Calculus Examples

$4 x^{3} - \frac{27}{2} x^{2} + 8 x$

Step 1

Write $4 x^{3} - \frac{27}{2} x^{2} + 8 x$ as a function.

$f (x) = 4 x^{3} - \frac{27}{2} x^{2} + 8 x$

Step 2

Find the first derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $4 x^{3} - \frac{27}{2} x^{2} + 8 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$

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

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$

Step 2.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$

Step 2.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{27}{2} x^{2}]$ .

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Step 2.3.1

Since $- \frac{27}{2}$ is constant with respect to $x$ , the derivative of $- \frac{27}{2} x^{2}$ with respect to $x$ is $- \frac{27}{2} \frac{d}{d x} [x^{2}]$ .

$12 x^{2} - \frac{27}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x]$

Step 2.3.2

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

$12 x^{2} - \frac{27}{2} (2 x) + \frac{d}{d x} [8 x]$

Step 2.3.3

Multiply $2$ by $- 1$ .

$12 x^{2} - 2 (\frac{27}{2}) x + \frac{d}{d x} [8 x]$

Step 2.3.4

Combine $- 2$ and $\frac{27}{2}$ .

$12 x^{2} + \frac{- 2 \cdot 27}{2} x + \frac{d}{d x} [8 x]$

Step 2.3.5

Multiply $- 2$ by $27$ .

$12 x^{2} + \frac{- 54}{2} x + \frac{d}{d x} [8 x]$

Step 2.3.6

Combine $\frac{- 54}{2}$ and $x$ .

$12 x^{2} + \frac{- 54 x}{2} + \frac{d}{d x} [8 x]$

Step 2.3.7

Cancel the common factor of $- 54$ and $2$ .

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Step 2.3.7.1

Factor $2$ out of $- 54 x$ .

$12 x^{2} + \frac{2 (- 27 x)}{2} + \frac{d}{d x} [8 x]$

Step 2.3.7.2

Cancel the common factors.

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Step 2.3.7.2.1

Factor $2$ out of $2$ .

$12 x^{2} + \frac{2 (- 27 x)}{2 (1)} + \frac{d}{d x} [8 x]$

Step 2.3.7.2.2

Cancel the common factor.

$12 x^{2} + \frac{2 (- 27 x)}{2 \cdot 1} + \frac{d}{d x} [8 x]$

Step 2.3.7.2.3

Rewrite the expression.

$12 x^{2} + \frac{- 27 x}{1} + \frac{d}{d x} [8 x]$

Step 2.3.7.2.4

Divide $- 27 x$ by $1$ .

$12 x^{2} - 27 x + \frac{d}{d x} [8 x]$

Step 2.4

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

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Step 2.4.1

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

$12 x^{2} - 27 x + 8 \frac{d}{d x} [x]$

Step 2.4.2

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

$12 x^{2} - 27 x + 8 \cdot 1$

Step 2.4.3

Multiply $8$ by $1$ .

$12 x^{2} - 27 x + 8$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $12 x^{2} - 27 x + 8$ with respect to $x$ is $\frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 27 x] + \frac{d}{d x} [8]$ .

$f'' (x) = \frac{d}{d x} (12 x^{2}) + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (8)$

Step 3.2

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 3.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 12 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (8)$

Step 3.2.2

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

$f'' (x) = 12 (2 x) + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (8)$

Step 3.2.3

Multiply $2$ by $12$ .

$f'' (x) = 24 x + \frac{d}{d x} (- 27 x) + \frac{d}{d x} (8)$

Step 3.3

Evaluate $\frac{d}{d x} [- 27 x]$ .

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Step 3.3.1

Since $- 27$ is constant with respect to $x$ , the derivative of $- 27 x$ with respect to $x$ is $- 27 \frac{d}{d x} [x]$ .

$f'' (x) = 24 x - 27 \frac{d}{d x} x + \frac{d}{d x} (8)$

Step 3.3.2

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

$f'' (x) = 24 x - 27 \cdot 1 + \frac{d}{d x} (8)$

Step 3.3.3

Multiply $- 27$ by $1$ .

$f'' (x) = 24 x - 27 + \frac{d}{d x} (8)$

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$f'' (x) = 24 x - 27 + 0$

Step 3.4.2

Add $24 x - 27$ and $0$ .

$f'' (x) = 24 x - 27$

Step 4

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

$12 x^{2} - 27 x + 8 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

By the Sum Rule, the derivative of $4 x^{3} - \frac{27}{2} x^{2} + 8 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$

Step 5.1.2

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

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Step 5.1.2.1

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

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$

Step 5.1.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$

Step 5.1.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- \frac{27}{2} x^{2}] + \frac{d}{d x} [8 x]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- \frac{27}{2} x^{2}]$ .

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Step 5.1.3.1

Since $- \frac{27}{2}$ is constant with respect to $x$ , the derivative of $- \frac{27}{2} x^{2}$ with respect to $x$ is $- \frac{27}{2} \frac{d}{d x} [x^{2}]$ .

$12 x^{2} - \frac{27}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x]$

Step 5.1.3.2

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

$12 x^{2} - \frac{27}{2} (2 x) + \frac{d}{d x} [8 x]$

Step 5.1.3.3

Multiply $2$ by $- 1$ .

$12 x^{2} - 2 (\frac{27}{2}) x + \frac{d}{d x} [8 x]$

Step 5.1.3.4

Combine $- 2$ and $\frac{27}{2}$ .

$12 x^{2} + \frac{- 2 \cdot 27}{2} x + \frac{d}{d x} [8 x]$

Step 5.1.3.5

Multiply $- 2$ by $27$ .

$12 x^{2} + \frac{- 54}{2} x + \frac{d}{d x} [8 x]$

Step 5.1.3.6

Combine $\frac{- 54}{2}$ and $x$ .

