Calculus Examples

$f (x) = 4 x^{3} - 34 x^{2} + 60 x$ , $0 < x < 2.5$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

By the Sum Rule, the derivative of $4 x^{3} - 34 x^{2} + 60 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 34 x^{2}] + \frac{d}{d x} [60 x]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 34 x^{2}] + \frac{d}{d x} [60 x]$

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 34 x^{2}] + \frac{d}{d x} [60 x]$

Step 1.1.1.2.2

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

$4 (3 x^{2}) + \frac{d}{d x} [- 34 x^{2}] + \frac{d}{d x} [60 x]$

Step 1.1.1.2.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [- 34 x^{2}] + \frac{d}{d x} [60 x]$

Step 1.1.1.3

Evaluate $\frac{d}{d x} [- 34 x^{2}]$ .

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Step 1.1.1.3.1

Since $- 34$ is constant with respect to $x$ , the derivative of $- 34 x^{2}$ with respect to $x$ is $- 34 \frac{d}{d x} [x^{2}]$ .

$12 x^{2} - 34 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [60 x]$

Step 1.1.1.3.2

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

$12 x^{2} - 34 (2 x) + \frac{d}{d x} [60 x]$

Step 1.1.1.3.3

Multiply $2$ by $- 34$ .

$12 x^{2} - 68 x + \frac{d}{d x} [60 x]$

Step 1.1.1.4

Evaluate $\frac{d}{d x} [60 x]$ .

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Step 1.1.1.4.1

Since $60$ is constant with respect to $x$ , the derivative of $60 x$ with respect to $x$ is $60 \frac{d}{d x} [x]$ .

$12 x^{2} - 68 x + 60 \frac{d}{d x} [x]$

Step 1.1.1.4.2

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

$12 x^{2} - 68 x + 60 \cdot 1$

Step 1.1.1.4.3

Multiply $60$ by $1$ .

$f' (x) = 12 x^{2} - 68 x + 60$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 68 x + 60$ .

$12 x^{2} - 68 x + 60$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $12 x^{2} - 68 x + 60 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$12 x^{2} - 68 x + 60 = 0$

Step 1.2.2

Factor $4$ out of $12 x^{2} - 68 x + 60$ .

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Step 1.2.2.1

Factor $4$ out of $12 x^{2}$ .

$4 (3 x^{2}) - 68 x + 60 = 0$

Step 1.2.2.2

Factor $4$ out of $- 68 x$ .

$4 (3 x^{2}) + 4 (- 17 x) + 60 = 0$

Step 1.2.2.3

Factor $4$ out of $60$ .

$4 (3 x^{2}) + 4 (- 17 x) + 4 (15) = 0$

Step 1.2.2.4

Factor $4$ out of $4 (3 x^{2}) + 4 (- 17 x)$ .

$4 (3 x^{2} - 17 x) + 4 (15) = 0$

Step 1.2.2.5

Factor $4$ out of $4 (3 x^{2} - 17 x) + 4 (15)$ .

$4 (3 x^{2} - 17 x + 15) = 0$

Step 1.2.3

Divide each term in $4 (3 x^{2} - 17 x + 15) = 0$ by $4$ and simplify.

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Step 1.2.3.1

Divide each term in $4 (3 x^{2} - 17 x + 15) = 0$ by $4$ .

$\frac{4 (3 x^{2} - 17 x + 15)}{4} = \frac{0}{4}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $4$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{4 (3 x^{2} - 17 x + 15)}{4} = \frac{0}{4}$

Step 1.2.3.2.1.2

Divide $3 x^{2} - 17 x + 15$ by $1$ .

$3 x^{2} - 17 x + 15 = \frac{0}{4}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $4$ .

$3 x^{2} - 17 x + 15 = 0$

Step 1.2.4

Use the quadratic formula to find the solutions.

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

Step 1.2.5

Substitute the values $a = 3$ , $b = - 17$ , and $c = 15$ into the quadratic formula and solve for $x$ .

$\frac{17 \pm \sqrt{{(- 17)}^{2} - 4 \cdot (3 \cdot 15)}}{2 \cdot 3}$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $- 17$ to the power of $2$ .

$x = \frac{17 \pm \sqrt{289 - 4 \cdot 3 \cdot 15}}{2 \cdot 3}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 3 \cdot 15$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{17 \pm \sqrt{289 - 12 \cdot 15}}{2 \cdot 3}$

Step 1.2.6.1.2.2

Multiply $- 12$ by $15$ .

$x = \frac{17 \pm \sqrt{289 - 180}}{2 \cdot 3}$

Step 1.2.6.1.3

Subtract $180$ from $289$ .

$x = \frac{17 \pm \sqrt{109}}{2 \cdot 3}$

Step 1.2.6.2

Multiply $2$ by $3$ .

$x = \frac{17 \pm \sqrt{109}}{6}$

Step 1.2.7

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

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

Raise $- 17$ to the power of $2$ .

$x = \frac{17 \pm \sqrt{289 - 4 \cdot 3 \cdot 15}}{2 \cdot 3}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 3 \cdot 15$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{17 \pm \sqrt{289 - 12 \cdot 15}}{2 \cdot 3}$

Step 1.2.7.1.2.2

Multiply $- 12$ by $15$ .

$x = \frac{17 \pm \sqrt{289 - 180}}{2 \cdot 3}$

Step 1.2.7.1.3

Subtract $180$ from $289$ .

$x = \frac{17 \pm \sqrt{109}}{2 \cdot 3}$

Step 1.2.7.2

Multiply $2$ by $3$ .

$x = \frac{17 \pm \sqrt{109}}{6}$

Step 1.2.7.3

Change the $\pm$ to $+$ .

