frac{2}{2}} + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.9.4

Cancel the common factor of $2$ .

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Step 19.2.1.12.9.4.1

Cancel the common factor.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 \cdot 73^{\frac{2}{2}} + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.9.4.2

Rewrite the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 \cdot 73 + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.9.5

Evaluate the exponent.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 9 \cdot 73 + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.10

Multiply $9$ by $73$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 + {(- \sqrt{73})}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.11

Apply the product rule to $- \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 + {(- 1)}^{3} {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.12

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 - {\sqrt{73}}^{3}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.13

Rewrite ${\sqrt{73}}^{3}$ as $\sqrt{73^{3}}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 - \sqrt{73^{3}}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.14

Raise $73$ to the power of $3$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 - \sqrt{389017}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.15

Rewrite $389017$ as $73^{2} \cdot 73$ .

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Step 19.2.1.12.15.1

Factor $5329$ out of $389017$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 - \sqrt{5329 (73)}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.15.2

Rewrite $5329$ as $73^{2}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 - \sqrt{73^{2} \cdot 73}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.16

Pull terms out from under the radical.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 - (73 \sqrt{73})}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.12.17

Multiply $73$ by $- 1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{27 - 27 \sqrt{73} + 657 - 73 \sqrt{73}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.13

Add $27$ and $657$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{684 - 27 \sqrt{73} - 73 \sqrt{73}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.14

Subtract $73 \sqrt{73}$ from $- 27 \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{684 - 100 \sqrt{73}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.15

Cancel the common factor of $684 - 100 \sqrt{73}$ and $512$ .

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Step 19.2.1.15.1

Factor $4$ out of $684$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{4 (171) - 100 \sqrt{73}}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.15.2

Factor $4$ out of $- 100 \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{4 (171) + 4 (- 25 \sqrt{73})}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.15.3

Factor $4$ out of $4 (171) + 4 (- 25 \sqrt{73})$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{4 (171 - 25 \sqrt{73})}{512} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.15.4

Cancel the common factors.

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Step 19.2.1.15.4.1

Factor $4$ out of $512$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{4 (171 - 25 \sqrt{73})}{4 \cdot 128} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.15.4.2

Cancel the common factor.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{4 (171 - 25 \sqrt{73})}{4 \cdot 128} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.15.4.3

Rewrite the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + 2 {(\frac{3 - \sqrt{73}}{8})}^{2}$

Step 19.2.1.16

Apply the product rule to $\frac{3 - \sqrt{73}}{8}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + 2 (\frac{{(3 - \sqrt{73})}^{2}}{8^{2}})$

Step 19.2.1.17

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + 2 (\frac{{(3 - \sqrt{73})}^{2}}{64})$

Step 19.2.1.18

Cancel the common factor of $2$ .

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Step 19.2.1.18.1

Factor $2$ out of $64$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + 2 (\frac{{(3 - \sqrt{73})}^{2}}{2 (32)})$

Step 19.2.1.18.2

Cancel the common factor.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + 2 (\frac{{(3 - \sqrt{73})}^{2}}{2 \cdot 32})$

Step 19.2.1.18.3

Rewrite the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{{(3 - \sqrt{73})}^{2}}{32}$

Step 19.2.1.19

Rewrite ${(3 - \sqrt{73})}^{2}$ as $(3 - \sqrt{73}) (3 - \sqrt{73})$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{(3 - \sqrt{73}) (3 - \sqrt{73})}{32}$

Step 19.2.1.20

Expand $(3 - \sqrt{73}) (3 - \sqrt{73})$ using the FOIL Method.

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Step 19.2.1.20.1

Apply the distributive property.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{3 (3 - \sqrt{73}) - \sqrt{73} (3 - \sqrt{73})}{32}$

Step 19.2.1.20.2

Apply the distributive property.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{3 \cdot 3 + 3 (- \sqrt{73}) - \sqrt{73} (3 - \sqrt{73})}{32}$

Step 19.2.1.20.3

Apply the distributive property.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{3 \cdot 3 + 3 (- \sqrt{73}) - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73})}{32}$

Step 19.2.1.21

Simplify and combine like terms.

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Step 19.2.1.21.1

Simplify each term.

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Step 19.2.1.21.1.1

Multiply $3$ by $3$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 + 3 (- \sqrt{73}) - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73})}{32}$

Step 19.2.1.21.1.2

Multiply $- 1$ by $3$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - \sqrt{73} \cdot 3 - \sqrt{73} (- \sqrt{73})}{32}$

Step 19.2.1.21.1.3

Multiply $3$ by $- 1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} - \sqrt{73} (- \sqrt{73})}{32}$

Step 19.2.1.21.1.4

Multiply $- \sqrt{73} (- \sqrt{73})$ .