$12 x^{2} + \frac{- 54 x}{2} + \frac{d}{d x} [8 x]$

Step 5.1.3.7

Cancel the common factor of $- 54$ and $2$ .

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Step 5.1.3.7.1

Factor $2$ out of $- 54 x$ .

$12 x^{2} + \frac{2 (- 27 x)}{2} + \frac{d}{d x} [8 x]$

Step 5.1.3.7.2

Cancel the common factors.

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Step 5.1.3.7.2.1

Factor $2$ out of $2$ .

$12 x^{2} + \frac{2 (- 27 x)}{2 (1)} + \frac{d}{d x} [8 x]$

Step 5.1.3.7.2.2

Cancel the common factor.

$12 x^{2} + \frac{2 (- 27 x)}{2 \cdot 1} + \frac{d}{d x} [8 x]$

Step 5.1.3.7.2.3

Rewrite the expression.

$12 x^{2} + \frac{- 27 x}{1} + \frac{d}{d x} [8 x]$

Step 5.1.3.7.2.4

Divide $- 27 x$ by $1$ .

$12 x^{2} - 27 x + \frac{d}{d x} [8 x]$

Step 5.1.4

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

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Step 5.1.4.1

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

$12 x^{2} - 27 x + 8 \frac{d}{d x} [x]$

Step 5.1.4.2

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

$12 x^{2} - 27 x + 8 \cdot 1$

Step 5.1.4.3

Multiply $8$ by $1$ .

$f' (x) = 12 x^{2} - 27 x + 8$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 27 x + 8$ .

$12 x^{2} - 27 x + 8$

Step 6

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 27 x + 8 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$12 x^{2} - 27 x + 8 = 0$

Step 6.2

Use the quadratic formula to find the solutions.

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

Step 6.3

Substitute the values $a = 12$ , $b = - 27$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{27 \pm \sqrt{{(- 27)}^{2} - 4 \cdot (12 \cdot 8)}}{2 \cdot 12}$

Step 6.4

Simplify.

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Step 6.4.1

Simplify the numerator.

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Step 6.4.1.1

Raise $- 27$ to the power of $2$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 12 \cdot 8}}{2 \cdot 12}$

Step 6.4.1.2

Multiply $- 4 \cdot 12 \cdot 8$ .

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Step 6.4.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{27 \pm \sqrt{729 - 48 \cdot 8}}{2 \cdot 12}$

Step 6.4.1.2.2

Multiply $- 48$ by $8$ .

$x = \frac{27 \pm \sqrt{729 - 384}}{2 \cdot 12}$

Step 6.4.1.3

Subtract $384$ from $729$ .

$x = \frac{27 \pm \sqrt{345}}{2 \cdot 12}$

Step 6.4.2

Multiply $2$ by $12$ .

$x = \frac{27 \pm \sqrt{345}}{24}$

Step 6.5

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

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Step 6.5.1

Simplify the numerator.

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Step 6.5.1.1

Raise $- 27$ to the power of $2$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 12 \cdot 8}}{2 \cdot 12}$

Step 6.5.1.2

Multiply $- 4 \cdot 12 \cdot 8$ .

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Step 6.5.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{27 \pm \sqrt{729 - 48 \cdot 8}}{2 \cdot 12}$

Step 6.5.1.2.2

Multiply $- 48$ by $8$ .

$x = \frac{27 \pm \sqrt{729 - 384}}{2 \cdot 12}$

Step 6.5.1.3

Subtract $384$ from $729$ .

$x = \frac{27 \pm \sqrt{345}}{2 \cdot 12}$

Step 6.5.2

Multiply $2$ by $12$ .

$x = \frac{27 \pm \sqrt{345}}{24}$

Step 6.5.3

Change the $\pm$ to $+$ .

$x = \frac{27 + \sqrt{345}}{24}$

Step 6.6

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

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Step 6.6.1

Simplify the numerator.

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Step 6.6.1.1

Raise $- 27$ to the power of $2$ .

$x = \frac{27 \pm \sqrt{729 - 4 \cdot 12 \cdot 8}}{2 \cdot 12}$

Step 6.6.1.2

Multiply $- 4 \cdot 12 \cdot 8$ .

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Step 6.6.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{27 \pm \sqrt{729 - 48 \cdot 8}}{2 \cdot 12}$

Step 6.6.1.2.2

Multiply $- 48$ by $8$ .

$x = \frac{27 \pm \sqrt{729 - 384}}{2 \cdot 12}$

Step 6.6.1.3

Subtract $384$ from $729$ .

$x = \frac{27 \pm \sqrt{345}}{2 \cdot 12}$

Step 6.6.2

Multiply $2$ by $12$ .

$x = \frac{27 \pm \sqrt{345}}{24}$

Step 6.6.3

Change the $\pm$ to $-$ .

$x = \frac{27 - \sqrt{345}}{24}$

Step 6.7

The final answer is the combination of both solutions.

$x = \frac{27 + \sqrt{345}}{24}, \frac{27 - \sqrt{345}}{24}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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 8

Critical points to evaluate.

$x = \frac{27 + \sqrt{345}}{24}, \frac{27 - \sqrt{345}}{24}$

Step 9

Evaluate the second derivative at $x = \frac{27 + \sqrt{345}}{24}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$24 (\frac{27 + \sqrt{345}}{24}) - 27$

Step 10

Evaluate the second derivative.

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Step 10.1

Cancel the common factor of $24$ .

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Step 10.1.1

Cancel the common factor.

$24 \frac{27 + \sqrt{345}}{24} - 27$

Step 10.1.2

Rewrite the expression.

$27 + \sqrt{345} - 27$

Step 10.2

Simplify by subtracting numbers.

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Step 10.2.1

Subtract $27$ from $27$ .

$0 + \sqrt{345}$

Step 10.2.2

Add $0$ and $\sqrt{345}$ .

