Calculus Examples

$p (x) = \frac{10 x^{3} - 250 x^{2} - 2320 x + 577000}{80000}$

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

Cancel the common factor of $10 x^{3} - 250 x^{2} - 2320 x + 577000$ and $80000$ .

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Step 1.1.1.1

Factor $10$ out of $10 x^{3}$ .

$\frac{d}{d x} [\frac{10 (x^{3}) - 250 x^{2} - 2320 x + 577000}{80000}]$

Step 1.1.1.2

Factor $10$ out of $- 250 x^{2}$ .

$\frac{d}{d x} [\frac{10 (x^{3}) + 10 (- 25 x^{2}) - 2320 x + 577000}{80000}]$

Step 1.1.1.3

Factor $10$ out of $10 (x^{3}) + 10 (- 25 x^{2})$ .

$\frac{d}{d x} [\frac{10 (x^{3} - 25 x^{2}) - 2320 x + 577000}{80000}]$

Step 1.1.1.4

Factor $10$ out of $- 2320 x$ .

$\frac{d}{d x} [\frac{10 (x^{3} - 25 x^{2}) + 10 (- 232 x) + 577000}{80000}]$

Step 1.1.1.5

Factor $10$ out of $10 (x^{3} - 25 x^{2}) + 10 (- 232 x)$ .

$\frac{d}{d x} [\frac{10 (x^{3} - 25 x^{2} - 232 x) + 577000}{80000}]$

Step 1.1.1.6

Factor $10$ out of $577000$ .

$\frac{d}{d x} [\frac{10 (x^{3} - 25 x^{2} - 232 x) + 10 \cdot 57700}{80000}]$

Step 1.1.1.7

Factor $10$ out of $10 (x^{3} - 25 x^{2} - 232 x) + 10 \cdot 57700$ .

$\frac{d}{d x} [\frac{10 (x^{3} - 25 x^{2} - 232 x + 57700)}{80000}]$

Step 1.1.1.8

Cancel the common factors.

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Step 1.1.1.8.1

Factor $10$ out of $80000$ .

$\frac{d}{d x} [\frac{10 (x^{3} - 25 x^{2} - 232 x + 57700)}{10 (8000)}]$

Step 1.1.1.8.2

Cancel the common factor.

$\frac{d}{d x} [\frac{10 (x^{3} - 25 x^{2} - 232 x + 57700)}{10 \cdot 8000}]$

Step 1.1.1.8.3

Rewrite the expression.

$\frac{d}{d x} [\frac{x^{3} - 25 x^{2} - 232 x + 57700}{8000}]$

Step 1.1.2

Since $\frac{1}{8000}$ is constant with respect to $x$ , the derivative of $\frac{x^{3} - 25 x^{2} - 232 x + 57700}{8000}$ with respect to $x$ is $\frac{1}{8000} \frac{d}{d x} [x^{3} - 25 x^{2} - 232 x + 57700]$ .

$\frac{1}{8000} \frac{d}{d x} [x^{3} - 25 x^{2} - 232 x + 57700]$

Step 1.1.3

By the Sum Rule, the derivative of $x^{3} - 25 x^{2} - 232 x + 57700$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 25 x^{2}] + \frac{d}{d x} [- 232 x] + \frac{d}{d x} [57700]$ .

$\frac{1}{8000} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 25 x^{2}] + \frac{d}{d x} [- 232 x] + \frac{d}{d x} [57700])$

Step 1.1.4

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

$\frac{1}{8000} (3 x^{2} + \frac{d}{d x} [- 25 x^{2}] + \frac{d}{d x} [- 232 x] + \frac{d}{d x} [57700])$

Step 1.1.5

Since $- 25$ is constant with respect to $x$ , the derivative of $- 25 x^{2}$ with respect to $x$ is $- 25 \frac{d}{d x} [x^{2}]$ .

$\frac{1}{8000} (3 x^{2} - 25 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 232 x] + \frac{d}{d x} [57700])$

Step 1.1.6

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

$\frac{1}{8000} (3 x^{2} - 25 (2 x) + \frac{d}{d x} [- 232 x] + \frac{d}{d x} [57700])$

Step 1.1.7

Multiply $2$ by $- 25$ .

$\frac{1}{8000} (3 x^{2} - 50 x + \frac{d}{d x} [- 232 x] + \frac{d}{d x} [57700])$

Step 1.1.8

Since $- 232$ is constant with respect to $x$ , the derivative of $- 232 x$ with respect to $x$ is $- 232 \frac{d}{d x} [x]$ .

$\frac{1}{8000} (3 x^{2} - 50 x - 232 \frac{d}{d x} [x] + \frac{d}{d x} [57700])$

Step 1.1.9

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

$\frac{1}{8000} (3 x^{2} - 50 x - 232 \cdot 1 + \frac{d}{d x} [57700])$

Step 1.1.10

Multiply $- 232$ by $1$ .