$x = \frac{17 + \sqrt{109}}{6}$

Step 1.2.8

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

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

Raise $- 17$ to the power of $2$ .

$x = \frac{17 \pm \sqrt{289 - 4 \cdot 3 \cdot 15}}{2 \cdot 3}$

Step 1.2.8.1.2

Multiply $- 4 \cdot 3 \cdot 15$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{17 \pm \sqrt{289 - 12 \cdot 15}}{2 \cdot 3}$

Step 1.2.8.1.2.2

Multiply $- 12$ by $15$ .

$x = \frac{17 \pm \sqrt{289 - 180}}{2 \cdot 3}$

Step 1.2.8.1.3

Subtract $180$ from $289$ .

$x = \frac{17 \pm \sqrt{109}}{2 \cdot 3}$

Step 1.2.8.2

Multiply $2$ by $3$ .

$x = \frac{17 \pm \sqrt{109}}{6}$

Step 1.2.8.3

Change the $\pm$ to $-$ .

$x = \frac{17 - \sqrt{109}}{6}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = \frac{17 + \sqrt{109}}{6}, \frac{17 - \sqrt{109}}{6}$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $4 x^{3} - 34 x^{2} + 60 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{17 + \sqrt{109}}{6}$ .

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Step 1.4.1.1

Substitute $\frac{17 + \sqrt{109}}{6}$ for $x$ .

$4 {(\frac{17 + \sqrt{109}}{6})}^{3} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

Apply the product rule to $\frac{17 + \sqrt{109}}{6}$ .

$4 \frac{{(17 + \sqrt{109})}^{3}}{6^{3}} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.2

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

$4 \frac{{(17 + \sqrt{109})}^{3}}{216} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.3

Cancel the common factor of $4$ .

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Step 1.4.1.2.1.3.1

Factor $4$ out of $216$ .

$4 \frac{{(17 + \sqrt{109})}^{3}}{4 (54)} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.3.2

Cancel the common factor.

$4 \frac{{(17 + \sqrt{109})}^{3}}{4 \cdot 54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.3.3

Rewrite the expression.

$\frac{{(17 + \sqrt{109})}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.4

Use the Binomial Theorem.

$\frac{17^{3} + 3 \cdot 17^{2} \sqrt{109} + 3 \cdot 17 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5

Simplify each term.

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Step 1.4.1.2.1.5.1

Raise $17$ to the power of $3$ .

$\frac{4913 + 3 \cdot 17^{2} \sqrt{109} + 3 \cdot 17 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.2

Raise $17$ to the power of $2$ .

$\frac{4913 + 3 \cdot 289 \sqrt{109} + 3 \cdot 17 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.3

Multiply $3$ by $289$ .

$\frac{4913 + 867 \sqrt{109} + 3 \cdot 17 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.4

Multiply $3$ by $17$ .

$\frac{4913 + 867 \sqrt{109} + 51 {\sqrt{109}}^{2} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.5

Rewrite ${\sqrt{109}}^{2}$ as $109$ .

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Step 1.4.1.2.1.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{109}$ as $109^{\frac{1}{2}}$ .

$\frac{4913 + 867 \sqrt{109} + 51 {(109^{\frac{1}{2}})}^{2} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.5.2

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

$\frac{4913 + 867 \sqrt{109} + 51 \cdot 109^{\frac{1}{2} \cdot 2} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.5.3

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

$\frac{4913 + 867 \sqrt{109} + 51 \cdot 109^{\frac{2}{2}} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.5.4

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.5.5.4.1

Cancel the common factor.

$\frac{4913 + 867 \sqrt{109} + 51 \cdot 109^{\frac{2}{2}} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.5.4.2

Rewrite the expression.

$\frac{4913 + 867 \sqrt{109} + 51 \cdot 109^{1} + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.5.5

Evaluate the exponent.

$\frac{4913 + 867 \sqrt{109} + 51 \cdot 109 + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.6

Multiply $51$ by $109$ .

$\frac{4913 + 867 \sqrt{109} + 5559 + {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.7

Rewrite ${\sqrt{109}}^{3}$ as $\sqrt{109^{3}}$ .

$\frac{4913 + 867 \sqrt{109} + 5559 + \sqrt{109^{3}}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.8

Raise $109$ to the power of $3$ .

$\frac{4913 + 867 \sqrt{109} + 5559 + \sqrt{1295029}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.9

Rewrite $1295029$ as $109^{2} \cdot 109$ .

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Step 1.4.1.2.1.5.9.1

Factor $11881$ out of $1295029$ .

$\frac{4913 + 867 \sqrt{109} + 5559 + \sqrt{11881 (109)}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.9.2

Rewrite $11881$ as $109^{2}$ .

$\frac{4913 + 867 \sqrt{109} + 5559 + \sqrt{109^{2} \cdot 109}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.5.10

Pull terms out from under the radical.

$\frac{4913 + 867 \sqrt{109} + 5559 + 109 \sqrt{109}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.6

Add $4913$ and $5559$ .

$\frac{10472 + 867 \sqrt{109} + 109 \sqrt{109}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.7

Add $867 \sqrt{109}$ and $109 \sqrt{109}$ .

$\frac{10472 + 976 \sqrt{109}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.8

Cancel the common factor of $10472 + 976 \sqrt{109}$ and $54$ .

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Step 1.4.1.2.1.8.1

Factor $2$ out of $10472$ .

$\frac{2 (5236) + 976 \sqrt{109}}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.8.2

Factor $2$ out of $976 \sqrt{109}$ .