$\sqrt{345}$

Step 11

$x = \frac{27 + \sqrt{345}}{24}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{27 + \sqrt{345}}{24}$ is a local minimum

Step 12

Find the y-value when $x = \frac{27 + \sqrt{345}}{24}$ .

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Step 12.1

Replace the variable $x$ with $\frac{27 + \sqrt{345}}{24}$ in the expression.

$f (\frac{27 + \sqrt{345}}{24}) = 4 {(\frac{27 + \sqrt{345}}{24})}^{3} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Apply the product rule to $\frac{27 + \sqrt{345}}{24}$ .

$f (\frac{27 + \sqrt{345}}{24}) = 4 (\frac{{(27 + \sqrt{345})}^{3}}{24^{3}}) - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.2

Raise $24$ to the power of $3$ .

$f (\frac{27 + \sqrt{345}}{24}) = 4 (\frac{{(27 + \sqrt{345})}^{3}}{13824}) - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.3

Cancel the common factor of $4$ .

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Step 12.2.1.3.1

Factor $4$ out of $13824$ .

$f (\frac{27 + \sqrt{345}}{24}) = 4 (\frac{{(27 + \sqrt{345})}^{3}}{4 (3456)}) - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.3.2

Cancel the common factor.

$f (\frac{27 + \sqrt{345}}{24}) = 4 (\frac{{(27 + \sqrt{345})}^{3}}{4 \cdot 3456}) - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.3.3

Rewrite the expression.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{{(27 + \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.4

Use the Binomial Theorem.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{27^{3} + 3 \cdot (27^{2} \sqrt{345}) + 3 \cdot (27 {\sqrt{345}}^{2}) + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5

Simplify each term.

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Step 12.2.1.5.1

Raise $27$ to the power of $3$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 3 \cdot (27^{2} \sqrt{345}) + 3 \cdot (27 {\sqrt{345}}^{2}) + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.2

Raise $27$ to the power of $2$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 3 \cdot (729 \sqrt{345}) + 3 \cdot (27 {\sqrt{345}}^{2}) + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.3

Multiply $3$ by $729$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 3 \cdot (27 {\sqrt{345}}^{2}) + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.4

Multiply $3$ by $27$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 81 {\sqrt{345}}^{2} + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.5

Rewrite ${\sqrt{345}}^{2}$ as $345$ .

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Step 12.2.1.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{345}$ as $345^{\frac{1}{2}}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 81 {(345^{\frac{1}{2}})}^{2} + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.5.2

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

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 81 \cdot 345^{\frac{1}{2} \cdot 2} + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.5.3

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

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 81 \cdot 345^{\frac{2}{2}} + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.5.4

Cancel the common factor of $2$ .

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Step 12.2.1.5.5.4.1

Cancel the common factor.

Step 12.2.1.5.5.4.2

Rewrite the expression.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 81 \cdot 345 + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.5.5

Evaluate the exponent.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 81 \cdot 345 + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.6

Multiply $81$ by $345$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 27945 + {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.7

Rewrite ${\sqrt{345}}^{3}$ as $\sqrt{345^{3}}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 27945 + \sqrt{345^{3}}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.8

Raise $345$ to the power of $3$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 27945 + \sqrt{41063625}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.9

Rewrite $41063625$ as $345^{2} \cdot 345$ .

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Step 12.2.1.5.9.1

Factor $119025$ out of $41063625$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 27945 + \sqrt{119025 (345)}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.9.2

Rewrite $119025$ as $345^{2}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 27945 + \sqrt{345^{2} \cdot 345}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.5.10

Pull terms out from under the radical.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{19683 + 2187 \sqrt{345} + 27945 + 345 \sqrt{345}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.6

Add $19683$ and $27945$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{47628 + 2187 \sqrt{345} + 345 \sqrt{345}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.7

Add $2187 \sqrt{345}$ and $345 \sqrt{345}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{47628 + 2532 \sqrt{345}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.8

Cancel the common factor of $47628 + 2532 \sqrt{345}$ and $3456$ .

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Step 12.2.1.8.1

Factor $12$ out of $47628$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{12 (3969) + 2532 \sqrt{345}}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.8.2

Factor $12$ out of $2532 \sqrt{345}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{12 (3969) + 12 (211 \sqrt{345})}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.8.3

Factor $12$ out of $12 (3969) + 12 (211 \sqrt{345})$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{12 (3969 + 211 \sqrt{345})}{3456} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.8.4

Cancel the common factors.

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Step 12.2.1.8.4.1

Factor $12$ out of $3456$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{12 (3969 + 211 \sqrt{345})}{12 \cdot 288} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.8.4.2

Cancel the common factor.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{12 (3969 + 211 \sqrt{345})}{12 \cdot 288} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.8.4.3

Rewrite the expression.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} - \frac{27}{2} \cdot {(\frac{27 + \sqrt{345}}{24})}^{2} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.9

Apply the product rule to $\frac{27 + \sqrt{345}}{24}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} - \frac{27}{2} \cdot \frac{{(27 + \sqrt{345})}^{2}}{24^{2}} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.10

Raise $24$ to the power of $2$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} - \frac{27}{2} \cdot \frac{{(27 + \sqrt{345})}^{2}}{576} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.11

Cancel the common factor of $9$ .