$\frac{1}{8000} (3 x^{2} - 50 x - 232 + \frac{d}{d x} [57700])$

Step 1.1.11

Since $57700$ is constant with respect to $x$ , the derivative of $57700$ with respect to $x$ is $0$ .

$\frac{1}{8000} (3 x^{2} - 50 x - 232 + 0)$

Step 1.1.12

Add $3 x^{2} - 50 x - 232$ and $0$ .

$\frac{1}{8000} (3 x^{2} - 50 x - 232)$

Step 1.1.13

Simplify.

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Step 1.1.13.1

Apply the distributive property.

$\frac{1}{8000} (3 x^{2}) + \frac{1}{8000} (- 50 x) + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2

Combine terms.

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Step 1.1.13.2.1

Combine $3$ and $\frac{1}{8000}$ .

$\frac{3}{8000} x^{2} + \frac{1}{8000} (- 50 x) + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.2

Combine $\frac{3}{8000}$ and $x^{2}$ .

$\frac{3 x^{2}}{8000} + \frac{1}{8000} (- 50 x) + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.3

Combine $- 50$ and $\frac{1}{8000}$ .

$\frac{3 x^{2}}{8000} + \frac{- 50}{8000} x + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.4

Combine $\frac{- 50}{8000}$ and $x$ .

$\frac{3 x^{2}}{8000} + \frac{- 50 x}{8000} + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.5

Cancel the common factor of $- 50$ and $8000$ .

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Step 1.1.13.2.5.1

Factor $50$ out of $- 50 x$ .

$\frac{3 x^{2}}{8000} + \frac{50 (- x)}{8000} + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.5.2

Cancel the common factors.

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Step 1.1.13.2.5.2.1

Factor $50$ out of $8000$ .

$\frac{3 x^{2}}{8000} + \frac{50 (- x)}{50 (160)} + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.5.2.2

Cancel the common factor.

$\frac{3 x^{2}}{8000} + \frac{50 (- x)}{50 \cdot 160} + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.5.2.3

Rewrite the expression.

$\frac{3 x^{2}}{8000} + \frac{- x}{160} + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.6

Move the negative in front of the fraction.

$\frac{3 x^{2}}{8000} - \frac{x}{160} + \frac{1}{8000} \cdot - 232$

Step 1.1.13.2.7

Combine $\frac{1}{8000}$ and $- 232$ .

$\frac{3 x^{2}}{8000} - \frac{x}{160} + \frac{- 232}{8000}$

Step 1.1.13.2.8

Cancel the common factor of $- 232$ and $8000$ .

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Step 1.1.13.2.8.1

Factor $8$ out of $- 232$ .

$\frac{3 x^{2}}{8000} - \frac{x}{160} + \frac{8 (- 29)}{8000}$

Step 1.1.13.2.8.2

Cancel the common factors.

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Step 1.1.13.2.8.2.1

Factor $8$ out of $8000$ .

$\frac{3 x^{2}}{8000} - \frac{x}{160} + \frac{8 \cdot - 29}{8 \cdot 1000}$

Step 1.1.13.2.8.2.2

Cancel the common factor.

$\frac{3 x^{2}}{8000} - \frac{x}{160} + \frac{8 \cdot - 29}{8 \cdot 1000}$

Step 1.1.13.2.8.2.3

Rewrite the expression.

$\frac{3 x^{2}}{8000} - \frac{x}{160} + \frac{- 29}{1000}$

Step 1.1.13.2.9

Move the negative in front of the fraction.

$f' (x) = \frac{3 x^{2}}{8000} - \frac{x}{160} - \frac{29}{1000}$

Step 1.2

The first derivative of $p (x)$ with respect to $x$ is $\frac{3 x^{2}}{8000} - \frac{x}{160} - \frac{29}{1000}$ .

$\frac{3 x^{2}}{8000} - \frac{x}{160} - \frac{29}{1000}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{3 x^{2}}{8000} - \frac{x}{160} - \frac{29}{1000} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{3 x^{2}}{8000} - \frac{x}{160} - \frac{29}{1000} = 0$

Step 2.2

Multiply through by the least common denominator $8000$ , then simplify.

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Step 2.2.1

Apply the distributive property.

$8000 (\frac{3 x^{2}}{8000}) + 8000 (- \frac{x}{160}) + 8000 (- \frac{29}{1000}) = 0$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of $8000$ .

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Step 2.2.2.1.1

Cancel the common factor.

$8000 (\frac{3 x^{2}}{8000}) + 8000 (- \frac{x}{160}) + 8000 (- \frac{29}{1000}) = 0$

Step 2.2.2.1.2

Rewrite the expression.

$3 x^{2} + 8000 (- \frac{x}{160}) + 8000 (- \frac{29}{1000}) = 0$

Step 2.2.2.2

Cancel the common factor of $160$ .

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Step 2.2.2.2.1

Move the leading negative in $- \frac{x}{160}$ into the numerator.