$\frac{2 (5236) + 2 (488 \sqrt{109})}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.8.3

Factor $2$ out of $2 (5236) + 2 (488 \sqrt{109})$ .

$\frac{2 (5236 + 488 \sqrt{109})}{54} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.8.4

Cancel the common factors.

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Step 1.4.1.2.1.8.4.1

Factor $2$ out of $54$ .

$\frac{2 (5236 + 488 \sqrt{109})}{2 \cdot 27} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.8.4.2

Cancel the common factor.

$\frac{2 (5236 + 488 \sqrt{109})}{2 \cdot 27} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.8.4.3

Rewrite the expression.

$\frac{5236 + 488 \sqrt{109}}{27} - 34 {(\frac{17 + \sqrt{109}}{6})}^{2} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.9

Apply the product rule to $\frac{17 + \sqrt{109}}{6}$ .

$\frac{5236 + 488 \sqrt{109}}{27} - 34 \frac{{(17 + \sqrt{109})}^{2}}{6^{2}} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.10

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

$\frac{5236 + 488 \sqrt{109}}{27} - 34 \frac{{(17 + \sqrt{109})}^{2}}{36} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.11

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.11.1

Factor $2$ out of $- 34$ .

$\frac{5236 + 488 \sqrt{109}}{27} + 2 (- 17) \frac{{(17 + \sqrt{109})}^{2}}{36} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.11.2

Factor $2$ out of $36$ .

$\frac{5236 + 488 \sqrt{109}}{27} + 2 \cdot - 17 \frac{{(17 + \sqrt{109})}^{2}}{2 \cdot 18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.11.3

Cancel the common factor.

$\frac{5236 + 488 \sqrt{109}}{27} + 2 \cdot - 17 \frac{{(17 + \sqrt{109})}^{2}}{2 \cdot 18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.11.4

Rewrite the expression.

$\frac{5236 + 488 \sqrt{109}}{27} - 17 \frac{{(17 + \sqrt{109})}^{2}}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.12

Combine $- 17$ and $\frac{{(17 + \sqrt{109})}^{2}}{18}$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 {(17 + \sqrt{109})}^{2}}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.13

Rewrite ${(17 + \sqrt{109})}^{2}$ as $(17 + \sqrt{109}) (17 + \sqrt{109})$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 ((17 + \sqrt{109}) (17 + \sqrt{109}))}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.14

Expand $(17 + \sqrt{109}) (17 + \sqrt{109})$ using the FOIL Method.

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Step 1.4.1.2.1.14.1

Apply the distributive property.

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (17 (17 + \sqrt{109}) + \sqrt{109} (17 + \sqrt{109}))}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.14.2

Apply the distributive property.

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (17 \cdot 17 + 17 \sqrt{109} + \sqrt{109} (17 + \sqrt{109}))}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.14.3

Apply the distributive property.

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (17 \cdot 17 + 17 \sqrt{109} + \sqrt{109} \cdot 17 + \sqrt{109} \sqrt{109})}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.15

Simplify and combine like terms.

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Step 1.4.1.2.1.15.1

Simplify each term.

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Step 1.4.1.2.1.15.1.1

Multiply $17$ by $17$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (289 + 17 \sqrt{109} + \sqrt{109} \cdot 17 + \sqrt{109} \sqrt{109})}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.15.1.2

Move $17$ to the left of $\sqrt{109}$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (289 + 17 \sqrt{109} + 17 \cdot \sqrt{109} + \sqrt{109} \sqrt{109})}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.15.1.3

Combine using the product rule for radicals.

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (289 + 17 \sqrt{109} + 17 \sqrt{109} + \sqrt{109 \cdot 109})}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.15.1.4

Multiply $109$ by $109$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (289 + 17 \sqrt{109} + 17 \sqrt{109} + \sqrt{11881})}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.15.1.5

Rewrite $11881$ as $109^{2}$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (289 + 17 \sqrt{109} + 17 \sqrt{109} + \sqrt{109^{2}})}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.15.1.6

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

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (289 + 17 \sqrt{109} + 17 \sqrt{109} + 109)}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.15.2

Add $289$ and $109$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (398 + 17 \sqrt{109} + 17 \sqrt{109})}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.15.3

Add $17 \sqrt{109}$ and $17 \sqrt{109}$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (398 + 34 \sqrt{109})}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.16

Cancel the common factor of $398 + 34 \sqrt{109}$ and $18$ .

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Step 1.4.1.2.1.16.1

Factor $2$ out of $- 17 (398 + 34 \sqrt{109})$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{2 (- 17 (199 + 17 \sqrt{109}))}{18} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.16.2

Cancel the common factors.

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Step 1.4.1.2.1.16.2.1

Factor $2$ out of $18$ .

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{2 (- 17 (199 + 17 \sqrt{109}))}{2 (9)} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.16.2.2

Cancel the common factor.

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{2 (- 17 (199 + 17 \sqrt{109}))}{2 \cdot 9} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.16.2.3

Rewrite the expression.

$\frac{5236 + 488 \sqrt{109}}{27} + \frac{- 17 (199 + 17 \sqrt{109})}{9} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.17

Move the negative in front of the fraction.

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109})}{9} + 60 (\frac{17 + \sqrt{109}}{6})$

Step 1.4.1.2.1.18

Cancel the common factor of $6$ .

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Step 1.4.1.2.1.18.1

Factor $6$ out of $60$ .

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109})}{9} + 6 (10) \frac{17 + \sqrt{109}}{6}$

Step 1.4.1.2.1.18.2

Cancel the common factor.

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109})}{9} + 6 \cdot 10 \frac{17 + \sqrt{109}}{6}$

Step 1.4.1.2.1.18.3

Rewrite the expression.

