Calculus Examples

$h (x) = x^{4} - x^{3} - 6 x^{2}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2

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

$4 x^{3} + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [- 6 x^{2}]$

Step 1.2

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

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Step 1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$4 x^{3} - \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}]$

Step 1.2.2

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

$4 x^{3} - (3 x^{2}) + \frac{d}{d x} [- 6 x^{2}]$

Step 1.2.3

Multiply $3$ by $- 1$ .

$4 x^{3} - 3 x^{2} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.3

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

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Step 1.3.1

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

$4 x^{3} - 3 x^{2} - 6 \frac{d}{d x} [x^{2}]$

Step 1.3.2

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

$4 x^{3} - 3 x^{2} - 6 (2 x)$

Step 1.3.3

Multiply $2$ by $- 6$ .

$4 x^{3} - 3 x^{2} - 12 x$

Step 2

Set the first derivative equal to $0$ and solve for $x$ .

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Step 2.1

Factor $x$ out of $4 x^{3} - 3 x^{2} - 12 x$ .

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Step 2.1.1

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

$x (4 x^{2}) - 3 x^{2} - 12 x = 0$

Step 2.1.2

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

$x (4 x^{2}) + x (- 3 x) - 12 x = 0$

Step 2.1.3

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

$x (4 x^{2}) + x (- 3 x) + x \cdot - 12 = 0$

Step 2.1.4

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

$x (4 x^{2} - 3 x) + x \cdot - 12 = 0$

Step 2.1.5

Factor $x$ out of $x (4 x^{2} - 3 x) + x \cdot - 12$ .

$x (4 x^{2} - 3 x - 12) = 0$

Step 2.2

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

$x = 0$

$4 x^{2} - 3 x - 12 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $4 x^{2} - 3 x - 12$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $4 x^{2} - 3 x - 12$ equal to $0$ .

$4 x^{2} - 3 x - 12 = 0$

Step 2.4.2

Solve $4 x^{2} - 3 x - 12 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 4$ , $b = - 3$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (4 \cdot - 12)}}{2 \cdot 4}$

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 12}}{2 \cdot 4}$

Step 2.4.2.3.1.2

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

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 12}}{2 \cdot 4}$

Step 2.4.2.3.1.2.2

Multiply $- 16$ by $- 12$ .

$x = \frac{3 \pm \sqrt{9 + 192}}{2 \cdot 4}$

Step 2.4.2.3.1.3

Add $9$ and $192$ .

$x = \frac{3 \pm \sqrt{201}}{2 \cdot 4}$

Step 2.4.2.3.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{201}}{8}$

Step 2.4.2.4

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

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Step 2.4.2.4.1

Simplify the numerator.

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Step 2.4.2.4.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 12}}{2 \cdot 4}$

Step 2.4.2.4.1.2

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

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Step 2.4.2.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 12}}{2 \cdot 4}$

Step 2.4.2.4.1.2.2

Multiply $- 16$ by $- 12$ .

$x = \frac{3 \pm \sqrt{9 + 192}}{2 \cdot 4}$

Step 2.4.2.4.1.3

Add $9$ and $192$ .

$x = \frac{3 \pm \sqrt{201}}{2 \cdot 4}$

Step 2.4.2.4.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{201}}{8}$

Step 2.4.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{201}}{8}$

Step 2.4.2.5

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

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Step 2.4.2.5.1

Simplify the numerator.

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Step 2.4.2.5.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 12}}{2 \cdot 4}$

Step 2.4.2.5.1.2

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

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Step 2.4.2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 12}}{2 \cdot 4}$

Step 2.4.2.5.1.2.2

Multiply $- 16$ by $- 12$ .

$x = \frac{3 \pm \sqrt{9 + 192}}{2 \cdot 4}$

Step 2.4.2.5.1.3

Add $9$ and $192$ .

$x = \frac{3 \pm \sqrt{201}}{2 \cdot 4}$

Step 2.4.2.5.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{201}}{8}$

Step 2.4.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{201}}{8}$

Step 2.4.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{201}}{8}, \frac{3 - \sqrt{201}}{8}$

Step 2.5

The final solution is all the values that make $x (4 x^{2} - 3 x - 12) = 0$ true.

$x = 0, \frac{3 + \sqrt{201}}{8}, \frac{3 - \sqrt{201}}{8}$

Step 3

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

$(- \infty, \frac{3 - \sqrt{201}}{8}) \cup (\frac{3 - \sqrt{201}}{8}, 0) \cup (0, \frac{3 + \sqrt{201}}{8}) \cup (\frac{3 + \sqrt{201}}{8}, \infty)$

Step 4

Substitute any number, such as $- 4$ , from the interval $(- \infty, \frac{3 - \sqrt{201}}{8})$ in the first derivative $4 x^{3} - 3 x^{2} - 12 x$ to check if the result is negative or positive.

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Step 4.1

Replace the variable $x$ with $- 4$ in the expression.

$h' (- 4) = 4 {(- 4)}^{3} - 3 {(- 4)}^{2} - 12 \cdot - 4$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$h' (- 4) = 4 \cdot - 64 - 3 {(- 4)}^{2} - 12 \cdot - 4$

Step 4.2.1.2

Multiply $4$ by $- 64$ .

$h' (- 4) = - 256 - 3 {(- 4)}^{2} - 12 \cdot - 4$

Step 4.2.1.3

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

$h' (- 4) = - 256 - 3 \cdot 16 - 12 \cdot - 4$

Step 4.2.1.4

Multiply $- 3$ by $16$ .

$h' (- 4) = - 256 - 48 - 12 \cdot - 4$

Step 4.2.1.5

Multiply $- 12$ by $- 4$ .

$h' (- 4) = - 256 - 48 + 48$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $48$ from $- 256$ .

$h' (- 4) = - 304 + 48$

Step 4.2.2.2

Add $- 304$ and $48$ .

$h' (- 4) = - 256$

Step 4.2.3

The final answer is $- 256$ .

$- 256$

Step 5

Substitute any number, such as $- 1$ , from the interval $(\frac{3 - \sqrt{201}}{8}, 0)$ in the first derivative $4 x^{3} - 3 x^{2} - 12 x$ to check if the result is negative or positive.

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Step 5.1

Replace the variable $x$ with $- 1$ in the expression.

$h' (- 1) = 4 {(- 1)}^{3} - 3 {(- 1)}^{2} - 12 \cdot - 1$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$h' (- 1) = 4 \cdot - 1 - 3 {(- 1)}^{2} - 12 \cdot - 1$

Step 5.2.1.2

Multiply $4$ by $- 1$ .

$h' (- 1) = - 4 - 3 {(- 1)}^{2} - 12 \cdot - 1$

Step 5.2.1.3

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

$h' (- 1) = - 4 - 3 \cdot 1 - 12 \cdot - 1$

Step 5.2.1.4

Multiply $- 3$ by $1$ .

$h' (- 1) = - 4 - 3 - 12 \cdot - 1$

Step 5.2.1.5

Multiply $- 12$ by $- 1$ .

$h' (- 1) = - 4 - 3 + 12$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $3$ from $- 4$ .

