Calculus Examples

f (x) = x^{3} - 8 x^{2} - 20 x

$f (x) = x^{3} - 8 x^{2} - 20 x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of

$x^{3} - 8 x^{2} - 20 x$ with respect to

$x$ is

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 8 x^{2}] + \frac{d}{d x} [- 20 x]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 8 x^{2}) + \frac{d}{d x} (- 20 x)$

Step 1.1.1.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 3$ .

$f' (x) = 3 x^{2} + \frac{d}{d x} (- 8 x^{2}) + \frac{d}{d x} (- 20 x)$

Step 1.1.2

Evaluate

$\frac{d}{d x} [- 8 x^{2}]$ .

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Step 1.1.2.1

Since

$- 8$ is constant with respect to

$x$ , the derivative of

$- 8 x^{2}$ with respect to

$x$ is

$- 8 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 3 x^{2} - 8 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 20 x)$

Step 1.1.2.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$f' (x) = 3 x^{2} - 8 (2 x) + \frac{d}{d x} (- 20 x)$

Step 1.1.2.3

Multiply

$2$ by

$- 8$ .

$f' (x) = 3 x^{2} - 16 x + \frac{d}{d x} (- 20 x)$

Step 1.1.3

Evaluate

$\frac{d}{d x} [- 20 x]$ .

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Step 1.1.3.1

Since

$- 20$ is constant with respect to

$x$ , the derivative of

$- 20 x$ with respect to

$x$ is

$- 20 \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} - 16 x - 20 \frac{d}{d x} x$

Step 1.1.3.2

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

$f' (x) = 3 x^{2} - 16 x - 20 \cdot 1$

Step 1.1.3.3

Multiply

$- 20$ by

$1$ .

$f' (x) = 3 x^{2} - 16 x - 20$

Step 1.2

The first derivative of

$f (x)$ with respect to

$x$ is

$3 x^{2} - 16 x - 20$ .

$3 x^{2} - 16 x - 20$

Step 2

Set the first derivative equal to

$0$ then solve the equation

$3 x^{2} - 16 x - 20 = 0$ .

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Step 2.1

Set the first derivative equal to

$0$ .

$3 x^{2} - 16 x - 20 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values

$a = 3$ ,

$b = - 16$ , and

$c = - 20$ into the quadratic formula and solve for

$x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (3 \cdot - 20)}}{2 \cdot 3}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise

$- 16$ to the power of

$2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 3 \cdot - 20}}{2 \cdot 3}$

Step 2.4.1.2

Multiply

$- 4 \cdot 3 \cdot - 20$ .

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Step 2.4.1.2.1

Multiply

$- 4$ by

$3$ .

$x = \frac{16 \pm \sqrt{256 - 12 \cdot - 20}}{2 \cdot 3}$

Step 2.4.1.2.2

Multiply

$- 12$ by

$- 20$ .

$x = \frac{16 \pm \sqrt{256 + 240}}{2 \cdot 3}$

Step 2.4.1.3

Add

$256$ and

$240$ .

$x = \frac{16 \pm \sqrt{496}}{2 \cdot 3}$

Step 2.4.1.4

Rewrite

$496$ as

$4^{2} \cdot 31$ .

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Step 2.4.1.4.1

Factor

$16$ out of

$496$ .

$x = \frac{16 \pm \sqrt{16 (31)}}{2 \cdot 3}$

Step 2.4.1.4.2

Rewrite

$16$ as

$4^{2}$ .

$x = \frac{16 \pm \sqrt{4^{2} \cdot 31}}{2 \cdot 3}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{31}}{2 \cdot 3}$

Step 2.4.2

Multiply

$2$ by

$3$ .

$x = \frac{16 \pm 4 \sqrt{31}}{6}$

Step 2.4.3

Simplify

$\frac{16 \pm 4 \sqrt{31}}{6}$ .

$x = \frac{8 \pm 2 \sqrt{31}}{3}$

Step 2.5

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise

$- 16$ to the power of

$2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 3 \cdot - 20}}{2 \cdot 3}$

Step 2.5.1.2

Multiply

$- 4 \cdot 3 \cdot - 20$ .

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Step 2.5.1.2.1

Multiply

$- 4$ by

$3$ .

$x = \frac{16 \pm \sqrt{256 - 12 \cdot - 20}}{2 \cdot 3}$

Step 2.5.1.2.2

Multiply

$- 12$ by

$- 20$ .

$x = \frac{16 \pm \sqrt{256 + 240}}{2 \cdot 3}$

Step 2.5.1.3

Add

$256$ and

$240$ .

$x = \frac{16 \pm \sqrt{496}}{2 \cdot 3}$

Step 2.5.1.4

Rewrite

$496$ as

$4^{2} \cdot 31$ .

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Step 2.5.1.4.1

Factor

$16$ out of

$496$ .

$x = \frac{16 \pm \sqrt{16 (31)}}{2 \cdot 3}$

Step 2.5.1.4.2

Rewrite

$16$ as

$4^{2}$ .