$3 x^{2} + 8000 (\frac{- x}{160}) + 8000 (- \frac{29}{1000}) = 0$

Step 2.2.2.2.2

Factor $160$ out of $8000$ .

$3 x^{2} + 160 (50) (\frac{- x}{160}) + 8000 (- \frac{29}{1000}) = 0$

Step 2.2.2.2.3

Cancel the common factor.

$3 x^{2} + 160 \cdot (50 (\frac{- x}{160})) + 8000 (- \frac{29}{1000}) = 0$

Step 2.2.2.2.4

Rewrite the expression.

$3 x^{2} + 50 (- x) + 8000 (- \frac{29}{1000}) = 0$

Step 2.2.2.3

Multiply $- 1$ by $50$ .

$3 x^{2} - 50 x + 8000 (- \frac{29}{1000}) = 0$

Step 2.2.2.4

Cancel the common factor of $1000$ .

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Step 2.2.2.4.1

Move the leading negative in $- \frac{29}{1000}$ into the numerator.

$3 x^{2} - 50 x + 8000 (\frac{- 29}{1000}) = 0$

Step 2.2.2.4.2

Factor $1000$ out of $8000$ .

$3 x^{2} - 50 x + 1000 (8) (\frac{- 29}{1000}) = 0$

Step 2.2.2.4.3

Cancel the common factor.

$3 x^{2} - 50 x + 1000 \cdot (8 (\frac{- 29}{1000})) = 0$

Step 2.2.2.4.4

Rewrite the expression.

$3 x^{2} - 50 x + 8 \cdot - 29 = 0$

Step 2.2.2.5

Multiply $8$ by $- 29$ .

$3 x^{2} - 50 x - 232 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = 3$ , $b = - 50$ , and $c = - 232$ into the quadratic formula and solve for $x$ .

$\frac{50 \pm \sqrt{{(- 50)}^{2} - 4 \cdot (3 \cdot - 232)}}{2 \cdot 3}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 50$ to the power of $2$ .

$x = \frac{50 \pm \sqrt{2500 - 4 \cdot 3 \cdot - 232}}{2 \cdot 3}$

Step 2.5.1.2

Multiply $- 4 \cdot 3 \cdot - 232$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{50 \pm \sqrt{2500 - 12 \cdot - 232}}{2 \cdot 3}$

Step 2.5.1.2.2

Multiply $- 12$ by $- 232$ .

$x = \frac{50 \pm \sqrt{2500 + 2784}}{2 \cdot 3}$

Step 2.5.1.3

Add $2500$ and $2784$ .

$x = \frac{50 \pm \sqrt{5284}}{2 \cdot 3}$

Step 2.5.1.4

Rewrite $5284$ as $2^{2} \cdot 1321$ .

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Step 2.5.1.4.1

Factor $4$ out of $5284$ .

$x = \frac{50 \pm \sqrt{4 (1321)}}{2 \cdot 3}$

Step 2.5.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{50 \pm \sqrt{2^{2} \cdot 1321}}{2 \cdot 3}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{50 \pm 2 \sqrt{1321}}{2 \cdot 3}$

Step 2.5.2

Multiply $2$ by $3$ .

$x = \frac{50 \pm 2 \sqrt{1321}}{6}$

Step 2.5.3

Simplify $\frac{50 \pm 2 \sqrt{1321}}{6}$ .

$x = \frac{25 \pm \sqrt{1321}}{3}$

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 $- 50$ to the power of $2$ .

$x = \frac{50 \pm \sqrt{2500 - 4 \cdot 3 \cdot - 232}}{2 \cdot 3}$

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot - 232$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{50 \pm \sqrt{2500 - 12 \cdot - 232}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply $- 12$ by $- 232$ .

$x = \frac{50 \pm \sqrt{2500 + 2784}}{2 \cdot 3}$

Step 2.6.1.3

Add $2500$ and $2784$ .

$x = \frac{50 \pm \sqrt{5284}}{2 \cdot 3}$

Step 2.6.1.4

Rewrite $5284$ as $2^{2} \cdot 1321$ .

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Step 2.6.1.4.1

Factor $4$ out of $5284$ .

$x = \frac{50 \pm \sqrt{4 (1321)}}{2 \cdot 3}$

Step 2.6.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{50 \pm \sqrt{2^{2} \cdot 1321}}{2 \cdot 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{50 \pm 2 \sqrt{1321}}{2 \cdot 3}$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{50 \pm 2 \sqrt{1321}}{6}$

Step 2.6.3

Simplify $\frac{50 \pm 2 \sqrt{1321}}{6}$ .

$x = \frac{25 \pm \sqrt{1321}}{3}$

Step 2.6.4

Change the $\pm$ to $+$ .

$x = \frac{25 + \sqrt{1321}}{3}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 50$ to the power of $2$ .

$x = \frac{50 \pm \sqrt{2500 - 4 \cdot 3 \cdot - 232}}{2 \cdot 3}$

Step 2.7.1.2

Multiply $- 4 \cdot 3 \cdot - 232$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{50 \pm \sqrt{2500 - 12 \cdot - 232}}{2 \cdot 3}$

Step 2.7.1.2.2

Multiply $- 12$ by $- 232$ .