$h' (- 1) = - 7 + 12$

Step 5.2.2.2

Add $- 7$ and $12$ .

$h' (- 1) = 5$

Step 5.2.3

The final answer is $5$ .

$5$

Step 6

Substitute any number, such as $1$ , from the interval $(0, \frac{3 + \sqrt{201}}{8})$ in the first derivative $4 x^{3} - 3 x^{2} - 12 x$ to check if the result is negative or positive.

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Step 6.1

Replace the variable $x$ with $1$ in the expression.

$h' (1) = 4 {(1)}^{3} - 3 {(1)}^{2} - 12 \cdot 1$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

One to any power is one.

$h' (1) = 4 \cdot 1 - 3 {(1)}^{2} - 12 \cdot 1$

Step 6.2.1.2

Multiply $4$ by $1$ .

$h' (1) = 4 - 3 {(1)}^{2} - 12 \cdot 1$

Step 6.2.1.3

One to any power is one.

$h' (1) = 4 - 3 \cdot 1 - 12 \cdot 1$

Step 6.2.1.4

Multiply $- 3$ by $1$ .

$h' (1) = 4 - 3 - 12 \cdot 1$

Step 6.2.1.5

Multiply $- 12$ by $1$ .

$h' (1) = 4 - 3 - 12$

Step 6.2.2

Simplify by subtracting numbers.

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Step 6.2.2.1

Subtract $3$ from $4$ .

$h' (1) = 1 - 12$

Step 6.2.2.2

Subtract $12$ from $1$ .

$h' (1) = - 11$

Step 6.2.3

The final answer is $- 11$ .

$- 11$

Step 7

Substitute any number, such as $5$ , from the interval $(\frac{3 + \sqrt{201}}{8}, \infty)$ in the first derivative $4 x^{3} - 3 x^{2} - 12 x$ to check if the result is negative or positive.

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Step 7.1

Replace the variable $x$ with $5$ in the expression.

$h' (5) = 4 {(5)}^{3} - 3 {(5)}^{2} - 12 \cdot 5$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $5$ to the power of $3$ .

$h' (5) = 4 \cdot 125 - 3 {(5)}^{2} - 12 \cdot 5$

Step 7.2.1.2

Multiply $4$ by $125$ .

$h' (5) = 500 - 3 {(5)}^{2} - 12 \cdot 5$

Step 7.2.1.3

Raise $5$ to the power of $2$ .

$h' (5) = 500 - 3 \cdot 25 - 12 \cdot 5$

Step 7.2.1.4

Multiply $- 3$ by $25$ .

$h' (5) = 500 - 75 - 12 \cdot 5$

Step 7.2.1.5

Multiply $- 12$ by $5$ .

$h' (5) = 500 - 75 - 60$

Step 7.2.2

Simplify by subtracting numbers.

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Step 7.2.2.1

Subtract $75$ from $500$ .

$h' (5) = 425 - 60$

Step 7.2.2.2

Subtract $60$ from $425$ .

$h' (5) = 365$

Step 7.2.3

The final answer is $365$ .

$365$

Step 8

Since the first derivative changed signs from negative to positive around $x = \frac{3 - \sqrt{201}}{8}$ , then there is a turning point at $x = \frac{3 - \sqrt{201}}{8}$ .

Step 9

Find the y-coordinate of $\frac{3 - \sqrt{201}}{8}$ to find the turning point.

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Step 9.1

Find $h (\frac{3 - \sqrt{201}}{8})$ to find the y-coordinate of $\frac{3 - \sqrt{201}}{8}$ .

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Step 9.1.1

Replace the variable $x$ with $\frac{3 - \sqrt{201}}{8}$ in the expression.

$h (\frac{3 - \sqrt{201}}{8}) = {(\frac{3 - \sqrt{201}}{8})}^{4} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2

Simplify ${(\frac{3 - \sqrt{201}}{8})}^{4} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$ .

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Step 9.1.2.1

Remove parentheses.

${(\frac{3 - \sqrt{201}}{8})}^{4} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2

Simplify each term.

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Step 9.1.2.2.1

Apply the product rule to $\frac{3 - \sqrt{201}}{8}$ .

$\frac{{(3 - \sqrt{201})}^{4}}{8^{4}} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.2

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

$\frac{{(3 - \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.3

Use the Binomial Theorem.

$\frac{3^{4} + 4 \cdot 3^{3} (- \sqrt{201}) + 6 \cdot 3^{2} {(- \sqrt{201})}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4

Simplify each term.

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Step 9.1.2.2.4.1

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

$\frac{81 + 4 \cdot 3^{3} (- \sqrt{201}) + 6 \cdot 3^{2} {(- \sqrt{201})}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.2

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

$\frac{81 + 4 \cdot 27 (- \sqrt{201}) + 6 \cdot 3^{2} {(- \sqrt{201})}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.3

Multiply $4$ by $27$ .

$\frac{81 + 108 (- \sqrt{201}) + 6 \cdot 3^{2} {(- \sqrt{201})}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.4

Multiply $- 1$ by $108$ .

$\frac{81 - 108 \sqrt{201} + 6 \cdot 3^{2} {(- \sqrt{201})}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.5

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

$\frac{81 - 108 \sqrt{201} + 6 \cdot 9 {(- \sqrt{201})}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.6

Multiply $6$ by $9$ .

$\frac{81 - 108 \sqrt{201} + 54 {(- \sqrt{201})}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.7

Apply the product rule to $- \sqrt{201}$ .

$\frac{81 - 108 \sqrt{201} + 54 ({(- 1)}^{2} {\sqrt{201}}^{2}) + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.8

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

$\frac{81 - 108 \sqrt{201} + 54 (1 {\sqrt{201}}^{2}) + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.9

Multiply ${\sqrt{201}}^{2}$ by $1$ .

$\frac{81 - 108 \sqrt{201} + 54 {\sqrt{201}}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.10

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Step 9.1.2.2.4.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$\frac{81 - 108 \sqrt{201} + 54 {(201^{\frac{1}{2}})}^{2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.10.2

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

$\frac{81 - 108 \sqrt{201} + 54 \cdot 201^{\frac{1}{2} \cdot 2} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.10.3

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

$\frac{81 - 108 \sqrt{201} + 54 \cdot 201^{\frac{2}{2}} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.10.4

Cancel the common factor of $2$ .

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Step 9.1.2.2.4.10.4.1

Cancel the common factor.

$\frac{81 - 108 \sqrt{201} + 54 \cdot 201^{\frac{2}{2}} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.10.4.2

Rewrite the expression.

$\frac{81 - 108 \sqrt{201} + 54 \cdot 201^{1} + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.10.5

Evaluate the exponent.

$\frac{81 - 108 \sqrt{201} + 54 \cdot 201 + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.11

Multiply $54$ by $201$ .

$\frac{81 - 108 \sqrt{201} + 10854 + 4 \cdot 3 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.12

Multiply $4$ by $3$ .

$\frac{81 - 108 \sqrt{201} + 10854 + 12 {(- \sqrt{201})}^{3} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.13

Apply the product rule to $- \sqrt{201}$ .