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109})}{9} + 10 (17 + \sqrt{109})$

Step 1.4.1.2.1.19

Apply the distributive property.

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109})}{9} + 10 \cdot 17 + 10 \sqrt{109}$

Step 1.4.1.2.1.20

Multiply $10$ by $17$ .

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109})}{9} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.2

To write $- \frac{17 (199 + 17 \sqrt{109})}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109})}{9} \cdot \frac{3}{3} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.3

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1.2.3.1

Multiply $\frac{17 (199 + 17 \sqrt{109})}{9}$ by $\frac{3}{3}$ .

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109}) \cdot 3}{9 \cdot 3} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.3.2

Multiply $9$ by $3$ .

$\frac{5236 + 488 \sqrt{109}}{27} - \frac{17 (199 + 17 \sqrt{109}) \cdot 3}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.4

Combine the numerators over the common denominator.

$\frac{5236 + 488 \sqrt{109} - 17 (199 + 17 \sqrt{109}) \cdot 3}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.5

Simplify the numerator.

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Step 1.4.1.2.5.1

Apply the distributive property.

$\frac{5236 + 488 \sqrt{109} + (- 17 \cdot 199 - 17 (17 \sqrt{109})) \cdot 3}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.5.2

Multiply $- 17$ by $199$ .

$\frac{5236 + 488 \sqrt{109} + (- 3383 - 17 (17 \sqrt{109})) \cdot 3}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.5.3

Multiply $17$ by $- 17$ .

$\frac{5236 + 488 \sqrt{109} + (- 3383 - 289 \sqrt{109}) \cdot 3}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.5.4

Apply the distributive property.

$\frac{5236 + 488 \sqrt{109} - 3383 \cdot 3 - 289 \sqrt{109} \cdot 3}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.5.5

Multiply $- 3383$ by $3$ .

$\frac{5236 + 488 \sqrt{109} - 10149 - 289 \sqrt{109} \cdot 3}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.5.6

Multiply $3$ by $- 289$ .

$\frac{5236 + 488 \sqrt{109} - 10149 - 867 \sqrt{109}}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.5.7

Subtract $10149$ from $5236$ .

$\frac{- 4913 + 488 \sqrt{109} - 867 \sqrt{109}}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.5.8

Subtract $867 \sqrt{109}$ from $488 \sqrt{109}$ .

$\frac{- 4913 - 379 \sqrt{109}}{27} + 170 + 10 \sqrt{109}$

Step 1.4.1.2.6

To write $170$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 4913 - 379 \sqrt{109}}{27} + 170 \cdot \frac{27}{27} + 10 \sqrt{109}$

Step 1.4.1.2.7

Combine $170$ and $\frac{27}{27}$ .

$\frac{- 4913 - 379 \sqrt{109}}{27} + \frac{170 \cdot 27}{27} + 10 \sqrt{109}$

Step 1.4.1.2.8

Simplify the expression.

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Step 1.4.1.2.8.1

Combine the numerators over the common denominator.

$\frac{- 4913 - 379 \sqrt{109} + 170 \cdot 27}{27} + 10 \sqrt{109}$

Step 1.4.1.2.8.2

Multiply $170$ by $27$ .

$\frac{- 4913 - 379 \sqrt{109} + 4590}{27} + 10 \sqrt{109}$

Step 1.4.1.2.8.3

Add $- 4913$ and $4590$ .

$\frac{- 323 - 379 \sqrt{109}}{27} + 10 \sqrt{109}$

Step 1.4.1.2.9

To write $10 \sqrt{109}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 323 - 379 \sqrt{109}}{27} + 10 \sqrt{109} \cdot \frac{27}{27}$

Step 1.4.1.2.10

Combine fractions.

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Step 1.4.1.2.10.1

Combine $10 \sqrt{109}$ and $\frac{27}{27}$ .

$\frac{- 323 - 379 \sqrt{109}}{27} + \frac{10 \sqrt{109} \cdot 27}{27}$

Step 1.4.1.2.10.2

Combine the numerators over the common denominator.

$\frac{- 323 - 379 \sqrt{109} + 10 \sqrt{109} \cdot 27}{27}$

Step 1.4.1.2.11

Simplify the numerator.

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Step 1.4.1.2.11.1

Multiply $27$ by $10$ .

$\frac{- 323 - 379 \sqrt{109} + 270 \sqrt{109}}{27}$

Step 1.4.1.2.11.2

Add $- 379 \sqrt{109}$ and $270 \sqrt{109}$ .

$\frac{- 323 - 109 \sqrt{109}}{27}$

Step 1.4.1.2.12

Simplify with factoring out.

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Step 1.4.1.2.12.1

Rewrite $- 323$ as $- 1 (323)$ .

$\frac{- 1 (323) - 109 \sqrt{109}}{27}$

Step 1.4.1.2.12.2

Factor $- 1$ out of $- 109 \sqrt{109}$ .

$\frac{- 1 (323) - (109 \sqrt{109})}{27}$

Step 1.4.1.2.12.3

Factor $- 1$ out of $- 1 (323) - (109 \sqrt{109})$ .

$\frac{- 1 (323 + 109 \sqrt{109})}{27}$

Step 1.4.1.2.12.4

Move the negative in front of the fraction.

$- \frac{323 + 109 \sqrt{109}}{27}$

Step 1.4.2

Evaluate at $x = \frac{17 - \sqrt{109}}{6}$ .

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Step 1.4.2.1

Substitute $\frac{17 - \sqrt{109}}{6}$ for $x$ .