$x = \frac{50 \pm \sqrt{2500 + 2784}}{2 \cdot 3}$

Step 2.7.1.3

Add $2500$ and $2784$ .

$x = \frac{50 \pm \sqrt{5284}}{2 \cdot 3}$

Step 2.7.1.4

Rewrite $5284$ as $2^{2} \cdot 1321$ .

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Step 2.7.1.4.1

Factor $4$ out of $5284$ .

$x = \frac{50 \pm \sqrt{4 (1321)}}{2 \cdot 3}$

Step 2.7.1.4.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{50 \pm \sqrt{2^{2} \cdot 1321}}{2 \cdot 3}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{50 \pm 2 \sqrt{1321}}{2 \cdot 3}$

Step 2.7.2

Multiply $2$ by $3$ .

$x = \frac{50 \pm 2 \sqrt{1321}}{6}$

Step 2.7.3

Simplify $\frac{50 \pm 2 \sqrt{1321}}{6}$ .

$x = \frac{25 \pm \sqrt{1321}}{3}$

Step 2.7.4

Change the $\pm$ to $-$ .

$x = \frac{25 - \sqrt{1321}}{3}$

Step 2.8

The final answer is the combination of both solutions.

$x = \frac{25 + \sqrt{1321}}{3}, \frac{25 - \sqrt{1321}}{3}$

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 $\frac{10 x^{3} - 250 x^{2} - 2320 x + 577000}{80000}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{25 + \sqrt{1321}}{3}$ .

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Step 4.1.1

Substitute $\frac{25 + \sqrt{1321}}{3}$ for $x$ .

$\frac{10 {(\frac{25 + \sqrt{1321}}{3})}^{3} - 250 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 2320 \frac{25 + \sqrt{1321}}{3} + 577000}{80000}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Cancel the common factor of $10 {(\frac{25 + \sqrt{1321}}{3})}^{3} - 250 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 2320 \frac{25 + \sqrt{1321}}{3} + 577000$ and $80000$ .

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Step 4.1.2.1.1

Factor $10$ out of $10 {(\frac{25 + \sqrt{1321}}{3})}^{3}$ .

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3}) - 250 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 2320 \frac{25 + \sqrt{1321}}{3} + 577000}{80000}$

Step 4.1.2.1.2

Factor $10$ out of $- 250 {(\frac{25 + \sqrt{1321}}{3})}^{2}$ .

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3}) + 10 (- 25 {(\frac{25 + \sqrt{1321}}{3})}^{2}) - 2320 \frac{25 + \sqrt{1321}}{3} + 577000}{80000}$

Step 4.1.2.1.3

Factor $10$ out of $10 ({(\frac{25 + \sqrt{1321}}{3})}^{3}) + 10 (- 25 {(\frac{25 + \sqrt{1321}}{3})}^{2})$ .

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2}) - 2320 \frac{25 + \sqrt{1321}}{3} + 577000}{80000}$

Step 4.1.2.1.4

Factor $10$ out of $- 2320 \frac{25 + \sqrt{1321}}{3}$ .

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2}) + 10 (- 232 \frac{25 + \sqrt{1321}}{3}) + 577000}{80000}$

Step 4.1.2.1.5

Factor $10$ out of $10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2}) + 10 (- 232 \frac{25 + \sqrt{1321}}{3})$ .

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3}) + 577000}{80000}$

Step 4.1.2.1.6

Factor $10$ out of $577000$ .

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3}) + 10 \cdot 57700}{80000}$

Step 4.1.2.1.7

Factor $10$ out of $10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3}) + 10 \cdot 57700$ .

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700)}{80000}$

Step 4.1.2.1.8

Cancel the common factors.

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Step 4.1.2.1.8.1

Factor $10$ out of $80000$ .

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700)}{10 (8000)}$

Step 4.1.2.1.8.2

Cancel the common factor.

$\frac{10 ({(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700)}{10 \cdot 8000}$

Step 4.1.2.1.8.3

Rewrite the expression.

$\frac{{(\frac{25 + \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2

Simplify the numerator.

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Step 4.1.2.2.1

Apply the product rule to $\frac{25 + \sqrt{1321}}{3}$ .

$\frac{\frac{{(25 + \sqrt{1321})}^{3}}{3^{3}} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.2

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

$\frac{\frac{{(25 + \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.3

Use the Binomial Theorem.

$\frac{\frac{25^{3} + 3 \cdot 25^{2} \sqrt{1321} + 3 \cdot 25 {\sqrt{1321}}^{2} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4

Simplify each term.

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Step 4.1.2.2.4.1

Raise $25$ to the power of $3$ .

$\frac{\frac{15625 + 3 \cdot 25^{2} \sqrt{1321} + 3 \cdot 25 {\sqrt{1321}}^{2} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.2

Raise $25$ to the power of $2$ .