$\frac{81 - 108 \sqrt{201} + 10854 + 12 ({(- 1)}^{3} {\sqrt{201}}^{3}) + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.14

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

$\frac{81 - 108 \sqrt{201} + 10854 + 12 (- {\sqrt{201}}^{3}) + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.15

Rewrite ${\sqrt{201}}^{3}$ as $\sqrt{201^{3}}$ .

$\frac{81 - 108 \sqrt{201} + 10854 + 12 (- \sqrt{201^{3}}) + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.16

Raise $201$ to the power of $3$ .

$\frac{81 - 108 \sqrt{201} + 10854 + 12 (- \sqrt{8120601}) + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.17

Rewrite $8120601$ as $201^{2} \cdot 201$ .

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Step 9.1.2.2.4.17.1

Factor $40401$ out of $8120601$ .

$\frac{81 - 108 \sqrt{201} + 10854 + 12 (- \sqrt{40401 (201)}) + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.17.2

Rewrite $40401$ as $201^{2}$ .

$\frac{81 - 108 \sqrt{201} + 10854 + 12 (- \sqrt{201^{2} \cdot 201}) + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.18

Pull terms out from under the radical.

$\frac{81 - 108 \sqrt{201} + 10854 + 12 (- (201 \sqrt{201})) + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.19

Multiply $201$ by $- 1$ .

$\frac{81 - 108 \sqrt{201} + 10854 + 12 (- 201 \sqrt{201}) + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.20

Multiply $- 201$ by $12$ .

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + {(- \sqrt{201})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.21

Apply the product rule to $- \sqrt{201}$ .

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + {(- 1)}^{4} {\sqrt{201}}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.22

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

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 1 {\sqrt{201}}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.23

Multiply ${\sqrt{201}}^{4}$ by $1$ .

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.24

Rewrite ${\sqrt{201}}^{4}$ as $201^{2}$ .

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Step 9.1.2.2.4.24.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + {(201^{\frac{1}{2}})}^{4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.24.2

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

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{1}{2} \cdot 4}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.24.3

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

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{4}{2}}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.24.4

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

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Step 9.1.2.2.4.24.4.1

Factor $2$ out of $4$ .

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2}}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.24.4.2

Cancel the common factors.

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Step 9.1.2.2.4.24.4.2.1

Factor $2$ out of $2$ .

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2 (1)}}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.24.4.2.2

Cancel the common factor.

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2 \cdot 1}}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.24.4.2.3

Rewrite the expression.

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{\frac{2}{1}}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.24.4.2.4

Divide $2$ by $1$ .

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 201^{2}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.4.25

Raise $201$ to the power of $2$ .

$\frac{81 - 108 \sqrt{201} + 10854 - 2412 \sqrt{201} + 40401}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.5

Add $81$ and $10854$ .

$\frac{10935 - 108 \sqrt{201} - 2412 \sqrt{201} + 40401}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.6

Add $10935$ and $40401$ .

$\frac{51336 - 108 \sqrt{201} - 2412 \sqrt{201}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.7

Subtract $2412 \sqrt{201}$ from $- 108 \sqrt{201}$ .

$\frac{51336 - 2520 \sqrt{201}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.8

Cancel the common factor of $51336 - 2520 \sqrt{201}$ and $4096$ .

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Step 9.1.2.2.8.1

Factor $8$ out of $51336$ .

$\frac{8 (6417) - 2520 \sqrt{201}}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.8.2

Factor $8$ out of $- 2520 \sqrt{201}$ .

$\frac{8 (6417) + 8 (- 315 \sqrt{201})}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.8.3

Factor $8$ out of $8 (6417) + 8 (- 315 \sqrt{201})$ .

$\frac{8 (6417 - 315 \sqrt{201})}{4096} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.8.4

Cancel the common factors.

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Step 9.1.2.2.8.4.1

Factor $8$ out of $4096$ .

$\frac{8 (6417 - 315 \sqrt{201})}{8 \cdot 512} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.8.4.2

Cancel the common factor.

$\frac{8 (6417 - 315 \sqrt{201})}{8 \cdot 512} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.8.4.3

Rewrite the expression.

$\frac{6417 - 315 \sqrt{201}}{512} - {(\frac{3 - \sqrt{201}}{8})}^{3} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.9

Apply the product rule to $\frac{3 - \sqrt{201}}{8}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{{(3 - \sqrt{201})}^{3}}{8^{3}} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.10

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{{(3 - \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.11

Use the Binomial Theorem.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{3^{3} + 3 \cdot 3^{2} (- \sqrt{201}) + 3 \cdot 3 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12

Simplify each term.

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Step 9.1.2.2.12.1

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 3 \cdot 3^{2} (- \sqrt{201}) + 3 \cdot 3 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.2

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

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Step 9.1.2.2.12.2.1

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

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Step 9.1.2.2.12.2.1.1

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 3^{1} \cdot 3^{2} (- \sqrt{201}) + 3 \cdot 3 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.2.1.2

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 3^{1 + 2} (- \sqrt{201}) + 3 \cdot 3 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.2.2

Add $1$ and $2$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 3^{3} (- \sqrt{201}) + 3 \cdot 3 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.3

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 + 27 (- \sqrt{201}) + 3 \cdot 3 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.4

Multiply $- 1$ by $27$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 3 \cdot 3 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.5

Multiply $3$ by $3$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 {(- \sqrt{201})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.6

Apply the product rule to $- \sqrt{201}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 ({(- 1)}^{2} {\sqrt{201}}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.7

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 (1 {\sqrt{201}}^{2}) + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.8

Multiply ${\sqrt{201}}^{2}$ by $1$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 {\sqrt{201}}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.9

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Step 9.1.2.2.12.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 {(201^{\frac{1}{2}})}^{2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.9.2

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201^{\frac{1}{2} \cdot 2} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.9.3

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201^{\frac{2}{2}} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.9.4

Cancel the common factor of $2$ .

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Step 9.1.2.2.12.9.4.1

Cancel the common factor.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201^{\frac{2}{2}} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.9.4.2

Rewrite the expression.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201^{1} + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.9.5

Evaluate the exponent.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 9 \cdot 201 + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.10

Multiply $9$ by $201$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 + {(- \sqrt{201})}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.11

Apply the product rule to $- \sqrt{201}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 + {(- 1)}^{3} {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.12

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.13

Rewrite ${\sqrt{201}}^{3}$ as $\sqrt{201^{3}}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - \sqrt{201^{3}}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.14

Raise $201$ to the power of $3$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - \sqrt{8120601}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.15

Rewrite $8120601$ as $201^{2} \cdot 201$ .

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Step 9.1.2.2.12.15.1

Factor $40401$ out of $8120601$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - \sqrt{40401 (201)}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.15.2

Rewrite $40401$ as $201^{2}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - \sqrt{201^{2} \cdot 201}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.16

Pull terms out from under the radical.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - (201 \sqrt{201})}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.12.17

Multiply $201$ by $- 1$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{27 - 27 \sqrt{201} + 1809 - 201 \sqrt{201}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.13

Add $27$ and $1809$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{1836 - 27 \sqrt{201} - 201 \sqrt{201}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.14

Subtract $201 \sqrt{201}$ from $- 27 \sqrt{201}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{1836 - 228 \sqrt{201}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.15

Cancel the common factor of $1836 - 228 \sqrt{201}$ and $512$ .