$4 {(\frac{17 - \sqrt{109}}{6})}^{3} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

Apply the product rule to $\frac{17 - \sqrt{109}}{6}$ .

$4 \frac{{(17 - \sqrt{109})}^{3}}{6^{3}} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.2

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

$4 \frac{{(17 - \sqrt{109})}^{3}}{216} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.3

Cancel the common factor of $4$ .

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Step 1.4.2.2.1.3.1

Factor $4$ out of $216$ .

$4 \frac{{(17 - \sqrt{109})}^{3}}{4 (54)} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.3.2

Cancel the common factor.

$4 \frac{{(17 - \sqrt{109})}^{3}}{4 \cdot 54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.3.3

Rewrite the expression.

$\frac{{(17 - \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.4

Use the Binomial Theorem.

$\frac{17^{3} + 3 \cdot 17^{2} (- \sqrt{109}) + 3 \cdot 17 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5

Simplify each term.

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Step 1.4.2.2.1.5.1

Raise $17$ to the power of $3$ .

$\frac{4913 + 3 \cdot 17^{2} (- \sqrt{109}) + 3 \cdot 17 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.2

Raise $17$ to the power of $2$ .

$\frac{4913 + 3 \cdot 289 (- \sqrt{109}) + 3 \cdot 17 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.3

Multiply $3$ by $289$ .

$\frac{4913 + 867 (- \sqrt{109}) + 3 \cdot 17 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.4

Multiply $- 1$ by $867$ .

$\frac{4913 - 867 \sqrt{109} + 3 \cdot 17 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.5

Multiply $3$ by $17$ .

$\frac{4913 - 867 \sqrt{109} + 51 {(- \sqrt{109})}^{2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.6

Apply the product rule to $- \sqrt{109}$ .

$\frac{4913 - 867 \sqrt{109} + 51 ({(- 1)}^{2} {\sqrt{109}}^{2}) + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.7

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

$\frac{4913 - 867 \sqrt{109} + 51 (1 {\sqrt{109}}^{2}) + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.8

Multiply ${\sqrt{109}}^{2}$ by $1$ .

$\frac{4913 - 867 \sqrt{109} + 51 {\sqrt{109}}^{2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.9

Rewrite ${\sqrt{109}}^{2}$ as $109$ .

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Step 1.4.2.2.1.5.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{109}$ as $109^{\frac{1}{2}}$ .

$\frac{4913 - 867 \sqrt{109} + 51 {(109^{\frac{1}{2}})}^{2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.9.2

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

$\frac{4913 - 867 \sqrt{109} + 51 \cdot 109^{\frac{1}{2} \cdot 2} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.9.3

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

$\frac{4913 - 867 \sqrt{109} + 51 \cdot 109^{\frac{2}{2}} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.9.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.5.9.4.1

Cancel the common factor.

$\frac{4913 - 867 \sqrt{109} + 51 \cdot 109^{\frac{2}{2}} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.9.4.2

Rewrite the expression.

$\frac{4913 - 867 \sqrt{109} + 51 \cdot 109^{1} + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.9.5

Evaluate the exponent.

$\frac{4913 - 867 \sqrt{109} + 51 \cdot 109 + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.10

Multiply $51$ by $109$ .

$\frac{4913 - 867 \sqrt{109} + 5559 + {(- \sqrt{109})}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.11

Apply the product rule to $- \sqrt{109}$ .

$\frac{4913 - 867 \sqrt{109} + 5559 + {(- 1)}^{3} {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.12

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

$\frac{4913 - 867 \sqrt{109} + 5559 - {\sqrt{109}}^{3}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.13

Rewrite ${\sqrt{109}}^{3}$ as $\sqrt{109^{3}}$ .

$\frac{4913 - 867 \sqrt{109} + 5559 - \sqrt{109^{3}}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.14

Raise $109$ to the power of $3$ .

$\frac{4913 - 867 \sqrt{109} + 5559 - \sqrt{1295029}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.15

Rewrite $1295029$ as $109^{2} \cdot 109$ .

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Step 1.4.2.2.1.5.15.1

Factor $11881$ out of $1295029$ .

$\frac{4913 - 867 \sqrt{109} + 5559 - \sqrt{11881 (109)}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.15.2

Rewrite $11881$ as $109^{2}$ .

$\frac{4913 - 867 \sqrt{109} + 5559 - \sqrt{109^{2} \cdot 109}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.16

Pull terms out from under the radical.

$\frac{4913 - 867 \sqrt{109} + 5559 - (109 \sqrt{109})}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.5.17

Multiply $109$ by $- 1$ .

$\frac{4913 - 867 \sqrt{109} + 5559 - 109 \sqrt{109}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.6

Add $4913$ and $5559$ .

$\frac{10472 - 867 \sqrt{109} - 109 \sqrt{109}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.7

Subtract $109 \sqrt{109}$ from $- 867 \sqrt{109}$ .

$\frac{10472 - 976 \sqrt{109}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.8

Cancel the common factor of $10472 - 976 \sqrt{109}$ and $54$ .

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Step 1.4.2.2.1.8.1

Factor $2$ out of $10472$ .

$\frac{2 (5236) - 976 \sqrt{109}}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.8.2

Factor $2$ out of $- 976 \sqrt{109}$ .

$\frac{2 (5236) + 2 (- 488 \sqrt{109})}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.8.3

Factor $2$ out of $2 (5236) + 2 (- 488 \sqrt{109})$ .

$\frac{2 (5236 - 488 \sqrt{109})}{54} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.8.4

Cancel the common factors.

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Step 1.4.2.2.1.8.4.1

Factor $2$ out of $54$ .