$x = \frac{16 \pm \sqrt{4^{2} \cdot 31}}{2 \cdot 3}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{31}}{2 \cdot 3}$

Step 2.5.2

Multiply

$2$ by

$3$ .

$x = \frac{16 \pm 4 \sqrt{31}}{6}$

Step 2.5.3

Simplify

$\frac{16 \pm 4 \sqrt{31}}{6}$ .

$x = \frac{8 \pm 2 \sqrt{31}}{3}$

Step 2.5.4

Change the

$\pm$ to

$+$ .

$x = \frac{8 + 2 \sqrt{31}}{3}$

Step 2.6

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise

$- 16$ to the power of

$2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 3 \cdot - 20}}{2 \cdot 3}$

Step 2.6.1.2

Multiply

$- 4 \cdot 3 \cdot - 20$ .

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Step 2.6.1.2.1

Multiply

$- 4$ by

$3$ .

$x = \frac{16 \pm \sqrt{256 - 12 \cdot - 20}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply

$- 12$ by

$- 20$ .

$x = \frac{16 \pm \sqrt{256 + 240}}{2 \cdot 3}$

Step 2.6.1.3

Add

$256$ and

$240$ .

$x = \frac{16 \pm \sqrt{496}}{2 \cdot 3}$

Step 2.6.1.4

Rewrite

$496$ as

$4^{2} \cdot 31$ .

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Step 2.6.1.4.1

Factor

$16$ out of

$496$ .

$x = \frac{16 \pm \sqrt{16 (31)}}{2 \cdot 3}$

Step 2.6.1.4.2

Rewrite

$16$ as

$4^{2}$ .

$x = \frac{16 \pm \sqrt{4^{2} \cdot 31}}{2 \cdot 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{31}}{2 \cdot 3}$

Step 2.6.2

Multiply

$2$ by

$3$ .

$x = \frac{16 \pm 4 \sqrt{31}}{6}$

Step 2.6.3

Simplify

$\frac{16 \pm 4 \sqrt{31}}{6}$ .

$x = \frac{8 \pm 2 \sqrt{31}}{3}$

Step 2.6.4

Change the

$\pm$ to

$-$ .

$x = \frac{8 - 2 \sqrt{31}}{3}$

Step 2.7

The final answer is the combination of both solutions.

$x = \frac{8 + 2 \sqrt{31}}{3}, \frac{8 - 2 \sqrt{31}}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

Step 4

Evaluate

$x^{3} - 8 x^{2} - 20 x$ at each

$x$ value where the derivative is

$0$ or undefined.

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Step 4.1

Evaluate at

$x = \frac{8 + 2 \sqrt{31}}{3}$ .

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Step 4.1.1

Substitute

$\frac{8 + 2 \sqrt{31}}{3}$ for

$x$ .

${(\frac{8 + 2 \sqrt{31}}{3})}^{3} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Apply the product rule to

$\frac{8 + 2 \sqrt{31}}{3}$ .

$\frac{{(8 + 2 \sqrt{31})}^{3}}{3^{3}} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.2

Raise

$3$ to the power of

$3$ .

$\frac{{(8 + 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.3

Use the Binomial Theorem.

$\frac{8^{3} + 3 \cdot 8^{2} (2 \sqrt{31}) + 3 \cdot 8 {(2 \sqrt{31})}^{2} + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4

Simplify each term.

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Step 4.1.2.1.4.1

Raise

$8$ to the power of

$3$ .

$\frac{512 + 3 \cdot 8^{2} (2 \sqrt{31}) + 3 \cdot 8 {(2 \sqrt{31})}^{2} + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.2

Raise

$8$ to the power of

$2$ .

$\frac{512 + 3 \cdot 64 (2 \sqrt{31}) + 3 \cdot 8 {(2 \sqrt{31})}^{2} + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.3

Multiply

$3$ by

$64$ .

$\frac{512 + 192 (2 \sqrt{31}) + 3 \cdot 8 {(2 \sqrt{31})}^{2} + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.4

Multiply

$2$ by

$192$ .

$\frac{512 + 384 \sqrt{31} + 3 \cdot 8 {(2 \sqrt{31})}^{2} + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.5

Multiply

$3$ by

$8$ .

$\frac{512 + 384 \sqrt{31} + 24 {(2 \sqrt{31})}^{2} + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.6

Apply the product rule to

$2 \sqrt{31}$ .

$\frac{512 + 384 \sqrt{31} + 24 (2^{2} {\sqrt{31}}^{2}) + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.7

Raise

$2$ to the power of

$2$ .

$\frac{512 + 384 \sqrt{31} + 24 (4 {\sqrt{31}}^{2}) + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.8

Rewrite

${\sqrt{31}}^{2}$ as

$31$ .

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Step 4.1.2.1.4.8.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{31}$ as

$31^{\frac{1}{2}}$ .

$\frac{512 + 384 \sqrt{31} + 24 (4 {(31^{\frac{1}{2}})}^{2}) + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.8.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{512 + 384 \sqrt{31} + 24 (4 \cdot 31^{\frac{1}{2} \cdot 2}) + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.8.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{512 + 384 \sqrt{31} + 24 (4 \cdot 31^{\frac{2}{2}}) + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.8.4

Cancel the common factor of

$2$ .