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Step 12.2.1.11.1

Move the leading negative in $- \frac{27}{2}$ into the numerator.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 27}{2} \cdot \frac{{(27 + \sqrt{345})}^{2}}{576} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.11.2

Factor $9$ out of $- 27$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{9 (- 3)}{2} \cdot \frac{{(27 + \sqrt{345})}^{2}}{576} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.11.3

Factor $9$ out of $576$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{9 \cdot - 3}{2} \cdot \frac{{(27 + \sqrt{345})}^{2}}{9 \cdot 64} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.11.4

Cancel the common factor.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{9 \cdot - 3}{2} \cdot \frac{{(27 + \sqrt{345})}^{2}}{9 \cdot 64} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.11.5

Rewrite the expression.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3}{2} \cdot \frac{{(27 + \sqrt{345})}^{2}}{64} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.12

Multiply $\frac{- 3}{2}$ by $\frac{{(27 + \sqrt{345})}^{2}}{64}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 {(27 + \sqrt{345})}^{2}}{2 \cdot 64} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.13

Multiply $2$ by $64$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 {(27 + \sqrt{345})}^{2}}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.14

Rewrite ${(27 + \sqrt{345})}^{2}$ as $(27 + \sqrt{345}) (27 + \sqrt{345})$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 ((27 + \sqrt{345}) (27 + \sqrt{345}))}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.15

Expand $(27 + \sqrt{345}) (27 + \sqrt{345})$ using the FOIL Method.

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Step 12.2.1.15.1

Apply the distributive property.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (27 (27 + \sqrt{345}) + \sqrt{345} (27 + \sqrt{345}))}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.15.2

Apply the distributive property.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (27 \cdot 27 + 27 \sqrt{345} + \sqrt{345} (27 + \sqrt{345}))}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.15.3

Apply the distributive property.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (27 \cdot 27 + 27 \sqrt{345} + \sqrt{345} \cdot 27 + \sqrt{345} \sqrt{345})}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.16

Simplify and combine like terms.

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Step 12.2.1.16.1

Simplify each term.

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Step 12.2.1.16.1.1

Multiply $27$ by $27$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (729 + 27 \sqrt{345} + \sqrt{345} \cdot 27 + \sqrt{345} \sqrt{345})}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.16.1.2

Move $27$ to the left of $\sqrt{345}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (729 + 27 \sqrt{345} + 27 \cdot \sqrt{345} + \sqrt{345} \sqrt{345})}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.16.1.3

Combine using the product rule for radicals.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (729 + 27 \sqrt{345} + 27 \sqrt{345} + \sqrt{345 \cdot 345})}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.16.1.4

Multiply $345$ by $345$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (729 + 27 \sqrt{345} + 27 \sqrt{345} + \sqrt{119025})}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.16.1.5

Rewrite $119025$ as $345^{2}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (729 + 27 \sqrt{345} + 27 \sqrt{345} + \sqrt{345^{2}})}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.16.1.6

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

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (729 + 27 \sqrt{345} + 27 \sqrt{345} + 345)}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.16.2

Add $729$ and $345$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (1074 + 27 \sqrt{345} + 27 \sqrt{345})}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.16.3

Add $27 \sqrt{345}$ and $27 \sqrt{345}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (1074 + 54 \sqrt{345})}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.17

Cancel the common factor of $1074 + 54 \sqrt{345}$ and $128$ .

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Step 12.2.1.17.1

Factor $2$ out of $- 3 (1074 + 54 \sqrt{345})$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{2 (- 3 (537 + 27 \sqrt{345}))}{128} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.17.2

Cancel the common factors.

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Step 12.2.1.17.2.1

Factor $2$ out of $128$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{2 (- 3 (537 + 27 \sqrt{345}))}{2 (64)} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.17.2.2

Cancel the common factor.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{2 (- 3 (537 + 27 \sqrt{345}))}{2 \cdot 64} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.17.2.3

Rewrite the expression.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} + \frac{- 3 (537 + 27 \sqrt{345})}{64} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.18

Move the negative in front of the fraction.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} - \frac{3 (537 + 27 \sqrt{345})}{64} + 8 (\frac{27 + \sqrt{345}}{24})$

Step 12.2.1.19

Cancel the common factor of $8$ .

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Step 12.2.1.19.1

Factor $8$ out of $24$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} - \frac{3 (537 + 27 \sqrt{345})}{64} + 8 (\frac{27 + \sqrt{345}}{8 (3)})$

Step 12.2.1.19.2

Cancel the common factor.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} - \frac{3 (537 + 27 \sqrt{345})}{64} + 8 (\frac{27 + \sqrt{345}}{8 \cdot 3})$

Step 12.2.1.19.3

Rewrite the expression.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} - \frac{3 (537 + 27 \sqrt{345})}{64} + \frac{27 + \sqrt{345}}{3}$

Step 12.2.2

Find the common denominator.

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Step 12.2.2.1

Multiply $\frac{3969 + 211 \sqrt{345}}{288}$ by $\frac{2}{2}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 + 211 \sqrt{345}}{288} \cdot \frac{2}{2} - \frac{3 (537 + 27 \sqrt{345})}{64} + \frac{27 + \sqrt{345}}{3}$

Step 12.2.2.2

Multiply $\frac{3969 + 211 \sqrt{345}}{288}$ by $\frac{2}{2}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - \frac{3 (537 + 27 \sqrt{345})}{64} + \frac{27 + \sqrt{345}}{3}$

Step 12.2.2.3

Multiply $\frac{3 (537 + 27 \sqrt{345})}{64}$ by $\frac{9}{9}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - (\frac{3 (537 + 27 \sqrt{345})}{64} \cdot \frac{9}{9}) + \frac{27 + \sqrt{345}}{3}$

Step 12.2.2.4

Multiply $\frac{3 (537 + 27 \sqrt{345})}{64}$ by $\frac{9}{9}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - \frac{3 (537 + 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{27 + \sqrt{345}}{3}$

Step 12.2.2.5

Multiply $\frac{27 + \sqrt{345}}{3}$ by $\frac{192}{192}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - \frac{3 (537 + 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{27 + \sqrt{345}}{3} \cdot \frac{192}{192}$

Step 12.2.2.6

Multiply $\frac{27 + \sqrt{345}}{3}$ by $\frac{192}{192}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - \frac{3 (537 + 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{(27 + \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 12.2.2.7

Reorder the factors of $288 \cdot 2$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{2 \cdot 288} - \frac{3 (537 + 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{(27 + \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 12.2.2.8

Multiply $2$ by $288$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{576} - \frac{3 (537 + 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{(27 + \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 12.2.2.9

Reorder the factors of $64 \cdot 9$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{576} - \frac{3 (537 + 27 \sqrt{345}) \cdot 9}{9 \cdot 64} + \frac{(27 + \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 12.2.2.10

Multiply $9$ by $64$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{576} - \frac{3 (537 + 27 \sqrt{345}) \cdot 9}{576} + \frac{(27 + \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 12.2.2.11

Multiply $3$ by $192$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2}{576} - \frac{3 (537 + 27 \sqrt{345}) \cdot 9}{576} + \frac{(27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.3

Combine the numerators over the common denominator.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{(3969 + 211 \sqrt{345}) \cdot 2 - 3 (537 + 27 \sqrt{345}) \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4

Simplify each term.