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Step 9.1.2.2.15.1

Factor $4$ out of $1836$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459) - 228 \sqrt{201}}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.15.2

Factor $4$ out of $- 228 \sqrt{201}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459) + 4 (- 57 \sqrt{201})}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.15.3

Factor $4$ out of $4 (459) + 4 (- 57 \sqrt{201})$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459 - 57 \sqrt{201})}{512} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.15.4

Cancel the common factors.

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Step 9.1.2.2.15.4.1

Factor $4$ out of $512$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459 - 57 \sqrt{201})}{4 \cdot 128} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.15.4.2

Cancel the common factor.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{4 (459 - 57 \sqrt{201})}{4 \cdot 128} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.15.4.3

Rewrite the expression.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 {(\frac{3 - \sqrt{201}}{8})}^{2}$

Step 9.1.2.2.16

Apply the product rule to $\frac{3 - \sqrt{201}}{8}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 \frac{{(3 - \sqrt{201})}^{2}}{8^{2}}$

Step 9.1.2.2.17

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 6 \frac{{(3 - \sqrt{201})}^{2}}{64}$

Step 9.1.2.2.18

Cancel the common factor of $2$ .

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Step 9.1.2.2.18.1

Factor $2$ out of $- 6$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + 2 (- 3) \frac{{(3 - \sqrt{201})}^{2}}{64}$

Step 9.1.2.2.18.2

Factor $2$ out of $64$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + 2 \cdot - 3 \frac{{(3 - \sqrt{201})}^{2}}{2 \cdot 32}$

Step 9.1.2.2.18.3

Cancel the common factor.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + 2 \cdot - 3 \frac{{(3 - \sqrt{201})}^{2}}{2 \cdot 32}$

Step 9.1.2.2.18.4

Rewrite the expression.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - 3 \frac{{(3 - \sqrt{201})}^{2}}{32}$

Step 9.1.2.2.19

Combine $- 3$ and $\frac{{(3 - \sqrt{201})}^{2}}{32}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 {(3 - \sqrt{201})}^{2}}{32}$

Step 9.1.2.2.20

Rewrite ${(3 - \sqrt{201})}^{2}$ as $(3 - \sqrt{201}) (3 - \sqrt{201})$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 ((3 - \sqrt{201}) (3 - \sqrt{201}))}{32}$

Step 9.1.2.2.21

Expand $(3 - \sqrt{201}) (3 - \sqrt{201})$ using the FOIL Method.

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Step 9.1.2.2.21.1

Apply the distributive property.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (3 (3 - \sqrt{201}) - \sqrt{201} (3 - \sqrt{201}))}{32}$

Step 9.1.2.2.21.2

Apply the distributive property.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (3 \cdot 3 + 3 (- \sqrt{201}) - \sqrt{201} (3 - \sqrt{201}))}{32}$

Step 9.1.2.2.21.3

Apply the distributive property.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (3 \cdot 3 + 3 (- \sqrt{201}) - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{32}$

Step 9.1.2.2.22

Simplify and combine like terms.

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Step 9.1.2.2.22.1

Simplify each term.

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Step 9.1.2.2.22.1.1

Multiply $3$ by $3$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 (- \sqrt{201}) - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{32}$

Step 9.1.2.2.22.1.2

Multiply $- 1$ by $3$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - \sqrt{201} \cdot 3 - \sqrt{201} (- \sqrt{201}))}{32}$

Step 9.1.2.2.22.1.3

Multiply $3$ by $- 1$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} - \sqrt{201} (- \sqrt{201}))}{32}$

Step 9.1.2.2.22.1.4

Multiply $- \sqrt{201} (- \sqrt{201})$ .

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Step 9.1.2.2.22.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 1 \sqrt{201} \sqrt{201})}{32}$

Step 9.1.2.2.22.1.4.2

Multiply $\sqrt{201}$ by $1$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + \sqrt{201} \sqrt{201})}{32}$

Step 9.1.2.2.22.1.4.3

Raise $\sqrt{201}$ to the power of $1$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{1} \sqrt{201})}{32}$

Step 9.1.2.2.22.1.4.4

Raise $\sqrt{201}$ to the power of $1$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{1} {\sqrt{201}}^{1})}{32}$

Step 9.1.2.2.22.1.4.5

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{1 + 1})}{32}$

Step 9.1.2.2.22.1.4.6

Add $1$ and $1$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {\sqrt{201}}^{2})}{32}$

Step 9.1.2.2.22.1.5

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Step 9.1.2.2.22.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + {(201^{\frac{1}{2}})}^{2})}{32}$

Step 9.1.2.2.22.1.5.2

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{1}{2} \cdot 2})}{32}$

Step 9.1.2.2.22.1.5.3

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

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{2}{2}})}{32}$

Step 9.1.2.2.22.1.5.4

Cancel the common factor of $2$ .

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Step 9.1.2.2.22.1.5.4.1

Cancel the common factor.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{\frac{2}{2}})}{32}$

Step 9.1.2.2.22.1.5.4.2

Rewrite the expression.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201^{1})}{32}$

Step 9.1.2.2.22.1.5.5

Evaluate the exponent.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (9 - 3 \sqrt{201} - 3 \sqrt{201} + 201)}{32}$

Step 9.1.2.2.22.2

Add $9$ and $201$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (210 - 3 \sqrt{201} - 3 \sqrt{201})}{32}$

Step 9.1.2.2.22.3

Subtract $3 \sqrt{201}$ from $- 3 \sqrt{201}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (210 - 6 \sqrt{201})}{32}$

Step 9.1.2.2.23

Cancel the common factor of $210 - 6 \sqrt{201}$ and $32$ .

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Step 9.1.2.2.23.1

Factor $2$ out of $- 3 (210 - 6 \sqrt{201})$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 - 3 \sqrt{201}))}{32}$

Step 9.1.2.2.23.2

Cancel the common factors.

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Step 9.1.2.2.23.2.1

Factor $2$ out of $32$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 - 3 \sqrt{201}))}{2 (16)}$

Step 9.1.2.2.23.2.2

Cancel the common factor.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 - 3 \sqrt{201}))}{2 \cdot 16}$

Step 9.1.2.2.23.2.3

Rewrite the expression.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} + \frac{- 3 (105 - 3 \sqrt{201})}{16}$

Step 9.1.2.2.24

Move the negative in front of the fraction.

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{459 - 57 \sqrt{201}}{128} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Step 9.1.2.3

Find the common denominator.