$\frac{2 (5236 - 488 \sqrt{109})}{2 \cdot 27} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.8.4.2

Cancel the common factor.

$\frac{2 (5236 - 488 \sqrt{109})}{2 \cdot 27} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.8.4.3

Rewrite the expression.

$\frac{5236 - 488 \sqrt{109}}{27} - 34 {(\frac{17 - \sqrt{109}}{6})}^{2} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.9

Apply the product rule to $\frac{17 - \sqrt{109}}{6}$ .

$\frac{5236 - 488 \sqrt{109}}{27} - 34 \frac{{(17 - \sqrt{109})}^{2}}{6^{2}} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.10

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

$\frac{5236 - 488 \sqrt{109}}{27} - 34 \frac{{(17 - \sqrt{109})}^{2}}{36} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.11

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.11.1

Factor $2$ out of $- 34$ .

$\frac{5236 - 488 \sqrt{109}}{27} + 2 (- 17) \frac{{(17 - \sqrt{109})}^{2}}{36} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.11.2

Factor $2$ out of $36$ .

$\frac{5236 - 488 \sqrt{109}}{27} + 2 \cdot - 17 \frac{{(17 - \sqrt{109})}^{2}}{2 \cdot 18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.11.3

Cancel the common factor.

$\frac{5236 - 488 \sqrt{109}}{27} + 2 \cdot - 17 \frac{{(17 - \sqrt{109})}^{2}}{2 \cdot 18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.11.4

Rewrite the expression.

$\frac{5236 - 488 \sqrt{109}}{27} - 17 \frac{{(17 - \sqrt{109})}^{2}}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.12

Combine $- 17$ and $\frac{{(17 - \sqrt{109})}^{2}}{18}$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 {(17 - \sqrt{109})}^{2}}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.13

Rewrite ${(17 - \sqrt{109})}^{2}$ as $(17 - \sqrt{109}) (17 - \sqrt{109})$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 ((17 - \sqrt{109}) (17 - \sqrt{109}))}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.14

Expand $(17 - \sqrt{109}) (17 - \sqrt{109})$ using the FOIL Method.

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Step 1.4.2.2.1.14.1

Apply the distributive property.

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (17 (17 - \sqrt{109}) - \sqrt{109} (17 - \sqrt{109}))}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.14.2

Apply the distributive property.

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (17 \cdot 17 + 17 (- \sqrt{109}) - \sqrt{109} (17 - \sqrt{109}))}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.14.3

Apply the distributive property.

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (17 \cdot 17 + 17 (- \sqrt{109}) - \sqrt{109} \cdot 17 - \sqrt{109} (- \sqrt{109}))}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15

Simplify and combine like terms.

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Step 1.4.2.2.1.15.1

Simplify each term.

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Step 1.4.2.2.1.15.1.1

Multiply $17$ by $17$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 + 17 (- \sqrt{109}) - \sqrt{109} \cdot 17 - \sqrt{109} (- \sqrt{109}))}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.2

Multiply $- 1$ by $17$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - \sqrt{109} \cdot 17 - \sqrt{109} (- \sqrt{109}))}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.3

Multiply $17$ by $- 1$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} - \sqrt{109} (- \sqrt{109}))}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.4

Multiply $- \sqrt{109} (- \sqrt{109})$ .

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Step 1.4.2.2.1.15.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + 1 \sqrt{109} \sqrt{109})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.4.2

Multiply $\sqrt{109}$ by $1$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + \sqrt{109} \sqrt{109})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.4.3

Raise $\sqrt{109}$ to the power of $1$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + {\sqrt{109}}^{1} \sqrt{109})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.4.4

Raise $\sqrt{109}$ to the power of $1$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + {\sqrt{109}}^{1} {\sqrt{109}}^{1})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.4.5

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

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + {\sqrt{109}}^{1 + 1})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.4.6

Add $1$ and $1$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + {\sqrt{109}}^{2})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.5

Rewrite ${\sqrt{109}}^{2}$ as $109$ .

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Step 1.4.2.2.1.15.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{109}$ as $109^{\frac{1}{2}}$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + {(109^{\frac{1}{2}})}^{2})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.5.2

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

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + 109^{\frac{1}{2} \cdot 2})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.5.3

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

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + 109^{\frac{2}{2}})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.5.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.15.1.5.4.1

Cancel the common factor.

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + 109^{\frac{2}{2}})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.5.4.2

Rewrite the expression.

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + 109^{1})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.1.5.5

Evaluate the exponent.

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (289 - 17 \sqrt{109} - 17 \sqrt{109} + 109)}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.2

Add $289$ and $109$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (398 - 17 \sqrt{109} - 17 \sqrt{109})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.15.3

Subtract $17 \sqrt{109}$ from $- 17 \sqrt{109}$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (398 - 34 \sqrt{109})}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.16

Cancel the common factor of $398 - 34 \sqrt{109}$ and $18$ .

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Step 1.4.2.2.1.16.1

Factor $2$ out of $- 17 (398 - 34 \sqrt{109})$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{2 (- 17 (199 - 17 \sqrt{109}))}{18} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.16.2

Cancel the common factors.

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Step 1.4.2.2.1.16.2.1

Factor $2$ out of $18$ .

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{2 (- 17 (199 - 17 \sqrt{109}))}{2 (9)} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.16.2.2

Cancel the common factor.

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{2 (- 17 (199 - 17 \sqrt{109}))}{2 \cdot 9} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.16.2.3

Rewrite the expression.

$\frac{5236 - 488 \sqrt{109}}{27} + \frac{- 17 (199 - 17 \sqrt{109})}{9} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.17

Move the negative in front of the fraction.