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Step 12.2.4.1

Apply the distributive property.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{3969 \cdot 2 + 211 \sqrt{345} \cdot 2 - 3 (537 + 27 \sqrt{345}) \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.2

Multiply $3969$ by $2$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 211 \sqrt{345} \cdot 2 - 3 (537 + 27 \sqrt{345}) \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.3

Multiply $2$ by $211$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} - 3 (537 + 27 \sqrt{345}) \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.4

Apply the distributive property.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} + (- 3 \cdot 537 - 3 (27 \sqrt{345})) \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.5

Multiply $- 3$ by $537$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} + (- 1611 - 3 (27 \sqrt{345})) \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.6

Multiply $27$ by $- 3$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} + (- 1611 - 81 \sqrt{345}) \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.7

Apply the distributive property.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} - 1611 \cdot 9 - 81 \sqrt{345} \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.8

Multiply $- 1611$ by $9$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} - 14499 - 81 \sqrt{345} \cdot 9 + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.9

Multiply $9$ by $- 81$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} - 14499 - 729 \sqrt{345} + (27 + \sqrt{345}) \cdot 192}{576}$

Step 12.2.4.10

Apply the distributive property.

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} - 14499 - 729 \sqrt{345} + 27 \cdot 192 + \sqrt{345} \cdot 192}{576}$

Step 12.2.4.11

Multiply $27$ by $192$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} - 14499 - 729 \sqrt{345} + 5184 + \sqrt{345} \cdot 192}{576}$

Step 12.2.4.12

Move $192$ to the left of $\sqrt{345}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{7938 + 422 \sqrt{345} - 14499 - 729 \sqrt{345} + 5184 + 192 \sqrt{345}}{576}$

Step 12.2.5

Simplify terms.

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Step 12.2.5.1

Subtract $14499$ from $7938$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{- 6561 + 422 \sqrt{345} - 729 \sqrt{345} + 5184 + 192 \sqrt{345}}{576}$

Step 12.2.5.2

Add $- 6561$ and $5184$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{- 1377 + 422 \sqrt{345} - 729 \sqrt{345} + 192 \sqrt{345}}{576}$

Step 12.2.5.3

Subtract $729 \sqrt{345}$ from $422 \sqrt{345}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{- 1377 - 307 \sqrt{345} + 192 \sqrt{345}}{576}$

Step 12.2.5.4

Add $- 307 \sqrt{345}$ and $192 \sqrt{345}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{- 1377 - 115 \sqrt{345}}{576}$

Step 12.2.5.5

Rewrite $- 1377$ as $- 1 (1377)$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{- 1 \cdot 1377 - 115 \sqrt{345}}{576}$

Step 12.2.5.6

Factor $- 1$ out of $- 115 \sqrt{345}$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{- 1 \cdot 1377 - (115 \sqrt{345})}{576}$

Step 12.2.5.7

Factor $- 1$ out of $- 1 (1377) - (115 \sqrt{345})$ .

$f (\frac{27 + \sqrt{345}}{24}) = \frac{- 1 (1377 + 115 \sqrt{345})}{576}$

Step 12.2.5.8

Move the negative in front of the fraction.

$f (\frac{27 + \sqrt{345}}{24}) = - \frac{1377 + 115 \sqrt{345}}{576}$

Step 12.2.6

The final answer is $- \frac{1377 + 115 \sqrt{345}}{576}$ .

$y = - \frac{1377 + 115 \sqrt{345}}{576}$

Step 13

Evaluate the second derivative at $x = \frac{27 - \sqrt{345}}{24}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$24 (\frac{27 - \sqrt{345}}{24}) - 27$

Step 14

Evaluate the second derivative.

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Step 14.1

Cancel the common factor of $24$ .

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Step 14.1.1

Cancel the common factor.

$24 \frac{27 - \sqrt{345}}{24} - 27$

Step 14.1.2

Rewrite the expression.

$27 - \sqrt{345} - 27$

Step 14.2

Simplify by subtracting numbers.

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Step 14.2.1

Subtract $27$ from $27$ .

$0 - \sqrt{345}$

Step 14.2.2

Subtract $\sqrt{345}$ from $0$ .

$- \sqrt{345}$

Step 15

$x = \frac{27 - \sqrt{345}}{24}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{27 - \sqrt{345}}{24}$ is a local maximum

Step 16

Find the y-value when $x = \frac{27 - \sqrt{345}}{24}$ .

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Step 16.1

Replace the variable $x$ with $\frac{27 - \sqrt{345}}{24}$ in the expression.

$f (\frac{27 - \sqrt{345}}{24}) = 4 {(\frac{27 - \sqrt{345}}{24})}^{3} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2

Simplify the result.

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Step 16.2.1

Simplify each term.

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Step 16.2.1.1

Apply the product rule to $\frac{27 - \sqrt{345}}{24}$ .

$f (\frac{27 - \sqrt{345}}{24}) = 4 (\frac{{(27 - \sqrt{345})}^{3}}{24^{3}}) - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.2

Raise $24$ to the power of $3$ .