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Step 4.1.2.1.4.8.4.1

Cancel the common factor.

$\frac{512 + 384 \sqrt{31} + 24 (4 \cdot 31^{\frac{2}{2}}) + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.8.4.2

Rewrite the expression.

$\frac{512 + 384 \sqrt{31} + 24 (4 \cdot 31^{1}) + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.8.5

Evaluate the exponent.

$\frac{512 + 384 \sqrt{31} + 24 (4 \cdot 31) + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.9

Multiply

$24 (4 \cdot 31)$ .

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Step 4.1.2.1.4.9.1

Multiply

$4$ by

$31$ .

$\frac{512 + 384 \sqrt{31} + 24 \cdot 124 + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.9.2

Multiply

$24$ by

$124$ .

$\frac{512 + 384 \sqrt{31} + 2976 + {(2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.10

Apply the product rule to

$2 \sqrt{31}$ .

$\frac{512 + 384 \sqrt{31} + 2976 + 2^{3} {\sqrt{31}}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.11

Raise

$2$ to the power of

$3$ .

$\frac{512 + 384 \sqrt{31} + 2976 + 8 {\sqrt{31}}^{3}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.12

Rewrite

${\sqrt{31}}^{3}$ as

$\sqrt{31^{3}}$ .

$\frac{512 + 384 \sqrt{31} + 2976 + 8 \sqrt{31^{3}}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.13

Raise

$31$ to the power of

$3$ .

$\frac{512 + 384 \sqrt{31} + 2976 + 8 \sqrt{29791}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.14

Rewrite

$29791$ as

$31^{2} \cdot 31$ .

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Step 4.1.2.1.4.14.1

Factor

$961$ out of

$29791$ .

$\frac{512 + 384 \sqrt{31} + 2976 + 8 \sqrt{961 (31)}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.14.2

Rewrite

$961$ as

$31^{2}$ .

$\frac{512 + 384 \sqrt{31} + 2976 + 8 \sqrt{31^{2} \cdot 31}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.15

Pull terms out from under the radical.

$\frac{512 + 384 \sqrt{31} + 2976 + 8 (31 \sqrt{31})}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.4.16

Multiply

$31$ by

$8$ .

$\frac{512 + 384 \sqrt{31} + 2976 + 248 \sqrt{31}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.5

Add

$512$ and

$2976$ .

$\frac{3488 + 384 \sqrt{31} + 248 \sqrt{31}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.6

Add

$384 \sqrt{31}$ and

$248 \sqrt{31}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 {(\frac{8 + 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.7

Apply the product rule to

$\frac{8 + 2 \sqrt{31}}{3}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{{(8 + 2 \sqrt{31})}^{2}}{3^{2}} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.8

Raise

$3$ to the power of

$2$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{{(8 + 2 \sqrt{31})}^{2}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.9

Rewrite

${(8 + 2 \sqrt{31})}^{2}$ as

$(8 + 2 \sqrt{31}) (8 + 2 \sqrt{31})$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{(8 + 2 \sqrt{31}) (8 + 2 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.10

Expand

$(8 + 2 \sqrt{31}) (8 + 2 \sqrt{31})$ using the FOIL Method.

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Step 4.1.2.1.10.1

Apply the distributive property.

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{8 (8 + 2 \sqrt{31}) + 2 \sqrt{31} (8 + 2 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.10.2

Apply the distributive property.

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{8 \cdot 8 + 8 (2 \sqrt{31}) + 2 \sqrt{31} (8 + 2 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.10.3

Apply the distributive property.

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{8 \cdot 8 + 8 (2 \sqrt{31}) + 2 \sqrt{31} \cdot 8 + 2 \sqrt{31} (2 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11

Simplify and combine like terms.

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Step 4.1.2.1.11.1

Simplify each term.

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Step 4.1.2.1.11.1.1

Multiply

$8$ by

$8$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 8 (2 \sqrt{31}) + 2 \sqrt{31} \cdot 8 + 2 \sqrt{31} (2 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.2

Multiply

$2$ by

$8$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 2 \sqrt{31} \cdot 8 + 2 \sqrt{31} (2 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.3

Multiply

$8$ by

$2$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 2 \sqrt{31} (2 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.4

Multiply

$2 \sqrt{31} (2 \sqrt{31})$ .

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Step 4.1.2.1.11.1.4.1

Multiply

$2$ by

$2$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 \sqrt{31} \sqrt{31}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.4.2

Raise

$\sqrt{31}$ to the power of

$1$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 ({\sqrt{31}}^{1} \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.4.3

Raise

$\sqrt{31}$ to the power of

$1$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 ({\sqrt{31}}^{1} {\sqrt{31}}^{1})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 {\sqrt{31}}^{1 + 1}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.4.5

Add

$1$ and

$1$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 {\sqrt{31}}^{2}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.5

Rewrite

${\sqrt{31}}^{2}$ as

$31$ .