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109})}{9} + 60 (\frac{17 - \sqrt{109}}{6})$

Step 1.4.2.2.1.18

Cancel the common factor of $6$ .

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Step 1.4.2.2.1.18.1

Factor $6$ out of $60$ .

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109})}{9} + 6 (10) \frac{17 - \sqrt{109}}{6}$

Step 1.4.2.2.1.18.2

Cancel the common factor.

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109})}{9} + 6 \cdot 10 \frac{17 - \sqrt{109}}{6}$

Step 1.4.2.2.1.18.3

Rewrite the expression.

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109})}{9} + 10 (17 - \sqrt{109})$

Step 1.4.2.2.1.19

Apply the distributive property.

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109})}{9} + 10 \cdot 17 + 10 (- \sqrt{109})$

Step 1.4.2.2.1.20

Multiply $10$ by $17$ .

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109})}{9} + 170 + 10 (- \sqrt{109})$

Step 1.4.2.2.1.21

Multiply $- 1$ by $10$ .

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109})}{9} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.2

To write $- \frac{17 (199 - 17 \sqrt{109})}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109})}{9} \cdot \frac{3}{3} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.3

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.2.2.3.1

Multiply $\frac{17 (199 - 17 \sqrt{109})}{9}$ by $\frac{3}{3}$ .

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109}) \cdot 3}{9 \cdot 3} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.3.2

Multiply $9$ by $3$ .

$\frac{5236 - 488 \sqrt{109}}{27} - \frac{17 (199 - 17 \sqrt{109}) \cdot 3}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.4

Combine the numerators over the common denominator.

$\frac{5236 - 488 \sqrt{109} - 17 (199 - 17 \sqrt{109}) \cdot 3}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.5

Simplify the numerator.

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Step 1.4.2.2.5.1

Apply the distributive property.

$\frac{5236 - 488 \sqrt{109} + (- 17 \cdot 199 - 17 (- 17 \sqrt{109})) \cdot 3}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.5.2

Multiply $- 17$ by $199$ .

$\frac{5236 - 488 \sqrt{109} + (- 3383 - 17 (- 17 \sqrt{109})) \cdot 3}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.5.3

Multiply $- 17$ by $- 17$ .

$\frac{5236 - 488 \sqrt{109} + (- 3383 + 289 \sqrt{109}) \cdot 3}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.5.4

Apply the distributive property.

$\frac{5236 - 488 \sqrt{109} - 3383 \cdot 3 + 289 \sqrt{109} \cdot 3}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.5.5

Multiply $- 3383$ by $3$ .

$\frac{5236 - 488 \sqrt{109} - 10149 + 289 \sqrt{109} \cdot 3}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.5.6

Multiply $3$ by $289$ .

$\frac{5236 - 488 \sqrt{109} - 10149 + 867 \sqrt{109}}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.5.7

Subtract $10149$ from $5236$ .

$\frac{- 4913 - 488 \sqrt{109} + 867 \sqrt{109}}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.5.8

Add $- 488 \sqrt{109}$ and $867 \sqrt{109}$ .

$\frac{- 4913 + 379 \sqrt{109}}{27} + 170 - 10 \sqrt{109}$

Step 1.4.2.2.6

To write $170$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 4913 + 379 \sqrt{109}}{27} + 170 \cdot \frac{27}{27} - 10 \sqrt{109}$

Step 1.4.2.2.7

Combine $170$ and $\frac{27}{27}$ .

$\frac{- 4913 + 379 \sqrt{109}}{27} + \frac{170 \cdot 27}{27} - 10 \sqrt{109}$

Step 1.4.2.2.8

Simplify the expression.

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Step 1.4.2.2.8.1

Combine the numerators over the common denominator.

$\frac{- 4913 + 379 \sqrt{109} + 170 \cdot 27}{27} - 10 \sqrt{109}$

Step 1.4.2.2.8.2

Multiply $170$ by $27$ .

$\frac{- 4913 + 379 \sqrt{109} + 4590}{27} - 10 \sqrt{109}$

Step 1.4.2.2.8.3

Add $- 4913$ and $4590$ .

$\frac{- 323 + 379 \sqrt{109}}{27} - 10 \sqrt{109}$

Step 1.4.2.2.9

To write $- 10 \sqrt{109}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 323 + 379 \sqrt{109}}{27} - 10 \sqrt{109} \cdot \frac{27}{27}$

Step 1.4.2.2.10

Combine fractions.

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Step 1.4.2.2.10.1

Combine $- 10 \sqrt{109}$ and $\frac{27}{27}$ .

$\frac{- 323 + 379 \sqrt{109}}{27} + \frac{- 10 \sqrt{109} \cdot 27}{27}$

Step 1.4.2.2.10.2

Combine the numerators over the common denominator.

$\frac{- 323 + 379 \sqrt{109} - 10 \sqrt{109} \cdot 27}{27}$

Step 1.4.2.2.11

Simplify the numerator.

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Step 1.4.2.2.11.1

Multiply $27$ by $- 10$ .

$\frac{- 323 + 379 \sqrt{109} - 270 \sqrt{109}}{27}$

Step 1.4.2.2.11.2

Subtract $270 \sqrt{109}$ from $379 \sqrt{109}$ .

$\frac{- 323 + 109 \sqrt{109}}{27}$

Step 1.4.2.2.12

Simplify with factoring out.

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Step 1.4.2.2.12.1

Rewrite $- 323$ as $- 1 (323)$ .

$\frac{- 1 (323) + 109 \sqrt{109}}{27}$

Step 1.4.2.2.12.2

Factor $- 1$ out of $109 \sqrt{109}$ .