$\frac{\frac{15625 + 3 \cdot 625 \sqrt{1321} + 3 \cdot 25 {\sqrt{1321}}^{2} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.3

Multiply $3$ by $625$ .

$\frac{\frac{15625 + 1875 \sqrt{1321} + 3 \cdot 25 {\sqrt{1321}}^{2} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.4

Multiply $3$ by $25$ .

$\frac{\frac{15625 + 1875 \sqrt{1321} + 75 {\sqrt{1321}}^{2} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.5

Rewrite ${\sqrt{1321}}^{2}$ as $1321$ .

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Step 4.1.2.2.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1321}$ as $1321^{\frac{1}{2}}$ .

$\frac{\frac{15625 + 1875 \sqrt{1321} + 75 {(1321^{\frac{1}{2}})}^{2} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.5.2

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

$\frac{\frac{15625 + 1875 \sqrt{1321} + 75 \cdot 1321^{\frac{1}{2} \cdot 2} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.5.3

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

$\frac{\frac{15625 + 1875 \sqrt{1321} + 75 \cdot 1321^{\frac{2}{2}} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.5.4

Cancel the common factor of $2$ .

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Step 4.1.2.2.4.5.4.1

Cancel the common factor.

$\frac{\frac{15625 + 1875 \sqrt{1321} + 75 \cdot 1321^{\frac{2}{2}} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.5.4.2

Rewrite the expression.

$\frac{\frac{15625 + 1875 \sqrt{1321} + 75 \cdot 1321^{1} + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.5.5

Evaluate the exponent.

$\frac{\frac{15625 + 1875 \sqrt{1321} + 75 \cdot 1321 + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.6

Multiply $75$ by $1321$ .

$\frac{\frac{15625 + 1875 \sqrt{1321} + 99075 + {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.7

Rewrite ${\sqrt{1321}}^{3}$ as $\sqrt{1321^{3}}$ .

$\frac{\frac{15625 + 1875 \sqrt{1321} + 99075 + \sqrt{1321^{3}}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.8

Raise $1321$ to the power of $3$ .

$\frac{\frac{15625 + 1875 \sqrt{1321} + 99075 + \sqrt{2305199161}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.9

Rewrite $2305199161$ as $1321^{2} \cdot 1321$ .

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Step 4.1.2.2.4.9.1

Factor $1745041$ out of $2305199161$ .

$\frac{\frac{15625 + 1875 \sqrt{1321} + 99075 + \sqrt{1745041 (1321)}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.9.2

Rewrite $1745041$ as $1321^{2}$ .

$\frac{\frac{15625 + 1875 \sqrt{1321} + 99075 + \sqrt{1321^{2} \cdot 1321}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.4.10

Pull terms out from under the radical.

$\frac{\frac{15625 + 1875 \sqrt{1321} + 99075 + 1321 \sqrt{1321}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.5

Add $15625$ and $99075$ .

$\frac{\frac{114700 + 1875 \sqrt{1321} + 1321 \sqrt{1321}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.6

Add $1875 \sqrt{1321}$ and $1321 \sqrt{1321}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 {(\frac{25 + \sqrt{1321}}{3})}^{2} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.7

Apply the product rule to $\frac{25 + \sqrt{1321}}{3}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{{(25 + \sqrt{1321})}^{2}}{3^{2}} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.8

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

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{{(25 + \sqrt{1321})}^{2}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.9

Rewrite ${(25 + \sqrt{1321})}^{2}$ as $(25 + \sqrt{1321}) (25 + \sqrt{1321})$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{(25 + \sqrt{1321}) (25 + \sqrt{1321})}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.10

Expand $(25 + \sqrt{1321}) (25 + \sqrt{1321})$ using the FOIL Method.

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Step 4.1.2.2.10.1

Apply the distributive property.

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{25 (25 + \sqrt{1321}) + \sqrt{1321} (25 + \sqrt{1321})}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.10.2

Apply the distributive property.

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{25 \cdot 25 + 25 \sqrt{1321} + \sqrt{1321} (25 + \sqrt{1321})}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.10.3

Apply the distributive property.

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{25 \cdot 25 + 25 \sqrt{1321} + \sqrt{1321} \cdot 25 + \sqrt{1321} \sqrt{1321}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.11

Simplify and combine like terms.

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Step 4.1.2.2.11.1

Simplify each term.

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Step 4.1.2.2.11.1.1

Multiply $25$ by $25$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{625 + 25 \sqrt{1321} + \sqrt{1321} \cdot 25 + \sqrt{1321} \sqrt{1321}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.11.1.2

Move $25$ to the left of $\sqrt{1321}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{625 + 25 \sqrt{1321} + 25 \cdot \sqrt{1321} + \sqrt{1321} \sqrt{1321}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.11.1.3

Combine using the product rule for radicals.