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Step 4.1.2.1.11.1.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{31}$ as

$31^{\frac{1}{2}}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 {(31^{\frac{1}{2}})}^{2}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 \cdot 31^{\frac{1}{2} \cdot 2}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.5.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 \cdot 31^{\frac{2}{2}}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.5.4

Cancel the common factor of

$2$ .

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Step 4.1.2.1.11.1.5.4.1

Cancel the common factor.

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 \cdot 31^{\frac{2}{2}}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.5.4.2

Rewrite the expression.

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 \cdot 31^{1}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.5.5

Evaluate the exponent.

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 4 \cdot 31}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.1.6

Multiply

$4$ by

$31$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{64 + 16 \sqrt{31} + 16 \sqrt{31} + 124}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.2

Add

$64$ and

$124$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{188 + 16 \sqrt{31} + 16 \sqrt{31}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.11.3

Add

$16 \sqrt{31}$ and

$16 \sqrt{31}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - 8 \frac{188 + 32 \sqrt{31}}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.12

Combine

$- 8$ and

$\frac{188 + 32 \sqrt{31}}{9}$ .

$\frac{3488 + 632 \sqrt{31}}{27} + \frac{- 8 (188 + 32 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.13

Move the negative in front of the fraction.

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{(8) (188 + 32 \sqrt{31})}{9} - 20 \frac{8 + 2 \sqrt{31}}{3}$

Step 4.1.2.1.14

Combine

$- 20$ and

$\frac{8 + 2 \sqrt{31}}{3}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{8 (188 + 32 \sqrt{31})}{9} + \frac{- 20 (8 + 2 \sqrt{31})}{3}$

Step 4.1.2.1.15

Move the negative in front of the fraction.

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{8 (188 + 32 \sqrt{31})}{9} - \frac{20 (8 + 2 \sqrt{31})}{3}$

Step 4.1.2.2

Find the common denominator.

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Step 4.1.2.2.1

Multiply

$\frac{8 (188 + 32 \sqrt{31})}{9}$ by

$\frac{3}{3}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - (\frac{8 (188 + 32 \sqrt{31})}{9} \cdot \frac{3}{3}) - \frac{20 (8 + 2 \sqrt{31})}{3}$

Step 4.1.2.2.2

Multiply

$\frac{8 (188 + 32 \sqrt{31})}{9}$ by

$\frac{3}{3}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{8 (188 + 32 \sqrt{31}) \cdot 3}{9 \cdot 3} - \frac{20 (8 + 2 \sqrt{31})}{3}$

Step 4.1.2.2.3

Multiply

$\frac{20 (8 + 2 \sqrt{31})}{3}$ by

$\frac{9}{9}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{8 (188 + 32 \sqrt{31}) \cdot 3}{9 \cdot 3} - (\frac{20 (8 + 2 \sqrt{31})}{3} \cdot \frac{9}{9})$

Step 4.1.2.2.4

Multiply

$\frac{20 (8 + 2 \sqrt{31})}{3}$ by

$\frac{9}{9}$ .

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{8 (188 + 32 \sqrt{31}) \cdot 3}{9 \cdot 3} - \frac{20 (8 + 2 \sqrt{31}) \cdot 9}{3 \cdot 9}$

Step 4.1.2.2.5

Reorder the factors of

$9 \cdot 3$ .

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{8 (188 + 32 \sqrt{31}) \cdot 3}{3 \cdot 9} - \frac{20 (8 + 2 \sqrt{31}) \cdot 9}{3 \cdot 9}$

Step 4.1.2.2.6

Multiply

$3$ by

$9$ .

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{8 (188 + 32 \sqrt{31}) \cdot 3}{27} - \frac{20 (8 + 2 \sqrt{31}) \cdot 9}{3 \cdot 9}$

Step 4.1.2.2.7

Multiply

$3$ by

$9$ .

$\frac{3488 + 632 \sqrt{31}}{27} - \frac{8 (188 + 32 \sqrt{31}) \cdot 3}{27} - \frac{20 (8 + 2 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.3

Combine the numerators over the common denominator.

$\frac{3488 + 632 \sqrt{31} - 8 (188 + 32 \sqrt{31}) \cdot 3 - 20 (8 + 2 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.4

Simplify each term.

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Step 4.1.2.4.1

Apply the distributive property.

$\frac{3488 + 632 \sqrt{31} + (- 8 \cdot 188 - 8 (32 \sqrt{31})) \cdot 3 - 20 (8 + 2 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.4.2

Multiply

$- 8$ by

$188$ .

$\frac{3488 + 632 \sqrt{31} + (- 1504 - 8 (32 \sqrt{31})) \cdot 3 - 20 (8 + 2 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.4.3

Multiply

$32$ by

$- 8$ .

$\frac{3488 + 632 \sqrt{31} + (- 1504 - 256 \sqrt{31}) \cdot 3 - 20 (8 + 2 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.4.4

Apply the distributive property.

$\frac{3488 + 632 \sqrt{31} - 1504 \cdot 3 - 256 \sqrt{31} \cdot 3 - 20 (8 + 2 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.4.5

Multiply

$- 1504$ by

$3$ .