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Step 9.1.2.3.1

Multiply $\frac{459 - 57 \sqrt{201}}{128}$ by $\frac{4}{4}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - (\frac{459 - 57 \sqrt{201}}{128} \cdot \frac{4}{4}) - \frac{3 (105 - 3 \sqrt{201})}{16}$

Step 9.1.2.3.2

Multiply $\frac{459 - 57 \sqrt{201}}{128}$ by $\frac{4}{4}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{(459 - 57 \sqrt{201}) \cdot 4}{128 \cdot 4} - \frac{3 (105 - 3 \sqrt{201})}{16}$

Step 9.1.2.3.3

Multiply $\frac{3 (105 - 3 \sqrt{201})}{16}$ by $\frac{32}{32}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{(459 - 57 \sqrt{201}) \cdot 4}{128 \cdot 4} - (\frac{3 (105 - 3 \sqrt{201})}{16} \cdot \frac{32}{32})$

Step 9.1.2.3.4

Multiply $\frac{3 (105 - 3 \sqrt{201})}{16}$ by $\frac{32}{32}$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{(459 - 57 \sqrt{201}) \cdot 4}{128 \cdot 4} - \frac{3 (105 - 3 \sqrt{201}) \cdot 32}{16 \cdot 32}$

Step 9.1.2.3.5

Reorder the factors of $128 \cdot 4$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{(459 - 57 \sqrt{201}) \cdot 4}{4 \cdot 128} - \frac{3 (105 - 3 \sqrt{201}) \cdot 32}{16 \cdot 32}$

Step 9.1.2.3.6

Multiply $4$ by $128$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{(459 - 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201}) \cdot 32}{16 \cdot 32}$

Step 9.1.2.3.7

Multiply $16$ by $32$ .

$\frac{6417 - 315 \sqrt{201}}{512} - \frac{(459 - 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.4

Combine the numerators over the common denominator.

$\frac{6417 - 315 \sqrt{201} - (459 - 57 \sqrt{201}) \cdot 4 - 3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.5

Simplify each term.

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Step 9.1.2.5.1

Apply the distributive property.

$\frac{6417 - 315 \sqrt{201} + (- 1 \cdot 459 - (- 57 \sqrt{201})) \cdot 4 - 3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.5.2

Multiply $- 1$ by $459$ .

$\frac{6417 - 315 \sqrt{201} + (- 459 - (- 57 \sqrt{201})) \cdot 4 - 3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.5.3

Multiply $- 57$ by $- 1$ .

$\frac{6417 - 315 \sqrt{201} + (- 459 + 57 \sqrt{201}) \cdot 4 - 3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.5.4

Apply the distributive property.

$\frac{6417 - 315 \sqrt{201} - 459 \cdot 4 + 57 \sqrt{201} \cdot 4 - 3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.5.5

Multiply $- 459$ by $4$ .

$\frac{6417 - 315 \sqrt{201} - 1836 + 57 \sqrt{201} \cdot 4 - 3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.5.6

Multiply $4$ by $57$ .

$\frac{6417 - 315 \sqrt{201} - 1836 + 228 \sqrt{201} - 3 (105 - 3 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.5.7

Apply the distributive property.

$\frac{6417 - 315 \sqrt{201} - 1836 + 228 \sqrt{201} + (- 3 \cdot 105 - 3 (- 3 \sqrt{201})) \cdot 32}{512}$

Step 9.1.2.5.8

Multiply $- 3$ by $105$ .

$\frac{6417 - 315 \sqrt{201} - 1836 + 228 \sqrt{201} + (- 315 - 3 (- 3 \sqrt{201})) \cdot 32}{512}$

Step 9.1.2.5.9

Multiply $- 3$ by $- 3$ .

$\frac{6417 - 315 \sqrt{201} - 1836 + 228 \sqrt{201} + (- 315 + 9 \sqrt{201}) \cdot 32}{512}$

Step 9.1.2.5.10

Apply the distributive property.

$\frac{6417 - 315 \sqrt{201} - 1836 + 228 \sqrt{201} - 315 \cdot 32 + 9 \sqrt{201} \cdot 32}{512}$

Step 9.1.2.5.11

Multiply $- 315$ by $32$ .

$\frac{6417 - 315 \sqrt{201} - 1836 + 228 \sqrt{201} - 10080 + 9 \sqrt{201} \cdot 32}{512}$

Step 9.1.2.5.12

Multiply $32$ by $9$ .

$\frac{6417 - 315 \sqrt{201} - 1836 + 228 \sqrt{201} - 10080 + 288 \sqrt{201}}{512}$

Step 9.1.2.6

Simplify terms.

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Step 9.1.2.6.1

Subtract $1836$ from $6417$ .

$\frac{4581 - 315 \sqrt{201} + 228 \sqrt{201} - 10080 + 288 \sqrt{201}}{512}$

Step 9.1.2.6.2

Subtract $10080$ from $4581$ .

$\frac{- 5499 - 315 \sqrt{201} + 228 \sqrt{201} + 288 \sqrt{201}}{512}$

Step 9.1.2.6.3

Add $- 315 \sqrt{201}$ and $228 \sqrt{201}$ .

$\frac{- 5499 - 87 \sqrt{201} + 288 \sqrt{201}}{512}$

Step 9.1.2.6.4

Add $- 87 \sqrt{201}$ and $288 \sqrt{201}$ .

$\frac{- 5499 + 201 \sqrt{201}}{512}$

Step 9.1.2.6.5

Rewrite $- 5499$ as $- 1 (5499)$ .

$\frac{- 1 (5499) + 201 \sqrt{201}}{512}$

Step 9.1.2.6.6

Factor $- 1$ out of $201 \sqrt{201}$ .

$\frac{- 1 (5499) - (- 201 \sqrt{201})}{512}$

Step 9.1.2.6.7

Factor $- 1$ out of $- 1 (5499) - (- 201 \sqrt{201})$ .

$\frac{- 1 (5499 - 201 \sqrt{201})}{512}$

Step 9.1.2.6.8

Move the negative in front of the fraction.

$- \frac{5499 - 201 \sqrt{201}}{512}$

Step 9.2

Write the $x$ and $y$ coordinates in point form.

$(\frac{3 - \sqrt{201}}{8}, - \frac{5499 - 201 \sqrt{201}}{512})$

Step 10

Since the first derivative changed signs from positive to negative around $x = 0$ , then there is a turning point at $x = 0$ .

Step 11

Find the y-coordinate of $0$ to find the turning point.

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Step 11.1

Find $h (0)$ to find the y-coordinate of $0$ .

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Step 11.1.1

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

$h (0) = {(0)}^{4} - {(0)}^{3} - 6 {(0)}^{2}$

Step 11.1.2

Simplify ${(0)}^{4} - {(0)}^{3} - 6 {(0)}^{2}$ .

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Step 11.1.2.1

Remove parentheses.

${(0)}^{4} - {(0)}^{3} - 6 {(0)}^{2}$

Step 11.1.2.2

Simplify each term.

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Step 11.1.2.2.1

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

$0 - {(0)}^{3} - 6 {(0)}^{2}$

Step 11.1.2.2.2

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

$0 - 0 - 6 {(0)}^{2}$

Step 11.1.2.2.3

Multiply $- 1$ by $0$ .

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

Step 11.1.2.2.4

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

$0 + 0 - 6 \cdot 0$

Step 11.1.2.2.5

Multiply $- 6$ by $0$ .

$0 + 0 + 0$

Step 11.1.2.3

Simplify by adding numbers.