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{625 + 25 \sqrt{1321} + 25 \sqrt{1321} + \sqrt{1321 \cdot 1321}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.11.1.4

Multiply $1321$ by $1321$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{625 + 25 \sqrt{1321} + 25 \sqrt{1321} + \sqrt{1745041}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.11.1.5

Rewrite $1745041$ as $1321^{2}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{625 + 25 \sqrt{1321} + 25 \sqrt{1321} + \sqrt{1321^{2}}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.11.1.6

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

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{625 + 25 \sqrt{1321} + 25 \sqrt{1321} + 1321}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.11.2

Add $625$ and $1321$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{1946 + 25 \sqrt{1321} + 25 \sqrt{1321}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.11.3

Add $25 \sqrt{1321}$ and $25 \sqrt{1321}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - 25 \frac{1946 + 50 \sqrt{1321}}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.12

Combine $- 25$ and $\frac{1946 + 50 \sqrt{1321}}{9}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} + \frac{- 25 (1946 + 50 \sqrt{1321})}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.13

Move the negative in front of the fraction.

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - \frac{25 (1946 + 50 \sqrt{1321})}{9} - 232 \frac{25 + \sqrt{1321}}{3} + 57700}{8000}$

Step 4.1.2.2.14

Combine $- 232$ and $\frac{25 + \sqrt{1321}}{3}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - \frac{25 (1946 + 50 \sqrt{1321})}{9} + \frac{- 232 (25 + \sqrt{1321})}{3} + 57700}{8000}$

Step 4.1.2.2.15

Move the negative in front of the fraction.

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - \frac{25 (1946 + 50 \sqrt{1321})}{9} - \frac{232 (25 + \sqrt{1321})}{3} + 57700}{8000}$

Step 4.1.2.2.16

To write $- \frac{25 (1946 + 50 \sqrt{1321})}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - \frac{25 (1946 + 50 \sqrt{1321})}{9} \cdot \frac{3}{3} - \frac{232 (25 + \sqrt{1321})}{3} + 57700}{8000}$

Step 4.1.2.2.17

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

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Step 4.1.2.2.17.1

Multiply $\frac{25 (1946 + 50 \sqrt{1321})}{9}$ by $\frac{3}{3}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - \frac{25 (1946 + 50 \sqrt{1321}) \cdot 3}{9 \cdot 3} - \frac{232 (25 + \sqrt{1321})}{3} + 57700}{8000}$

Step 4.1.2.2.17.2

Multiply $9$ by $3$ .

$\frac{\frac{114700 + 3196 \sqrt{1321}}{27} - \frac{25 (1946 + 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 + \sqrt{1321})}{3} + 57700}{8000}$

Step 4.1.2.2.18

Combine the numerators over the common denominator.

$\frac{\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 + \sqrt{1321})}{3} + 57700}{8000}$

Step 4.1.2.2.19

To write $- \frac{232 (25 + \sqrt{1321})}{3}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 + \sqrt{1321})}{3} \cdot \frac{9}{9} + 57700}{8000}$

Step 4.1.2.2.20

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

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Step 4.1.2.2.20.1

Multiply $\frac{232 (25 + \sqrt{1321})}{3}$ by $\frac{9}{9}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 + \sqrt{1321}) \cdot 9}{3 \cdot 9} + 57700}{8000}$

Step 4.1.2.2.20.2

Multiply $3$ by $9$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 + \sqrt{1321}) \cdot 9}{27} + 57700}{8000}$

Step 4.1.2.2.21

Combine the numerators over the common denominator.

$\frac{\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9}{27} + 57700}{8000}$

Step 4.1.2.2.22

To write $57700$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9}{27} + 57700 \cdot \frac{27}{27}}{8000}$

Step 4.1.2.2.23

Combine $57700$ and $\frac{27}{27}$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9}{27} + \frac{57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.24

Combine the numerators over the common denominator.

$\frac{\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25

Rewrite $\frac{114700 + 3196 \sqrt{1321} - 25 (1946 + 50 \sqrt{1321}) \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}$ in a factored form.

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Step 4.1.2.2.25.1

Apply the distributive property.

$\frac{\frac{114700 + 3196 \sqrt{1321} + (- 25 \cdot 1946 - 25 (50 \sqrt{1321})) \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.2

Multiply $- 25$ by $1946$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} + (- 48650 - 25 (50 \sqrt{1321})) \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.3

Multiply $50$ by $- 25$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} + (- 48650 - 1250 \sqrt{1321}) \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.4

Apply the distributive property.

$\frac{\frac{114700 + 3196 \sqrt{1321} - 48650 \cdot 3 - 1250 \sqrt{1321} \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.5

Multiply $- 48650$ by $3$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 145950 - 1250 \sqrt{1321} \cdot 3 - 232 (25 + \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.6

Multiply $3$ by $- 1250$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 145950 - 3750 \sqrt{1321} - 232 (25 + \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.7

Apply the distributive property.