$\frac{3488 + 632 \sqrt{31} - 4512 - 256 \sqrt{31} \cdot 3 - 20 (8 + 2 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.4.6

Multiply

$3$ by

$- 256$ .

$\frac{3488 + 632 \sqrt{31} - 4512 - 768 \sqrt{31} - 20 (8 + 2 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.4.7

Apply the distributive property.

$\frac{3488 + 632 \sqrt{31} - 4512 - 768 \sqrt{31} + (- 20 \cdot 8 - 20 (2 \sqrt{31})) \cdot 9}{27}$

Step 4.1.2.4.8

Multiply

$- 20$ by

$8$ .

$\frac{3488 + 632 \sqrt{31} - 4512 - 768 \sqrt{31} + (- 160 - 20 (2 \sqrt{31})) \cdot 9}{27}$

Step 4.1.2.4.9

Multiply

$2$ by

$- 20$ .

$\frac{3488 + 632 \sqrt{31} - 4512 - 768 \sqrt{31} + (- 160 - 40 \sqrt{31}) \cdot 9}{27}$

Step 4.1.2.4.10

Apply the distributive property.

$\frac{3488 + 632 \sqrt{31} - 4512 - 768 \sqrt{31} - 160 \cdot 9 - 40 \sqrt{31} \cdot 9}{27}$

Step 4.1.2.4.11

Multiply

$- 160$ by

$9$ .

$\frac{3488 + 632 \sqrt{31} - 4512 - 768 \sqrt{31} - 1440 - 40 \sqrt{31} \cdot 9}{27}$

Step 4.1.2.4.12

Multiply

$9$ by

$- 40$ .

$\frac{3488 + 632 \sqrt{31} - 4512 - 768 \sqrt{31} - 1440 - 360 \sqrt{31}}{27}$

Step 4.1.2.5

Simplify terms.

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Step 4.1.2.5.1

Subtract

$4512$ from

$3488$ .

$\frac{- 1024 + 632 \sqrt{31} - 768 \sqrt{31} - 1440 - 360 \sqrt{31}}{27}$

Step 4.1.2.5.2

Subtract

$1440$ from

$- 1024$ .

$\frac{- 2464 + 632 \sqrt{31} - 768 \sqrt{31} - 360 \sqrt{31}}{27}$

Step 4.1.2.5.3

Subtract

$768 \sqrt{31}$ from

$632 \sqrt{31}$ .

$\frac{- 2464 - 136 \sqrt{31} - 360 \sqrt{31}}{27}$

Step 4.1.2.5.4

Subtract

$360 \sqrt{31}$ from

$- 136 \sqrt{31}$ .

$\frac{- 2464 - 496 \sqrt{31}}{27}$

Step 4.1.2.5.5

Rewrite

$- 2464$ as

$- 1 (2464)$ .

$\frac{- 1 (2464) - 496 \sqrt{31}}{27}$

Step 4.1.2.5.6

Factor

$- 1$ out of

$- 496 \sqrt{31}$ .

$\frac{- 1 (2464) - (496 \sqrt{31})}{27}$

Step 4.1.2.5.7

Factor

$- 1$ out of

$- 1 (2464) - (496 \sqrt{31})$ .

$\frac{- 1 (2464 + 496 \sqrt{31})}{27}$

Step 4.1.2.5.8

Move the negative in front of the fraction.

$- \frac{2464 + 496 \sqrt{31}}{27}$

Step 4.2

Evaluate at

$x = \frac{8 - 2 \sqrt{31}}{3}$ .

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Step 4.2.1

Substitute

$\frac{8 - 2 \sqrt{31}}{3}$ for

$x$ .

${(\frac{8 - 2 \sqrt{31}}{3})}^{3} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Apply the product rule to

$\frac{8 - 2 \sqrt{31}}{3}$ .

$\frac{{(8 - 2 \sqrt{31})}^{3}}{3^{3}} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.2

Raise

$3$ to the power of

$3$ .

$\frac{{(8 - 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.3

Use the Binomial Theorem.

$\frac{8^{3} + 3 \cdot 8^{2} (- 2 \sqrt{31}) + 3 \cdot 8 {(- 2 \sqrt{31})}^{2} + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4

Simplify each term.

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Step 4.2.2.1.4.1

Raise

$8$ to the power of

$3$ .

$\frac{512 + 3 \cdot 8^{2} (- 2 \sqrt{31}) + 3 \cdot 8 {(- 2 \sqrt{31})}^{2} + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.2

Raise

$8$ to the power of

$2$ .

$\frac{512 + 3 \cdot 64 (- 2 \sqrt{31}) + 3 \cdot 8 {(- 2 \sqrt{31})}^{2} + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.3

Multiply

$3$ by

$64$ .

$\frac{512 + 192 (- 2 \sqrt{31}) + 3 \cdot 8 {(- 2 \sqrt{31})}^{2} + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.4

Multiply

$- 2$ by

$192$ .

