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

$201^{2} \cdot 201$ .

Tap for more steps...

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

$201^{2}$ .

Tap for more steps...

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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