$\frac{- 1 (323) - (- 109 \sqrt{109})}{27}$

Step 1.4.2.2.12.3

Factor $- 1$ out of $- 1 (323) - (- 109 \sqrt{109})$ .

$\frac{- 1 (323 - 109 \sqrt{109})}{27}$

Step 1.4.2.2.12.4

Move the negative in front of the fraction.

$- \frac{323 - 109 \sqrt{109}}{27}$

Step 1.4.3

List all of the points.

$(\frac{17 + \sqrt{109}}{6}, - \frac{323 + 109 \sqrt{109}}{27}), (\frac{17 - \sqrt{109}}{6}, - \frac{323 - 109 \sqrt{109}}{27})$

Step 2

Exclude the points that are not on the interval.

$(\frac{17 - \sqrt{109}}{6}, - \frac{323 - 109 \sqrt{109}}{27})$

Step 3

Use the first derivative test to determine which points can be maxima or minima.

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Step 3.1

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

$(- \infty, \frac{17 - \sqrt{109}}{6}) \cup (\frac{17 - \sqrt{109}}{6}, \frac{17 + \sqrt{109}}{6}) \cup (\frac{17 + \sqrt{109}}{6}, \infty)$

Step 3.2

Substitute any number, such as $0$ , from the interval $(- \infty, \frac{17 - \sqrt{109}}{6})$ in the first derivative $12 x^{2} - 68 x + 60$ to check if the result is negative or positive.

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Step 3.2.1

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

$f' (0) = 12 {(0)}^{2} - 68 \cdot 0 + 60$

Step 3.2.2

Simplify the result.

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

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

$f' (0) = 12 \cdot 0 - 68 \cdot 0 + 60$

Step 3.2.2.1.2

Multiply $12$ by $0$ .

$f' (0) = 0 - 68 \cdot 0 + 60$

Step 3.2.2.1.3

Multiply $- 68$ by $0$ .

$f' (0) = 0 + 0 + 60$

Step 3.2.2.2

Simplify by adding numbers.

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Step 3.2.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 + 60$

Step 3.2.2.2.2

Add $0$ and $60$ .

$f' (0) = 60$

Step 3.2.2.3

The final answer is $60$ .

$60$

Step 3.3

Substitute any number, such as $3$ , from the interval $(\frac{17 - \sqrt{109}}{6}, \frac{17 + \sqrt{109}}{6})$ in the first derivative $12 x^{2} - 68 x + 60$ to check if the result is negative or positive.

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Step 3.3.1

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

$f' (3) = 12 {(3)}^{2} - 68 \cdot 3 + 60$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

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

$f' (3) = 12 \cdot 9 - 68 \cdot 3 + 60$

Step 3.3.2.1.2

Multiply $12$ by $9$ .

$f' (3) = 108 - 68 \cdot 3 + 60$

Step 3.3.2.1.3

Multiply $- 68$ by $3$ .

$f' (3) = 108 - 204 + 60$

Step 3.3.2.2

Simplify by adding and subtracting.

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Step 3.3.2.2.1

Subtract $204$ from $108$ .

$f' (3) = - 96 + 60$

Step 3.3.2.2.2

Add $- 96$ and $60$ .

$f' (3) = - 36$

Step 3.3.2.3

The final answer is $- 36$ .

$- 36$

Step 3.4

Substitute any number, such as $7$ , from the interval $(\frac{17 + \sqrt{109}}{6}, \infty)$ in the first derivative $12 x^{2} - 68 x + 60$ to check if the result is negative or positive.

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Step 3.4.1

Replace the variable $x$ with $7$ in the expression.

$f' (7) = 12 {(7)}^{2} - 68 \cdot 7 + 60$

Step 3.4.2

Simplify the result.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Raise $7$ to the power of $2$ .

$f' (7) = 12 \cdot 49 - 68 \cdot 7 + 60$

Step 3.4.2.1.2

Multiply $12$ by $49$ .

$f' (7) = 588 - 68 \cdot 7 + 60$

Step 3.4.2.1.3

Multiply $- 68$ by $7$ .

$f' (7) = 588 - 476 + 60$

Step 3.4.2.2

Simplify by adding and subtracting.

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Step 3.4.2.2.1

Subtract $476$ from $588$ .

$f' (7) = 112 + 60$

Step 3.4.2.2.2

Add $112$ and $60$ .

$f' (7) = 172$

Step 3.4.2.3

The final answer is $172$ .

$172$

Step 3.5

Since the first derivative changed signs from positive to negative around $x = \frac{17 - \sqrt{109}}{6}$ , then $x = \frac{17 - \sqrt{109}}{6}$ is a local maximum.

$x = \frac{17 - \sqrt{109}}{6}$ is a local maximum

Step 3.6

Since the first derivative changed signs from negative to positive around $x = \frac{17 + \sqrt{109}}{6}$ , then $x = \frac{17 + \sqrt{109}}{6}$ is a local minimum.

$x = \frac{17 + \sqrt{109}}{6}$ is a local minimum

Step 3.7

These are the local extrema for $f (x) = 4 x^{3} - 34 x^{2} + 60 x$ .

$x = \frac{17 - \sqrt{109}}{6}$ is a local maximum

$x = \frac{17 + \sqrt{109}}{6}$ is a local minimum

$x = \frac{17 - \sqrt{109}}{6}$ is a local maximum

$x = \frac{17 + \sqrt{109}}{6}$ is a local minimum

Step 4

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(\frac{17 - \sqrt{109}}{6}, - \frac{323 - 109 \sqrt{109}}{27})$

No absolute minimum

Step 5