$\frac{512 - 384 \sqrt{31} + 3 \cdot 8 {(- 2 \sqrt{31})}^{2} + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.5

Multiply

$3$ by

$8$ .

$\frac{512 - 384 \sqrt{31} + 24 {(- 2 \sqrt{31})}^{2} + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.6

Apply the product rule to

$- 2 \sqrt{31}$ .

$\frac{512 - 384 \sqrt{31} + 24 ({(- 2)}^{2} {\sqrt{31}}^{2}) + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.7

Raise

$- 2$ to the power of

$2$ .

$\frac{512 - 384 \sqrt{31} + 24 (4 {\sqrt{31}}^{2}) + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.8

Rewrite

${\sqrt{31}}^{2}$ as

$31$ .

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Step 4.2.2.1.4.8.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{31}$ as

$31^{\frac{1}{2}}$ .

$\frac{512 - 384 \sqrt{31} + 24 (4 {(31^{\frac{1}{2}})}^{2}) + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.8.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{512 - 384 \sqrt{31} + 24 (4 \cdot 31^{\frac{1}{2} \cdot 2}) + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.8.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{512 - 384 \sqrt{31} + 24 (4 \cdot 31^{\frac{2}{2}}) + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.8.4

Cancel the common factor of

$2$ .

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Step 4.2.2.1.4.8.4.1

Cancel the common factor.

$\frac{512 - 384 \sqrt{31} + 24 (4 \cdot 31^{\frac{2}{2}}) + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.8.4.2

Rewrite the expression.

$\frac{512 - 384 \sqrt{31} + 24 (4 \cdot 31^{1}) + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.8.5

Evaluate the exponent.

$\frac{512 - 384 \sqrt{31} + 24 (4 \cdot 31) + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.9

Multiply

$24 (4 \cdot 31)$ .

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Step 4.2.2.1.4.9.1

Multiply

$4$ by

$31$ .

$\frac{512 - 384 \sqrt{31} + 24 \cdot 124 + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.9.2

Multiply

$24$ by

$124$ .

$\frac{512 - 384 \sqrt{31} + 2976 + {(- 2 \sqrt{31})}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.10

Apply the product rule to

$- 2 \sqrt{31}$ .

$\frac{512 - 384 \sqrt{31} + 2976 + {(- 2)}^{3} {\sqrt{31}}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.11

Raise

$- 2$ to the power of

$3$ .

$\frac{512 - 384 \sqrt{31} + 2976 - 8 {\sqrt{31}}^{3}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.12

Rewrite

${\sqrt{31}}^{3}$ as

$\sqrt{31^{3}}$ .

$\frac{512 - 384 \sqrt{31} + 2976 - 8 \sqrt{31^{3}}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.13

Raise

$31$ to the power of

$3$ .

$\frac{512 - 384 \sqrt{31} + 2976 - 8 \sqrt{29791}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.14

Rewrite

$29791$ as

$31^{2} \cdot 31$ .

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Step 4.2.2.1.4.14.1

Factor

$961$ out of

$29791$ .

$\frac{512 - 384 \sqrt{31} + 2976 - 8 \sqrt{961 (31)}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.14.2

Rewrite

$961$ as

$31^{2}$ .

$\frac{512 - 384 \sqrt{31} + 2976 - 8 \sqrt{31^{2} \cdot 31}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.15

Pull terms out from under the radical.

$\frac{512 - 384 \sqrt{31} + 2976 - 8 (31 \sqrt{31})}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.4.16

Multiply

$31$ by

$- 8$ .

$\frac{512 - 384 \sqrt{31} + 2976 - 248 \sqrt{31}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.5

Add

$512$ and

$2976$ .

$\frac{3488 - 384 \sqrt{31} - 248 \sqrt{31}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.6

Subtract

$248 \sqrt{31}$ from

$- 384 \sqrt{31}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 {(\frac{8 - 2 \sqrt{31}}{3})}^{2} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.7

Apply the product rule to

$\frac{8 - 2 \sqrt{31}}{3}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{{(8 - 2 \sqrt{31})}^{2}}{3^{2}} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.8

Raise

$3$ to the power of

$2$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{{(8 - 2 \sqrt{31})}^{2}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.9

Rewrite

${(8 - 2 \sqrt{31})}^{2}$ as

$(8 - 2 \sqrt{31}) (8 - 2 \sqrt{31})$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{(8 - 2 \sqrt{31}) (8 - 2 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.10

Expand

$(8 - 2 \sqrt{31}) (8 - 2 \sqrt{31})$ using the FOIL Method.

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Step 4.2.2.1.10.1

Apply the distributive property.

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{8 (8 - 2 \sqrt{31}) - 2 \sqrt{31} (8 - 2 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.10.2

Apply the distributive property.

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{8 \cdot 8 + 8 (- 2 \sqrt{31}) - 2 \sqrt{31} (8 - 2 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.10.3

Apply the distributive property.

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{8 \cdot 8 + 8 (- 2 \sqrt{31}) - 2 \sqrt{31} \cdot 8 - 2 \sqrt{31} (- 2 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11

Simplify and combine like terms.