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Step 11.1.2.3.1

Add $0$ and $0$ .

$0 + 0$

Step 11.1.2.3.2

Add $0$ and $0$ .

$0$

Step 11.2

Write the $x$ and $y$ coordinates in point form.

$(0, 0)$

Step 12

Since the first derivative changed signs from negative to positive around $x = \frac{3 + \sqrt{201}}{8}$ , then there is a turning point at $x = \frac{3 + \sqrt{201}}{8}$ .

Step 13

Find the y-coordinate of $\frac{3 + \sqrt{201}}{8}$ to find the turning point.

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Step 13.1

Find $h (\frac{3 + \sqrt{201}}{8})$ to find the y-coordinate of $\frac{3 + \sqrt{201}}{8}$ .

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Step 13.1.1

Replace the variable $x$ with $\frac{3 + \sqrt{201}}{8}$ in the expression.

$h (\frac{3 + \sqrt{201}}{8}) = {(\frac{3 + \sqrt{201}}{8})}^{4} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2

Simplify ${(\frac{3 + \sqrt{201}}{8})}^{4} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$ .

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Step 13.1.2.1

Remove parentheses.

${(\frac{3 + \sqrt{201}}{8})}^{4} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2

Simplify each term.

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Step 13.1.2.2.1

Apply the product rule to $\frac{3 + \sqrt{201}}{8}$ .

$\frac{{(3 + \sqrt{201})}^{4}}{8^{4}} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.2

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

$\frac{{(3 + \sqrt{201})}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.3

Use the Binomial Theorem.

$\frac{3^{4} + 4 \cdot 3^{3} \sqrt{201} + 6 \cdot 3^{2} {\sqrt{201}}^{2} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4

Simplify each term.

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Step 13.1.2.2.4.1

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

$\frac{81 + 4 \cdot 3^{3} \sqrt{201} + 6 \cdot 3^{2} {\sqrt{201}}^{2} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.2

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

$\frac{81 + 4 \cdot 27 \sqrt{201} + 6 \cdot 3^{2} {\sqrt{201}}^{2} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.3

Multiply $4$ by $27$ .

$\frac{81 + 108 \sqrt{201} + 6 \cdot 3^{2} {\sqrt{201}}^{2} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.4

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

$\frac{81 + 108 \sqrt{201} + 6 \cdot 9 {\sqrt{201}}^{2} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.5

Multiply $6$ by $9$ .

$\frac{81 + 108 \sqrt{201} + 54 {\sqrt{201}}^{2} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.6

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Step 13.1.2.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$\frac{81 + 108 \sqrt{201} + 54 {(201^{\frac{1}{2}})}^{2} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.6.2

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

$\frac{81 + 108 \sqrt{201} + 54 \cdot 201^{\frac{1}{2} \cdot 2} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.6.3

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

$\frac{81 + 108 \sqrt{201} + 54 \cdot 201^{\frac{2}{2}} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.6.4

Cancel the common factor of $2$ .

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Step 13.1.2.2.4.6.4.1

Cancel the common factor.

$\frac{81 + 108 \sqrt{201} + 54 \cdot 201^{\frac{2}{2}} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.6.4.2

Rewrite the expression.

$\frac{81 + 108 \sqrt{201} + 54 \cdot 201^{1} + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.6.5

Evaluate the exponent.

$\frac{81 + 108 \sqrt{201} + 54 \cdot 201 + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.7

Multiply $54$ by $201$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 4 \cdot 3 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.8

Multiply $4$ by $3$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 12 {\sqrt{201}}^{3} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.9

Rewrite ${\sqrt{201}}^{3}$ as $\sqrt{201^{3}}$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 12 \sqrt{201^{3}} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.10

Raise $201$ to the power of $3$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 12 \sqrt{8120601} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.11

Rewrite $8120601$ as $201^{2} \cdot 201$ .

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Step 13.1.2.2.4.11.1

Factor $40401$ out of $8120601$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 12 \sqrt{40401 (201)} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.11.2

Rewrite $40401$ as $201^{2}$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 12 \sqrt{201^{2} \cdot 201} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.12

Pull terms out from under the radical.

$\frac{81 + 108 \sqrt{201} + 10854 + 12 (201 \sqrt{201}) + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.13

Multiply $201$ by $12$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + {\sqrt{201}}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.14

Rewrite ${\sqrt{201}}^{4}$ as $201^{2}$ .

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Step 13.1.2.2.4.14.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + {(201^{\frac{1}{2}})}^{4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.14.2

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

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{1}{2} \cdot 4}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.14.3

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

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{4}{2}}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.14.4

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

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Step 13.1.2.2.4.14.4.1

Factor $2$ out of $4$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2}}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.14.4.2

Cancel the common factors.

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Step 13.1.2.2.4.14.4.2.1

Factor $2$ out of $2$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2 (1)}}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.14.4.2.2

Cancel the common factor.

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{2 \cdot 2}{2 \cdot 1}}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.14.4.2.3

Rewrite the expression.

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{\frac{2}{1}}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.14.4.2.4

Divide $2$ by $1$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 201^{2}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.4.15

Raise $201$ to the power of $2$ .

$\frac{81 + 108 \sqrt{201} + 10854 + 2412 \sqrt{201} + 40401}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.5

Add $81$ and $10854$ .

$\frac{10935 + 108 \sqrt{201} + 2412 \sqrt{201} + 40401}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.6

Add $10935$ and $40401$ .

$\frac{51336 + 108 \sqrt{201} + 2412 \sqrt{201}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.7

Add $108 \sqrt{201}$ and $2412 \sqrt{201}$ .

$\frac{51336 + 2520 \sqrt{201}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.8

Cancel the common factor of $51336 + 2520 \sqrt{201}$ and $4096$ .

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Step 13.1.2.2.8.1

Factor $8$ out of $51336$ .

$\frac{8 (6417) + 2520 \sqrt{201}}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.8.2

Factor $8$ out of $2520 \sqrt{201}$ .

$\frac{8 (6417) + 8 (315 \sqrt{201})}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.8.3

Factor $8$ out of $8 (6417) + 8 (315 \sqrt{201})$ .

$\frac{8 (6417 + 315 \sqrt{201})}{4096} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.8.4

Cancel the common factors.

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Step 13.1.2.2.8.4.1

Factor $8$ out of $4096$ .

$\frac{8 (6417 + 315 \sqrt{201})}{8 \cdot 512} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.8.4.2

Cancel the common factor.

$\frac{8 (6417 + 315 \sqrt{201})}{8 \cdot 512} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.8.4.3

Rewrite the expression.

$\frac{6417 + 315 \sqrt{201}}{512} - {(\frac{3 + \sqrt{201}}{8})}^{3} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.9

Apply the product rule to $\frac{3 + \sqrt{201}}{8}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{{(3 + \sqrt{201})}^{3}}{8^{3}} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.10

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{{(3 + \sqrt{201})}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.11

Use the Binomial Theorem.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{3^{3} + 3 \cdot 3^{2} \sqrt{201} + 3 \cdot 3 {\sqrt{201}}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12

Simplify each term.