$\frac{\frac{114700 + 3196 \sqrt{1321} - 145950 - 3750 \sqrt{1321} + (- 232 \cdot 25 - 232 \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.8

Multiply $- 232$ by $25$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 145950 - 3750 \sqrt{1321} + (- 5800 - 232 \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.9

Apply the distributive property.

$\frac{\frac{114700 + 3196 \sqrt{1321} - 145950 - 3750 \sqrt{1321} - 5800 \cdot 9 - 232 \sqrt{1321} \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.10

Multiply $- 5800$ by $9$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 145950 - 3750 \sqrt{1321} - 52200 - 232 \sqrt{1321} \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.11

Multiply $9$ by $- 232$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 145950 - 3750 \sqrt{1321} - 52200 - 2088 \sqrt{1321} + 57700 \cdot 27}{27}}{8000}$

Step 4.1.2.2.25.12

Multiply $57700$ by $27$ .

$\frac{\frac{114700 + 3196 \sqrt{1321} - 145950 - 3750 \sqrt{1321} - 52200 - 2088 \sqrt{1321} + 1557900}{27}}{8000}$

Step 4.1.2.2.25.13

Subtract $145950$ from $114700$ .

$\frac{\frac{- 31250 + 3196 \sqrt{1321} - 3750 \sqrt{1321} - 52200 - 2088 \sqrt{1321} + 1557900}{27}}{8000}$

Step 4.1.2.2.25.14

Subtract $52200$ from $- 31250$ .

$\frac{\frac{- 83450 + 3196 \sqrt{1321} - 3750 \sqrt{1321} - 2088 \sqrt{1321} + 1557900}{27}}{8000}$

Step 4.1.2.2.25.15

Add $- 83450$ and $1557900$ .

$\frac{\frac{1474450 + 3196 \sqrt{1321} - 3750 \sqrt{1321} - 2088 \sqrt{1321}}{27}}{8000}$

Step 4.1.2.2.25.16

Subtract $3750 \sqrt{1321}$ from $3196 \sqrt{1321}$ .

$\frac{\frac{1474450 - 554 \sqrt{1321} - 2088 \sqrt{1321}}{27}}{8000}$

Step 4.1.2.2.25.17

Subtract $2088 \sqrt{1321}$ from $- 554 \sqrt{1321}$ .

$\frac{\frac{1474450 - 2642 \sqrt{1321}}{27}}{8000}$

Step 4.1.2.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1474450 - 2642 \sqrt{1321}}{27} \cdot \frac{1}{8000}$

Step 4.1.2.4

Multiply $\frac{1474450 - 2642 \sqrt{1321}}{27} \cdot \frac{1}{8000}$ .

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Step 4.1.2.4.1

Multiply $\frac{1474450 - 2642 \sqrt{1321}}{27}$ by $\frac{1}{8000}$ .

$\frac{1474450 - 2642 \sqrt{1321}}{27 \cdot 8000}$

Step 4.1.2.4.2

Multiply $27$ by $8000$ .

$\frac{1474450 - 2642 \sqrt{1321}}{216000}$

Step 4.1.2.5

Cancel the common factor of $1474450 - 2642 \sqrt{1321}$ and $216000$ .

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Step 4.1.2.5.1

Factor $2$ out of $1474450$ .

$\frac{2 (737225) - 2642 \sqrt{1321}}{216000}$

Step 4.1.2.5.2

Factor $2$ out of $- 2642 \sqrt{1321}$ .

$\frac{2 (737225) + 2 (- 1321 \sqrt{1321})}{216000}$

Step 4.1.2.5.3

Factor $2$ out of $2 (737225) + 2 (- 1321 \sqrt{1321})$ .

$\frac{2 (737225 - 1321 \sqrt{1321})}{216000}$

Step 4.1.2.5.4

Cancel the common factors.

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Step 4.1.2.5.4.1

Factor $2$ out of $216000$ .

$\frac{2 (737225 - 1321 \sqrt{1321})}{2 \cdot 108000}$

Step 4.1.2.5.4.2

Cancel the common factor.

$\frac{2 (737225 - 1321 \sqrt{1321})}{2 \cdot 108000}$

Step 4.1.2.5.4.3

Rewrite the expression.

$\frac{737225 - 1321 \sqrt{1321}}{108000}$

Step 4.2

Evaluate at $x = \frac{25 - \sqrt{1321}}{3}$ .

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Step 4.2.1

Substitute $\frac{25 - \sqrt{1321}}{3}$ for $x$ .

$\frac{10 {(\frac{25 - \sqrt{1321}}{3})}^{3} - 250 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 2320 \frac{25 - \sqrt{1321}}{3} + 577000}{80000}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Cancel the common factor of $10 {(\frac{25 - \sqrt{1321}}{3})}^{3} - 250 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 2320 \frac{25 - \sqrt{1321}}{3} + 577000$ and $80000$ .

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Step 4.2.2.1.1

Factor $10$ out of $10 {(\frac{25 - \sqrt{1321}}{3})}^{3}$ .