$f (\frac{27 - \sqrt{345}}{24}) = 4 (\frac{{(27 - \sqrt{345})}^{3}}{13824}) - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.3

Cancel the common factor of $4$ .

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Step 16.2.1.3.1

Factor $4$ out of $13824$ .

$f (\frac{27 - \sqrt{345}}{24}) = 4 (\frac{{(27 - \sqrt{345})}^{3}}{4 (3456)}) - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.3.2

Cancel the common factor.

$f (\frac{27 - \sqrt{345}}{24}) = 4 (\frac{{(27 - \sqrt{345})}^{3}}{4 \cdot 3456}) - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.3.3

Rewrite the expression.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{{(27 - \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.4

Use the Binomial Theorem.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{27^{3} + 3 \cdot (27^{2} (- \sqrt{345})) + 3 \cdot (27 {(- \sqrt{345})}^{2}) + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5

Simplify each term.

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Step 16.2.1.5.1

Raise $27$ to the power of $3$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 + 3 \cdot (27^{2} (- \sqrt{345})) + 3 \cdot (27 {(- \sqrt{345})}^{2}) + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.2

Raise $27$ to the power of $2$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 + 3 \cdot (729 (- \sqrt{345})) + 3 \cdot (27 {(- \sqrt{345})}^{2}) + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.3

Multiply $3$ by $729$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 + 2187 (- \sqrt{345}) + 3 \cdot (27 {(- \sqrt{345})}^{2}) + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.4

Multiply $- 1$ by $2187$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 3 \cdot (27 {(- \sqrt{345})}^{2}) + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.5

Multiply $3$ by $27$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 {(- \sqrt{345})}^{2} + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.6

Apply the product rule to $- \sqrt{345}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 ({(- 1)}^{2} {\sqrt{345}}^{2}) + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.7

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

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 (1 {\sqrt{345}}^{2}) + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.8

Multiply ${\sqrt{345}}^{2}$ by $1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 {\sqrt{345}}^{2} + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.9

Rewrite ${\sqrt{345}}^{2}$ as $345$ .

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Step 16.2.1.5.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{345}$ as $345^{\frac{1}{2}}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 {(345^{\frac{1}{2}})}^{2} + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.9.2

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

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 \cdot 345^{\frac{1}{2} \cdot 2} + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.9.3

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

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 \cdot 345^{\frac{2}{2}} + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.9.4

Cancel the common factor of $2$ .

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Step 16.2.1.5.9.4.1

Cancel the common factor.

Step 16.2.1.5.9.4.2

Rewrite the expression.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 \cdot 345 + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.9.5

Evaluate the exponent.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 81 \cdot 345 + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.10

Multiply $81$ by $345$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 + {(- \sqrt{345})}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.11

Apply the product rule to $- \sqrt{345}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 + {(- 1)}^{3} {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.12

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

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 - {\sqrt{345}}^{3}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.13

Rewrite ${\sqrt{345}}^{3}$ as $\sqrt{345^{3}}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 - \sqrt{345^{3}}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.14

Raise $345$ to the power of $3$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 - \sqrt{41063625}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.15

Rewrite $41063625$ as $345^{2} \cdot 345$ .

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Step 16.2.1.5.15.1

Factor $119025$ out of $41063625$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 - \sqrt{119025 (345)}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.15.2

Rewrite $119025$ as $345^{2}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 - \sqrt{345^{2} \cdot 345}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.16

Pull terms out from under the radical.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 - (345 \sqrt{345})}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.5.17

Multiply $345$ by $- 1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{19683 - 2187 \sqrt{345} + 27945 - 345 \sqrt{345}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.6

Add $19683$ and $27945$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{47628 - 2187 \sqrt{345} - 345 \sqrt{345}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.7

Subtract $345 \sqrt{345}$ from $- 2187 \sqrt{345}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{47628 - 2532 \sqrt{345}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.8

Cancel the common factor of $47628 - 2532 \sqrt{345}$ and $3456$ .

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Step 16.2.1.8.1

Factor $12$ out of $47628$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{12 (3969) - 2532 \sqrt{345}}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.8.2

Factor $12$ out of $- 2532 \sqrt{345}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{12 (3969) + 12 (- 211 \sqrt{345})}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.8.3

Factor $12$ out of $12 (3969) + 12 (- 211 \sqrt{345})$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{12 (3969 - 211 \sqrt{345})}{3456} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.8.4

Cancel the common factors.

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Step 16.2.1.8.4.1

Factor $12$ out of $3456$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{12 (3969 - 211 \sqrt{345})}{12 \cdot 288} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.8.4.2

Cancel the common factor.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{12 (3969 - 211 \sqrt{345})}{12 \cdot 288} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.8.4.3

Rewrite the expression.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} - \frac{27}{2} \cdot {(\frac{27 - \sqrt{345}}{24})}^{2} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.9

Apply the product rule to $\frac{27 - \sqrt{345}}{24}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} - \frac{27}{2} \cdot \frac{{(27 - \sqrt{345})}^{2}}{24^{2}} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.10

Raise $24$ to the power of $2$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} - \frac{27}{2} \cdot \frac{{(27 - \sqrt{345})}^{2}}{576} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.11

Cancel the common factor of $9$ .