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Step 19.2.1.21.1.4.1

Multiply $- 1$ by $- 1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 1 \sqrt{73} \sqrt{73}}{32}$

Step 19.2.1.21.1.4.2

Multiply $\sqrt{73}$ by $1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + \sqrt{73} \sqrt{73}}{32}$

Step 19.2.1.21.1.4.3

Raise $\sqrt{73}$ to the power of $1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + \sqrt{73} \sqrt{73}}{32}$

Step 19.2.1.21.1.4.4

Raise $\sqrt{73}$ to the power of $1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + \sqrt{73} \sqrt{73}}{32}$

Step 19.2.1.21.1.4.5

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{1 + 1}}{32}$

Step 19.2.1.21.1.4.6

Add $1$ and $1$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + {\sqrt{73}}^{2}}{32}$

Step 19.2.1.21.1.5

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 19.2.1.21.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + {(73^{\frac{1}{2}})}^{2}}{32}$

Step 19.2.1.21.1.5.2

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{1}{2} \cdot 2}}{32}$

Step 19.2.1.21.1.5.3

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

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{2}{2}}}{32}$

Step 19.2.1.21.1.5.4

Cancel the common factor of $2$ .

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Step 19.2.1.21.1.5.4.1

Cancel the common factor.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73^{\frac{2}{2}}}{32}$

Step 19.2.1.21.1.5.4.2

Rewrite the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73}{32}$

Step 19.2.1.21.1.5.5

Evaluate the exponent.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{9 - 3 \sqrt{73} - 3 \sqrt{73} + 73}{32}$

Step 19.2.1.21.2

Add $9$ and $73$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{82 - 3 \sqrt{73} - 3 \sqrt{73}}{32}$

Step 19.2.1.21.3

Subtract $3 \sqrt{73}$ from $- 3 \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{82 - 6 \sqrt{73}}{32}$

Step 19.2.1.22

Cancel the common factor of $82 - 6 \sqrt{73}$ and $32$ .

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Step 19.2.1.22.1

Factor $2$ out of $82$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{2 (41) - 6 \sqrt{73}}{32}$

Step 19.2.1.22.2

Factor $2$ out of $- 6 \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{2 (41) + 2 (- 3 \sqrt{73})}{32}$

Step 19.2.1.22.3

Factor $2$ out of $2 (41) + 2 (- 3 \sqrt{73})$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{2 (41 - 3 \sqrt{73})}{32}$

Step 19.2.1.22.4

Cancel the common factors.

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Step 19.2.1.22.4.1

Factor $2$ out of $32$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{2 (41 - 3 \sqrt{73})}{2 \cdot 16}$

Step 19.2.1.22.4.2

Cancel the common factor.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{2 (41 - 3 \sqrt{73})}{2 \cdot 16}$

Step 19.2.1.22.4.3

Rewrite the expression.

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} + \frac{41 - 3 \sqrt{73}}{16}$

Step 19.2.2

Find the common denominator.

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Step 19.2.2.1

Multiply $\frac{171 - 25 \sqrt{73}}{128}$ by $\frac{4}{4}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{171 - 25 \sqrt{73}}{128} \cdot \frac{4}{4} + \frac{41 - 3 \sqrt{73}}{16}$

Step 19.2.2.2

Multiply $\frac{171 - 25 \sqrt{73}}{128}$ by $\frac{4}{4}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{(171 - 25 \sqrt{73}) \cdot 4}{128 \cdot 4} + \frac{41 - 3 \sqrt{73}}{16}$

Step 19.2.2.3

Multiply $\frac{41 - 3 \sqrt{73}}{16}$ by $\frac{32}{32}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{(171 - 25 \sqrt{73}) \cdot 4}{128 \cdot 4} + \frac{41 - 3 \sqrt{73}}{16} \cdot \frac{32}{32}$

Step 19.2.2.4

Multiply $\frac{41 - 3 \sqrt{73}}{16}$ by $\frac{32}{32}$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{(171 - 25 \sqrt{73}) \cdot 4}{128 \cdot 4} + \frac{(41 - 3 \sqrt{73}) \cdot 32}{16 \cdot 32}$

Step 19.2.2.5

Reorder the factors of $128 \cdot 4$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{(171 - 25 \sqrt{73}) \cdot 4}{4 \cdot 128} + \frac{(41 - 3 \sqrt{73}) \cdot 32}{16 \cdot 32}$

Step 19.2.2.6

Multiply $4$ by $128$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{(171 - 25 \sqrt{73}) \cdot 4}{512} + \frac{(41 - 3 \sqrt{73}) \cdot 32}{16 \cdot 32}$

Step 19.2.2.7

Multiply $16$ by $32$ .