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Step 4.2.2.1.11.1

Simplify each term.

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Step 4.2.2.1.11.1.1

Multiply

$8$ by

$8$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 + 8 (- 2 \sqrt{31}) - 2 \sqrt{31} \cdot 8 - 2 \sqrt{31} (- 2 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.2

Multiply

$- 2$ by

$8$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 2 \sqrt{31} \cdot 8 - 2 \sqrt{31} (- 2 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.3

Multiply

$8$ by

$- 2$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} - 2 \sqrt{31} (- 2 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.4

Multiply

$- 2 \sqrt{31} (- 2 \sqrt{31})$ .

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Step 4.2.2.1.11.1.4.1

Multiply

$- 2$ by

$- 2$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 \sqrt{31} \sqrt{31}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.4.2

Raise

$\sqrt{31}$ to the power of

$1$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 ({\sqrt{31}}^{1} \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.4.3

Raise

$\sqrt{31}$ to the power of

$1$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 ({\sqrt{31}}^{1} {\sqrt{31}}^{1})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 {\sqrt{31}}^{1 + 1}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.4.5

Add

$1$ and

$1$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 {\sqrt{31}}^{2}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.5

Rewrite

${\sqrt{31}}^{2}$ as

$31$ .

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Step 4.2.2.1.11.1.5.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{31}$ as

$31^{\frac{1}{2}}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 {(31^{\frac{1}{2}})}^{2}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.5.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 \cdot 31^{\frac{1}{2} \cdot 2}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.5.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 \cdot 31^{\frac{2}{2}}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.5.4

Cancel the common factor of

$2$ .

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Step 4.2.2.1.11.1.5.4.1

Cancel the common factor.

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 \cdot 31^{\frac{2}{2}}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.5.4.2

Rewrite the expression.

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 \cdot 31^{1}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.5.5

Evaluate the exponent.

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 4 \cdot 31}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.1.6

Multiply

$4$ by

$31$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{64 - 16 \sqrt{31} - 16 \sqrt{31} + 124}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.2

Add

$64$ and

$124$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{188 - 16 \sqrt{31} - 16 \sqrt{31}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.11.3

Subtract

$16 \sqrt{31}$ from

$- 16 \sqrt{31}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - 8 \frac{188 - 32 \sqrt{31}}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.12

Combine

$- 8$ and

$\frac{188 - 32 \sqrt{31}}{9}$ .

$\frac{3488 - 632 \sqrt{31}}{27} + \frac{- 8 (188 - 32 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.13

Move the negative in front of the fraction.

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{(8) (188 - 32 \sqrt{31})}{9} - 20 \frac{8 - 2 \sqrt{31}}{3}$

Step 4.2.2.1.14

Combine

$- 20$ and

$\frac{8 - 2 \sqrt{31}}{3}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{8 (188 - 32 \sqrt{31})}{9} + \frac{- 20 (8 - 2 \sqrt{31})}{3}$

Step 4.2.2.1.15

Move the negative in front of the fraction.

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{8 (188 - 32 \sqrt{31})}{9} - \frac{20 (8 - 2 \sqrt{31})}{3}$

Step 4.2.2.2

Find the common denominator.

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Step 4.2.2.2.1

Multiply

$\frac{8 (188 - 32 \sqrt{31})}{9}$ by

$\frac{3}{3}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - (\frac{8 (188 - 32 \sqrt{31})}{9} \cdot \frac{3}{3}) - \frac{20 (8 - 2 \sqrt{31})}{3}$

Step 4.2.2.2.2

Multiply

$\frac{8 (188 - 32 \sqrt{31})}{9}$ by

$\frac{3}{3}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{8 (188 - 32 \sqrt{31}) \cdot 3}{9 \cdot 3} - \frac{20 (8 - 2 \sqrt{31})}{3}$

Step 4.2.2.2.3

Multiply

$\frac{20 (8 - 2 \sqrt{31})}{3}$ by

$\frac{9}{9}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{8 (188 - 32 \sqrt{31}) \cdot 3}{9 \cdot 3} - (\frac{20 (8 - 2 \sqrt{31})}{3} \cdot \frac{9}{9})$

Step 4.2.2.2.4

Multiply

$\frac{20 (8 - 2 \sqrt{31})}{3}$ by

$\frac{9}{9}$ .

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{8 (188 - 32 \sqrt{31}) \cdot 3}{9 \cdot 3} - \frac{20 (8 - 2 \sqrt{31}) \cdot 9}{3 \cdot 9}$

Step 4.2.2.2.5

Reorder the factors of

$9 \cdot 3$ .

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{8 (188 - 32 \sqrt{31}) \cdot 3}{3 \cdot 9} - \frac{20 (8 - 2 \sqrt{31}) \cdot 9}{3 \cdot 9}$

Step 4.2.2.2.6

Multiply

$3$ by

$9$ .

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{8 (188 - 32 \sqrt{31}) \cdot 3}{27} - \frac{20 (8 - 2 \sqrt{31}) \cdot 9}{3 \cdot 9}$

Step 4.2.2.2.7

Multiply

$3$ by

$9$ .