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Step 16.2.1.11.1

Move the leading negative in $- \frac{27}{2}$ into the numerator.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 27}{2} \cdot \frac{{(27 - \sqrt{345})}^{2}}{576} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.11.2

Factor $9$ out of $- 27$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{9 (- 3)}{2} \cdot \frac{{(27 - \sqrt{345})}^{2}}{576} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.11.3

Factor $9$ out of $576$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{9 \cdot - 3}{2} \cdot \frac{{(27 - \sqrt{345})}^{2}}{9 \cdot 64} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.11.4

Cancel the common factor.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{9 \cdot - 3}{2} \cdot \frac{{(27 - \sqrt{345})}^{2}}{9 \cdot 64} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.11.5

Rewrite the expression.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3}{2} \cdot \frac{{(27 - \sqrt{345})}^{2}}{64} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.12

Multiply $\frac{- 3}{2}$ by $\frac{{(27 - \sqrt{345})}^{2}}{64}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 {(27 - \sqrt{345})}^{2}}{2 \cdot 64} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.13

Multiply $2$ by $64$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 {(27 - \sqrt{345})}^{2}}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.14

Rewrite ${(27 - \sqrt{345})}^{2}$ as $(27 - \sqrt{345}) (27 - \sqrt{345})$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 ((27 - \sqrt{345}) (27 - \sqrt{345}))}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.15

Expand $(27 - \sqrt{345}) (27 - \sqrt{345})$ using the FOIL Method.

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Step 16.2.1.15.1

Apply the distributive property.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (27 (27 - \sqrt{345}) - \sqrt{345} (27 - \sqrt{345}))}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.15.2

Apply the distributive property.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (27 \cdot 27 + 27 (- \sqrt{345}) - \sqrt{345} (27 - \sqrt{345}))}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.15.3

Apply the distributive property.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (27 \cdot 27 + 27 (- \sqrt{345}) - \sqrt{345} \cdot 27 - \sqrt{345} (- \sqrt{345}))}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16

Simplify and combine like terms.

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Step 16.2.1.16.1

Simplify each term.

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Step 16.2.1.16.1.1

Multiply $27$ by $27$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 + 27 (- \sqrt{345}) - \sqrt{345} \cdot 27 - \sqrt{345} (- \sqrt{345}))}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.2

Multiply $- 1$ by $27$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - \sqrt{345} \cdot 27 - \sqrt{345} (- \sqrt{345}))}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.3

Multiply $27$ by $- 1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} - \sqrt{345} (- \sqrt{345}))}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.4

Multiply $- \sqrt{345} (- \sqrt{345})$ .

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Step 16.2.1.16.1.4.1

Multiply $- 1$ by $- 1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + 1 \sqrt{345} \sqrt{345})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.4.2

Multiply $\sqrt{345}$ by $1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + \sqrt{345} \sqrt{345})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.4.3

Raise $\sqrt{345}$ to the power of $1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + \sqrt{345} \sqrt{345})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.4.4

Raise $\sqrt{345}$ to the power of $1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + \sqrt{345} \sqrt{345})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.4.5

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

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + {\sqrt{345}}^{1 + 1})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.4.6

Add $1$ and $1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + {\sqrt{345}}^{2})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.5

Rewrite ${\sqrt{345}}^{2}$ as $345$ .

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Step 16.2.1.16.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{345}$ as $345^{\frac{1}{2}}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + {(345^{\frac{1}{2}})}^{2})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.5.2

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

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + 345^{\frac{1}{2} \cdot 2})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.5.3

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

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + 345^{\frac{2}{2}})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.5.4

Cancel the common factor of $2$ .

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Step 16.2.1.16.1.5.4.1

Cancel the common factor.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + 345^{\frac{2}{2}})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.5.4.2

Rewrite the expression.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + 345)}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.1.5.5

Evaluate the exponent.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (729 - 27 \sqrt{345} - 27 \sqrt{345} + 345)}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.2

Add $729$ and $345$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (1074 - 27 \sqrt{345} - 27 \sqrt{345})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.16.3

Subtract $27 \sqrt{345}$ from $- 27 \sqrt{345}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (1074 - 54 \sqrt{345})}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.17

Cancel the common factor of $1074 - 54 \sqrt{345}$ and $128$ .

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Step 16.2.1.17.1

Factor $2$ out of $- 3 (1074 - 54 \sqrt{345})$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{2 (- 3 (537 - 27 \sqrt{345}))}{128} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.17.2

Cancel the common factors.

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Step 16.2.1.17.2.1

Factor $2$ out of $128$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{2 (- 3 (537 - 27 \sqrt{345}))}{2 (64)} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.17.2.2

Cancel the common factor.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{2 (- 3 (537 - 27 \sqrt{345}))}{2 \cdot 64} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.17.2.3

Rewrite the expression.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} + \frac{- 3 (537 - 27 \sqrt{345})}{64} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.18

Move the negative in front of the fraction.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} - \frac{3 (537 - 27 \sqrt{345})}{64} + 8 (\frac{27 - \sqrt{345}}{24})$

Step 16.2.1.19

Cancel the common factor of $8$ .

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Step 16.2.1.19.1

Factor $8$ out of $24$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} - \frac{3 (537 - 27 \sqrt{345})}{64} + 8 (\frac{27 - \sqrt{345}}{8 (3)})$

Step 16.2.1.19.2

Cancel the common factor.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} - \frac{3 (537 - 27 \sqrt{345})}{64} + 8 (\frac{27 - \sqrt{345}}{8 \cdot 3})$

Step 16.2.1.19.3

Rewrite the expression.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} - \frac{3 (537 - 27 \sqrt{345})}{64} + \frac{27 - \sqrt{345}}{3}$

Step 16.2.2

Find the common denominator.