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Step 13.1.2.2.12.1

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 3 \cdot 3^{2} \sqrt{201} + 3 \cdot 3 {\sqrt{201}}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.2

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

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Step 13.1.2.2.12.2.1

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

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Step 13.1.2.2.12.2.1.1

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 3^{1} \cdot 3^{2} \sqrt{201} + 3 \cdot 3 {\sqrt{201}}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.2.1.2

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 3^{1 + 2} \sqrt{201} + 3 \cdot 3 {\sqrt{201}}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.2.2

Add $1$ and $2$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 3^{3} \sqrt{201} + 3 \cdot 3 {\sqrt{201}}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.3

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 3 \cdot 3 {\sqrt{201}}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.4

Multiply $3$ by $3$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 {\sqrt{201}}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.5

Rewrite ${\sqrt{201}}^{2}$ as $201$ .

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Step 13.1.2.2.12.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{201}$ as $201^{\frac{1}{2}}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 {(201^{\frac{1}{2}})}^{2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.5.2

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201^{\frac{1}{2} \cdot 2} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.5.3

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201^{\frac{2}{2}} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.5.4

Cancel the common factor of $2$ .

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Step 13.1.2.2.12.5.4.1

Cancel the common factor.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201^{\frac{2}{2}} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.5.4.2

Rewrite the expression.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201^{1} + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.5.5

Evaluate the exponent.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 9 \cdot 201 + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.6

Multiply $9$ by $201$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + {\sqrt{201}}^{3}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.7

Rewrite ${\sqrt{201}}^{3}$ as $\sqrt{201^{3}}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + \sqrt{201^{3}}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.8

Raise $201$ to the power of $3$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + \sqrt{8120601}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.9

Rewrite $8120601$ as $201^{2} \cdot 201$ .

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Step 13.1.2.2.12.9.1

Factor $40401$ out of $8120601$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + \sqrt{40401 (201)}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.9.2

Rewrite $40401$ as $201^{2}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + \sqrt{201^{2} \cdot 201}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.12.10

Pull terms out from under the radical.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{27 + 27 \sqrt{201} + 1809 + 201 \sqrt{201}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.13

Add $27$ and $1809$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{1836 + 27 \sqrt{201} + 201 \sqrt{201}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.14

Add $27 \sqrt{201}$ and $201 \sqrt{201}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{1836 + 228 \sqrt{201}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.15

Cancel the common factor of $1836 + 228 \sqrt{201}$ and $512$ .

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Step 13.1.2.2.15.1

Factor $4$ out of $1836$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459) + 228 \sqrt{201}}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.15.2

Factor $4$ out of $228 \sqrt{201}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459) + 4 (57 \sqrt{201})}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.15.3

Factor $4$ out of $4 (459) + 4 (57 \sqrt{201})$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459 + 57 \sqrt{201})}{512} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.15.4

Cancel the common factors.

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Step 13.1.2.2.15.4.1

Factor $4$ out of $512$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459 + 57 \sqrt{201})}{4 \cdot 128} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.15.4.2

Cancel the common factor.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{4 (459 + 57 \sqrt{201})}{4 \cdot 128} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.15.4.3

Rewrite the expression.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 {(\frac{3 + \sqrt{201}}{8})}^{2}$

Step 13.1.2.2.16

Apply the product rule to $\frac{3 + \sqrt{201}}{8}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 \frac{{(3 + \sqrt{201})}^{2}}{8^{2}}$

Step 13.1.2.2.17

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 6 \frac{{(3 + \sqrt{201})}^{2}}{64}$

Step 13.1.2.2.18

Cancel the common factor of $2$ .

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Step 13.1.2.2.18.1

Factor $2$ out of $- 6$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + 2 (- 3) \frac{{(3 + \sqrt{201})}^{2}}{64}$

Step 13.1.2.2.18.2

Factor $2$ out of $64$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + 2 \cdot - 3 \frac{{(3 + \sqrt{201})}^{2}}{2 \cdot 32}$

Step 13.1.2.2.18.3

Cancel the common factor.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + 2 \cdot - 3 \frac{{(3 + \sqrt{201})}^{2}}{2 \cdot 32}$

Step 13.1.2.2.18.4

Rewrite the expression.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - 3 \frac{{(3 + \sqrt{201})}^{2}}{32}$

Step 13.1.2.2.19

Combine $- 3$ and $\frac{{(3 + \sqrt{201})}^{2}}{32}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 {(3 + \sqrt{201})}^{2}}{32}$

Step 13.1.2.2.20

Rewrite ${(3 + \sqrt{201})}^{2}$ as $(3 + \sqrt{201}) (3 + \sqrt{201})$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 ((3 + \sqrt{201}) (3 + \sqrt{201}))}{32}$

Step 13.1.2.2.21

Expand $(3 + \sqrt{201}) (3 + \sqrt{201})$ using the FOIL Method.

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Step 13.1.2.2.21.1

Apply the distributive property.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (3 (3 + \sqrt{201}) + \sqrt{201} (3 + \sqrt{201}))}{32}$

Step 13.1.2.2.21.2

Apply the distributive property.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (3 \cdot 3 + 3 \sqrt{201} + \sqrt{201} (3 + \sqrt{201}))}{32}$

Step 13.1.2.2.21.3

Apply the distributive property.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (3 \cdot 3 + 3 \sqrt{201} + \sqrt{201} \cdot 3 + \sqrt{201} \sqrt{201})}{32}$

Step 13.1.2.2.22

Simplify and combine like terms.

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Step 13.1.2.2.22.1

Simplify each term.

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Step 13.1.2.2.22.1.1

Multiply $3$ by $3$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + \sqrt{201} \cdot 3 + \sqrt{201} \sqrt{201})}{32}$

Step 13.1.2.2.22.1.2

Move $3$ to the left of $\sqrt{201}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \cdot \sqrt{201} + \sqrt{201} \sqrt{201})}{32}$

Step 13.1.2.2.22.1.3

Combine using the product rule for radicals.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{201 \cdot 201})}{32}$

Step 13.1.2.2.22.1.4

Multiply $201$ by $201$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{40401})}{32}$

Step 13.1.2.2.22.1.5

Rewrite $40401$ as $201^{2}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + \sqrt{201^{2}})}{32}$

Step 13.1.2.2.22.1.6

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

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (9 + 3 \sqrt{201} + 3 \sqrt{201} + 201)}{32}$

Step 13.1.2.2.22.2

Add $9$ and $201$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (210 + 3 \sqrt{201} + 3 \sqrt{201})}{32}$

Step 13.1.2.2.22.3

Add $3 \sqrt{201}$ and $3 \sqrt{201}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (210 + 6 \sqrt{201})}{32}$

Step 13.1.2.2.23

Cancel the common factor of $210 + 6 \sqrt{201}$ and $32$ .

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Step 13.1.2.2.23.1

Factor $2$ out of $- 3 (210 + 6 \sqrt{201})$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 + 3 \sqrt{201}))}{32}$

Step 13.1.2.2.23.2

Cancel the common factors.