$\frac{3488 - 632 \sqrt{31}}{27} - \frac{8 (188 - 32 \sqrt{31}) \cdot 3}{27} - \frac{20 (8 - 2 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.3

Combine the numerators over the common denominator.

$\frac{3488 - 632 \sqrt{31} - 8 (188 - 32 \sqrt{31}) \cdot 3 - 20 (8 - 2 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.4

Simplify each term.

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Step 4.2.2.4.1

Apply the distributive property.

$\frac{3488 - 632 \sqrt{31} + (- 8 \cdot 188 - 8 (- 32 \sqrt{31})) \cdot 3 - 20 (8 - 2 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.4.2

Multiply

$- 8$ by

$188$ .

$\frac{3488 - 632 \sqrt{31} + (- 1504 - 8 (- 32 \sqrt{31})) \cdot 3 - 20 (8 - 2 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.4.3

Multiply

$- 32$ by

$- 8$ .

$\frac{3488 - 632 \sqrt{31} + (- 1504 + 256 \sqrt{31}) \cdot 3 - 20 (8 - 2 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.4.4

Apply the distributive property.

$\frac{3488 - 632 \sqrt{31} - 1504 \cdot 3 + 256 \sqrt{31} \cdot 3 - 20 (8 - 2 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.4.5

Multiply

$- 1504$ by

$3$ .

$\frac{3488 - 632 \sqrt{31} - 4512 + 256 \sqrt{31} \cdot 3 - 20 (8 - 2 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.4.6

Multiply

$3$ by

$256$ .

$\frac{3488 - 632 \sqrt{31} - 4512 + 768 \sqrt{31} - 20 (8 - 2 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.4.7

Apply the distributive property.

$\frac{3488 - 632 \sqrt{31} - 4512 + 768 \sqrt{31} + (- 20 \cdot 8 - 20 (- 2 \sqrt{31})) \cdot 9}{27}$

Step 4.2.2.4.8

Multiply

$- 20$ by

$8$ .

$\frac{3488 - 632 \sqrt{31} - 4512 + 768 \sqrt{31} + (- 160 - 20 (- 2 \sqrt{31})) \cdot 9}{27}$

Step 4.2.2.4.9

Multiply

$- 2$ by

$- 20$ .

$\frac{3488 - 632 \sqrt{31} - 4512 + 768 \sqrt{31} + (- 160 + 40 \sqrt{31}) \cdot 9}{27}$

Step 4.2.2.4.10

Apply the distributive property.

$\frac{3488 - 632 \sqrt{31} - 4512 + 768 \sqrt{31} - 160 \cdot 9 + 40 \sqrt{31} \cdot 9}{27}$

Step 4.2.2.4.11

Multiply

$- 160$ by

$9$ .

$\frac{3488 - 632 \sqrt{31} - 4512 + 768 \sqrt{31} - 1440 + 40 \sqrt{31} \cdot 9}{27}$

Step 4.2.2.4.12

Multiply

$9$ by

$40$ .

$\frac{3488 - 632 \sqrt{31} - 4512 + 768 \sqrt{31} - 1440 + 360 \sqrt{31}}{27}$

Step 4.2.2.5

Simplify terms.

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Step 4.2.2.5.1

Subtract

$4512$ from

$3488$ .

$\frac{- 1024 - 632 \sqrt{31} + 768 \sqrt{31} - 1440 + 360 \sqrt{31}}{27}$

Step 4.2.2.5.2

Subtract

$1440$ from

$- 1024$ .

$\frac{- 2464 - 632 \sqrt{31} + 768 \sqrt{31} + 360 \sqrt{31}}{27}$

Step 4.2.2.5.3

Add

$- 632 \sqrt{31}$ and

$768 \sqrt{31}$ .

$\frac{- 2464 + 136 \sqrt{31} + 360 \sqrt{31}}{27}$

Step 4.2.2.5.4

Add

$136 \sqrt{31}$ and

$360 \sqrt{31}$ .

$\frac{- 2464 + 496 \sqrt{31}}{27}$

Step 4.2.2.5.5

Rewrite

$- 2464$ as

$- 1 (2464)$ .

$\frac{- 1 (2464) + 496 \sqrt{31}}{27}$

Step 4.2.2.5.6

Factor

$- 1$ out of

$496 \sqrt{31}$ .

$\frac{- 1 (2464) - (- 496 \sqrt{31})}{27}$

Step 4.2.2.5.7

Factor

$- 1$ out of

$- 1 (2464) - (- 496 \sqrt{31})$ .

$\frac{- 1 (2464 - 496 \sqrt{31})}{27}$

Step 4.2.2.5.8

Move the negative in front of the fraction.

$- \frac{2464 - 496 \sqrt{31}}{27}$

Step 4.3

List all of the points.

$(\frac{8 + 2 \sqrt{31}}{3}, - \frac{2464 + 496 \sqrt{31}}{27}), (\frac{8 - 2 \sqrt{31}}{3}, - \frac{2464 - 496 \sqrt{31}}{27})$

Step 5