$f (\frac{3 - \sqrt{73}}{8}) = - \frac{1169 - 123 \sqrt{73}}{512} + \frac{(171 - 25 \sqrt{73}) \cdot 4}{512} + \frac{(41 - 3 \sqrt{73}) \cdot 32}{512}$

Step 19.2.3

Combine the numerators over the common denominator.

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- (1169 - 123 \sqrt{73}) + (171 - 25 \sqrt{73}) \cdot 4 + (41 - 3 \sqrt{73}) \cdot 32}{512}$

Step 19.2.4

Simplify each term.

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Step 19.2.4.1

Apply the distributive property.

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1 \cdot 1169 - (- 123 \sqrt{73}) + (171 - 25 \sqrt{73}) \cdot 4 + (41 - 3 \sqrt{73}) \cdot 32}{512}$

Step 19.2.4.2

Multiply $- 1$ by $1169$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1169 - (- 123 \sqrt{73}) + (171 - 25 \sqrt{73}) \cdot 4 + (41 - 3 \sqrt{73}) \cdot 32}{512}$

Step 19.2.4.3

Multiply $- 123$ by $- 1$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1169 + 123 \sqrt{73} + (171 - 25 \sqrt{73}) \cdot 4 + (41 - 3 \sqrt{73}) \cdot 32}{512}$

Step 19.2.4.4

Apply the distributive property.

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1169 + 123 \sqrt{73} + 171 \cdot 4 - 25 \sqrt{73} \cdot 4 + (41 - 3 \sqrt{73}) \cdot 32}{512}$

Step 19.2.4.5

Multiply $171$ by $4$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1169 + 123 \sqrt{73} + 684 - 25 \sqrt{73} \cdot 4 + (41 - 3 \sqrt{73}) \cdot 32}{512}$

Step 19.2.4.6

Multiply $4$ by $- 25$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1169 + 123 \sqrt{73} + 684 - 100 \sqrt{73} + (41 - 3 \sqrt{73}) \cdot 32}{512}$

Step 19.2.4.7

Apply the distributive property.

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1169 + 123 \sqrt{73} + 684 - 100 \sqrt{73} + 41 \cdot 32 - 3 \sqrt{73} \cdot 32}{512}$

Step 19.2.4.8

Multiply $41$ by $32$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1169 + 123 \sqrt{73} + 684 - 100 \sqrt{73} + 1312 - 3 \sqrt{73} \cdot 32}{512}$

Step 19.2.4.9

Multiply $32$ by $- 3$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 1169 + 123 \sqrt{73} + 684 - 100 \sqrt{73} + 1312 - 96 \sqrt{73}}{512}$

Step 19.2.5

Simplify by adding terms.

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Step 19.2.5.1

Add $- 1169$ and $684$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{- 485 + 123 \sqrt{73} - 100 \sqrt{73} + 1312 - 96 \sqrt{73}}{512}$

Step 19.2.5.2

Add $- 485$ and $1312$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{827 + 123 \sqrt{73} - 100 \sqrt{73} - 96 \sqrt{73}}{512}$

Step 19.2.5.3

Subtract $100 \sqrt{73}$ from $123 \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{827 + 23 \sqrt{73} - 96 \sqrt{73}}{512}$

Step 19.2.5.4

Subtract $96 \sqrt{73}$ from $23 \sqrt{73}$ .

$f (\frac{3 - \sqrt{73}}{8}) = \frac{827 - 73 \sqrt{73}}{512}$

Step 19.2.6

The final answer is $\frac{827 - 73 \sqrt{73}}{512}$ .

$y = \frac{827 - 73 \sqrt{73}}{512}$

Step 20

These are the local extrema for $f (x) = - x^{4} + x^{3} + 2 x^{2}$ .

$(0, 0)$ is a local minima

$(\frac{3 + \sqrt{73}}{8}, \frac{827 + 73 \sqrt{73}}{512})$ is a local maxima

$(\frac{3 - \sqrt{73}}{8}, \frac{827 - 73 \sqrt{73}}{512})$ is a local maxima

Step 21