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Step 13.1.2.2.23.2.1

Factor $2$ out of $32$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 + 3 \sqrt{201}))}{2 (16)}$

Step 13.1.2.2.23.2.2

Cancel the common factor.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{2 (- 3 (105 + 3 \sqrt{201}))}{2 \cdot 16}$

Step 13.1.2.2.23.2.3

Rewrite the expression.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} + \frac{- 3 (105 + 3 \sqrt{201})}{16}$

Step 13.1.2.2.24

Move the negative in front of the fraction.

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{459 + 57 \sqrt{201}}{128} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Step 13.1.2.3

Find the common denominator.

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Step 13.1.2.3.1

Multiply $\frac{459 + 57 \sqrt{201}}{128}$ by $\frac{4}{4}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - (\frac{459 + 57 \sqrt{201}}{128} \cdot \frac{4}{4}) - \frac{3 (105 + 3 \sqrt{201})}{16}$

Step 13.1.2.3.2

Multiply $\frac{459 + 57 \sqrt{201}}{128}$ by $\frac{4}{4}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{(459 + 57 \sqrt{201}) \cdot 4}{128 \cdot 4} - \frac{3 (105 + 3 \sqrt{201})}{16}$

Step 13.1.2.3.3

Multiply $\frac{3 (105 + 3 \sqrt{201})}{16}$ by $\frac{32}{32}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{(459 + 57 \sqrt{201}) \cdot 4}{128 \cdot 4} - (\frac{3 (105 + 3 \sqrt{201})}{16} \cdot \frac{32}{32})$

Step 13.1.2.3.4

Multiply $\frac{3 (105 + 3 \sqrt{201})}{16}$ by $\frac{32}{32}$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{(459 + 57 \sqrt{201}) \cdot 4}{128 \cdot 4} - \frac{3 (105 + 3 \sqrt{201}) \cdot 32}{16 \cdot 32}$

Step 13.1.2.3.5

Reorder the factors of $128 \cdot 4$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{(459 + 57 \sqrt{201}) \cdot 4}{4 \cdot 128} - \frac{3 (105 + 3 \sqrt{201}) \cdot 32}{16 \cdot 32}$

Step 13.1.2.3.6

Multiply $4$ by $128$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{(459 + 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201}) \cdot 32}{16 \cdot 32}$

Step 13.1.2.3.7

Multiply $16$ by $32$ .

$\frac{6417 + 315 \sqrt{201}}{512} - \frac{(459 + 57 \sqrt{201}) \cdot 4}{512} - \frac{3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.4

Combine the numerators over the common denominator.

$\frac{6417 + 315 \sqrt{201} - (459 + 57 \sqrt{201}) \cdot 4 - 3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.5

Simplify each term.

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Step 13.1.2.5.1

Apply the distributive property.

$\frac{6417 + 315 \sqrt{201} + (- 1 \cdot 459 - (57 \sqrt{201})) \cdot 4 - 3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.5.2

Multiply $- 1$ by $459$ .

$\frac{6417 + 315 \sqrt{201} + (- 459 - (57 \sqrt{201})) \cdot 4 - 3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.5.3

Multiply $57$ by $- 1$ .

$\frac{6417 + 315 \sqrt{201} + (- 459 - 57 \sqrt{201}) \cdot 4 - 3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.5.4

Apply the distributive property.

$\frac{6417 + 315 \sqrt{201} - 459 \cdot 4 - 57 \sqrt{201} \cdot 4 - 3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.5.5

Multiply $- 459$ by $4$ .

$\frac{6417 + 315 \sqrt{201} - 1836 - 57 \sqrt{201} \cdot 4 - 3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.5.6

Multiply $4$ by $- 57$ .

$\frac{6417 + 315 \sqrt{201} - 1836 - 228 \sqrt{201} - 3 (105 + 3 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.5.7

Apply the distributive property.

$\frac{6417 + 315 \sqrt{201} - 1836 - 228 \sqrt{201} + (- 3 \cdot 105 - 3 (3 \sqrt{201})) \cdot 32}{512}$

Step 13.1.2.5.8

Multiply $- 3$ by $105$ .

$\frac{6417 + 315 \sqrt{201} - 1836 - 228 \sqrt{201} + (- 315 - 3 (3 \sqrt{201})) \cdot 32}{512}$

Step 13.1.2.5.9

Multiply $3$ by $- 3$ .

$\frac{6417 + 315 \sqrt{201} - 1836 - 228 \sqrt{201} + (- 315 - 9 \sqrt{201}) \cdot 32}{512}$

Step 13.1.2.5.10

Apply the distributive property.

$\frac{6417 + 315 \sqrt{201} - 1836 - 228 \sqrt{201} - 315 \cdot 32 - 9 \sqrt{201} \cdot 32}{512}$

Step 13.1.2.5.11

Multiply $- 315$ by $32$ .

$\frac{6417 + 315 \sqrt{201} - 1836 - 228 \sqrt{201} - 10080 - 9 \sqrt{201} \cdot 32}{512}$

Step 13.1.2.5.12

Multiply $32$ by $- 9$ .

$\frac{6417 + 315 \sqrt{201} - 1836 - 228 \sqrt{201} - 10080 - 288 \sqrt{201}}{512}$

Step 13.1.2.6

Simplify terms.

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Step 13.1.2.6.1

Subtract $1836$ from $6417$ .

$\frac{4581 + 315 \sqrt{201} - 228 \sqrt{201} - 10080 - 288 \sqrt{201}}{512}$

Step 13.1.2.6.2

Subtract $10080$ from $4581$ .

$\frac{- 5499 + 315 \sqrt{201} - 228 \sqrt{201} - 288 \sqrt{201}}{512}$

Step 13.1.2.6.3

Subtract $228 \sqrt{201}$ from $315 \sqrt{201}$ .

$\frac{- 5499 + 87 \sqrt{201} - 288 \sqrt{201}}{512}$

Step 13.1.2.6.4

Subtract $288 \sqrt{201}$ from $87 \sqrt{201}$ .

$\frac{- 5499 - 201 \sqrt{201}}{512}$

Step 13.1.2.6.5

Rewrite $- 5499$ as $- 1 (5499)$ .

$\frac{- 1 (5499) - 201 \sqrt{201}}{512}$

Step 13.1.2.6.6

Factor $- 1$ out of $- 201 \sqrt{201}$ .

$\frac{- 1 (5499) - (201 \sqrt{201})}{512}$

Step 13.1.2.6.7

Factor $- 1$ out of $- 1 (5499) - (201 \sqrt{201})$ .

$\frac{- 1 (5499 + 201 \sqrt{201})}{512}$

Step 13.1.2.6.8

Move the negative in front of the fraction.

$- \frac{5499 + 201 \sqrt{201}}{512}$

Step 13.2

Write the $x$ and $y$ coordinates in point form.

$(\frac{3 + \sqrt{201}}{8}, - \frac{5499 + 201 \sqrt{201}}{512})$

Step 14

These are the turning points.

$(\frac{3 - \sqrt{201}}{8}, - \frac{5499 - 201 \sqrt{201}}{512})$

$(0, 0)$

$(\frac{3 + \sqrt{201}}{8}, - \frac{5499 + 201 \sqrt{201}}{512})$

Step 15

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