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Step 16.2.2.1

Multiply $\frac{3969 - 211 \sqrt{345}}{288}$ by $\frac{2}{2}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 - 211 \sqrt{345}}{288} \cdot \frac{2}{2} - \frac{3 (537 - 27 \sqrt{345})}{64} + \frac{27 - \sqrt{345}}{3}$

Step 16.2.2.2

Multiply $\frac{3969 - 211 \sqrt{345}}{288}$ by $\frac{2}{2}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - \frac{3 (537 - 27 \sqrt{345})}{64} + \frac{27 - \sqrt{345}}{3}$

Step 16.2.2.3

Multiply $\frac{3 (537 - 27 \sqrt{345})}{64}$ by $\frac{9}{9}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - (\frac{3 (537 - 27 \sqrt{345})}{64} \cdot \frac{9}{9}) + \frac{27 - \sqrt{345}}{3}$

Step 16.2.2.4

Multiply $\frac{3 (537 - 27 \sqrt{345})}{64}$ by $\frac{9}{9}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - \frac{3 (537 - 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{27 - \sqrt{345}}{3}$

Step 16.2.2.5

Multiply $\frac{27 - \sqrt{345}}{3}$ by $\frac{192}{192}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - \frac{3 (537 - 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{27 - \sqrt{345}}{3} \cdot \frac{192}{192}$

Step 16.2.2.6

Multiply $\frac{27 - \sqrt{345}}{3}$ by $\frac{192}{192}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{288 \cdot 2} - \frac{3 (537 - 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{(27 - \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 16.2.2.7

Reorder the factors of $288 \cdot 2$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{2 \cdot 288} - \frac{3 (537 - 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{(27 - \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 16.2.2.8

Multiply $2$ by $288$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{576} - \frac{3 (537 - 27 \sqrt{345}) \cdot 9}{64 \cdot 9} + \frac{(27 - \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 16.2.2.9

Reorder the factors of $64 \cdot 9$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{576} - \frac{3 (537 - 27 \sqrt{345}) \cdot 9}{9 \cdot 64} + \frac{(27 - \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 16.2.2.10

Multiply $9$ by $64$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{576} - \frac{3 (537 - 27 \sqrt{345}) \cdot 9}{576} + \frac{(27 - \sqrt{345}) \cdot 192}{3 \cdot 192}$

Step 16.2.2.11

Multiply $3$ by $192$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2}{576} - \frac{3 (537 - 27 \sqrt{345}) \cdot 9}{576} + \frac{(27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.3

Combine the numerators over the common denominator.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{(3969 - 211 \sqrt{345}) \cdot 2 - 3 (537 - 27 \sqrt{345}) \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4

Simplify each term.

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Step 16.2.4.1

Apply the distributive property.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{3969 \cdot 2 - 211 \sqrt{345} \cdot 2 - 3 (537 - 27 \sqrt{345}) \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.2

Multiply $3969$ by $2$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 211 \sqrt{345} \cdot 2 - 3 (537 - 27 \sqrt{345}) \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.3

Multiply $2$ by $- 211$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} - 3 (537 - 27 \sqrt{345}) \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.4

Apply the distributive property.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} + (- 3 \cdot 537 - 3 (- 27 \sqrt{345})) \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.5

Multiply $- 3$ by $537$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} + (- 1611 - 3 (- 27 \sqrt{345})) \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.6

Multiply $- 27$ by $- 3$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} + (- 1611 + 81 \sqrt{345}) \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.7

Apply the distributive property.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} - 1611 \cdot 9 + 81 \sqrt{345} \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.8

Multiply $- 1611$ by $9$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} - 14499 + 81 \sqrt{345} \cdot 9 + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.9

Multiply $9$ by $81$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} - 14499 + 729 \sqrt{345} + (27 - \sqrt{345}) \cdot 192}{576}$

Step 16.2.4.10

Apply the distributive property.

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} - 14499 + 729 \sqrt{345} + 27 \cdot 192 - \sqrt{345} \cdot 192}{576}$

Step 16.2.4.11

Multiply $27$ by $192$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} - 14499 + 729 \sqrt{345} + 5184 - \sqrt{345} \cdot 192}{576}$

Step 16.2.4.12

Multiply $192$ by $- 1$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{7938 - 422 \sqrt{345} - 14499 + 729 \sqrt{345} + 5184 - 192 \sqrt{345}}{576}$

Step 16.2.5

Simplify terms.

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Step 16.2.5.1

Subtract $14499$ from $7938$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{- 6561 - 422 \sqrt{345} + 729 \sqrt{345} + 5184 - 192 \sqrt{345}}{576}$

Step 16.2.5.2

Add $- 6561$ and $5184$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{- 1377 - 422 \sqrt{345} + 729 \sqrt{345} - 192 \sqrt{345}}{576}$

Step 16.2.5.3

Add $- 422 \sqrt{345}$ and $729 \sqrt{345}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{- 1377 + 307 \sqrt{345} - 192 \sqrt{345}}{576}$

Step 16.2.5.4

Subtract $192 \sqrt{345}$ from $307 \sqrt{345}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{- 1377 + 115 \sqrt{345}}{576}$

Step 16.2.5.5

Rewrite $- 1377$ as $- 1 (1377)$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{- 1 \cdot 1377 + 115 \sqrt{345}}{576}$

Step 16.2.5.6

Factor $- 1$ out of $115 \sqrt{345}$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{- 1 \cdot 1377 - (- 115 \sqrt{345})}{576}$

Step 16.2.5.7

Factor $- 1$ out of $- 1 (1377) - (- 115 \sqrt{345})$ .

$f (\frac{27 - \sqrt{345}}{24}) = \frac{- 1 (1377 - 115 \sqrt{345})}{576}$

Step 16.2.5.8

Move the negative in front of the fraction.

$f (\frac{27 - \sqrt{345}}{24}) = - \frac{1377 - 115 \sqrt{345}}{576}$

Step 16.2.6

The final answer is $- \frac{1377 - 115 \sqrt{345}}{576}$ .

$y = - \frac{1377 - 115 \sqrt{345}}{576}$

Step 17

These are the local extrema for $f (x) = 4 x^{3} - \frac{27}{2} x^{2} + 8 x$ .

$(\frac{27 + \sqrt{345}}{24}, - \frac{1377 + 115 \sqrt{345}}{576})$ is a local minima

$(\frac{27 - \sqrt{345}}{24}, - \frac{1377 - 115 \sqrt{345}}{576})$ is a local maxima

Step 18