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3}) - 250 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 2320 \frac{25 - \sqrt{1321}}{3} + 577000}{80000}$

Step 4.2.2.1.2

Factor $10$ out of $- 250 {(\frac{25 - \sqrt{1321}}{3})}^{2}$ .

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3}) + 10 (- 25 {(\frac{25 - \sqrt{1321}}{3})}^{2}) - 2320 \frac{25 - \sqrt{1321}}{3} + 577000}{80000}$

Step 4.2.2.1.3

Factor $10$ out of $10 ({(\frac{25 - \sqrt{1321}}{3})}^{3}) + 10 (- 25 {(\frac{25 - \sqrt{1321}}{3})}^{2})$ .

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2}) - 2320 \frac{25 - \sqrt{1321}}{3} + 577000}{80000}$

Step 4.2.2.1.4

Factor $10$ out of $- 2320 \frac{25 - \sqrt{1321}}{3}$ .

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2}) + 10 (- 232 \frac{25 - \sqrt{1321}}{3}) + 577000}{80000}$

Step 4.2.2.1.5

Factor $10$ out of $10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2}) + 10 (- 232 \frac{25 - \sqrt{1321}}{3})$ .

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3}) + 577000}{80000}$

Step 4.2.2.1.6

Factor $10$ out of $577000$ .

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3}) + 10 \cdot 57700}{80000}$

Step 4.2.2.1.7

Factor $10$ out of $10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3}) + 10 \cdot 57700$ .

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700)}{80000}$

Step 4.2.2.1.8

Cancel the common factors.

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Step 4.2.2.1.8.1

Factor $10$ out of $80000$ .

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700)}{10 (8000)}$

Step 4.2.2.1.8.2

Cancel the common factor.

$\frac{10 ({(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700)}{10 \cdot 8000}$

Step 4.2.2.1.8.3

Rewrite the expression.

$\frac{{(\frac{25 - \sqrt{1321}}{3})}^{3} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2

Simplify the numerator.

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Step 4.2.2.2.1

Apply the product rule to $\frac{25 - \sqrt{1321}}{3}$ .

$\frac{\frac{{(25 - \sqrt{1321})}^{3}}{3^{3}} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.2

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

$\frac{\frac{{(25 - \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.3

Use the Binomial Theorem.

$\frac{\frac{25^{3} + 3 \cdot 25^{2} (- \sqrt{1321}) + 3 \cdot 25 {(- \sqrt{1321})}^{2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4

Simplify each term.

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Step 4.2.2.2.4.1

Raise $25$ to the power of $3$ .

$\frac{\frac{15625 + 3 \cdot 25^{2} (- \sqrt{1321}) + 3 \cdot 25 {(- \sqrt{1321})}^{2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.2

Raise $25$ to the power of $2$ .

$\frac{\frac{15625 + 3 \cdot 625 (- \sqrt{1321}) + 3 \cdot 25 {(- \sqrt{1321})}^{2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.3

Multiply $3$ by $625$ .

$\frac{\frac{15625 + 1875 (- \sqrt{1321}) + 3 \cdot 25 {(- \sqrt{1321})}^{2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.4

Multiply $- 1$ by $1875$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 3 \cdot 25 {(- \sqrt{1321})}^{2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.5

Multiply $3$ by $25$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 {(- \sqrt{1321})}^{2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.6

Apply the product rule to $- \sqrt{1321}$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 ({(- 1)}^{2} {\sqrt{1321}}^{2}) + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.7

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

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 (1 {\sqrt{1321}}^{2}) + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.8

Multiply ${\sqrt{1321}}^{2}$ by $1$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 {\sqrt{1321}}^{2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.9

Rewrite ${\sqrt{1321}}^{2}$ as $1321$ .

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Step 4.2.2.2.4.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1321}$ as $1321^{\frac{1}{2}}$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 {(1321^{\frac{1}{2}})}^{2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.9.2

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

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 \cdot 1321^{\frac{1}{2} \cdot 2} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.9.3

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

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 \cdot 1321^{\frac{2}{2}} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.9.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.4.9.4.1

Cancel the common factor.

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 \cdot 1321^{\frac{2}{2}} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.9.4.2

Rewrite the expression.

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 \cdot 1321^{1} + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.9.5

Evaluate the exponent.

$\frac{\frac{15625 - 1875 \sqrt{1321} + 75 \cdot 1321 + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.10

Multiply $75$ by $1321$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 + {(- \sqrt{1321})}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.11

Apply the product rule to $- \sqrt{1321}$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 + {(- 1)}^{3} {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.12

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

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 - {\sqrt{1321}}^{3}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.13

Rewrite ${\sqrt{1321}}^{3}$ as $\sqrt{1321^{3}}$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 - \sqrt{1321^{3}}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.14

Raise $1321$ to the power of $3$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 - \sqrt{2305199161}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.15

Rewrite $2305199161$ as $1321^{2} \cdot 1321$ .

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Step 4.2.2.2.4.15.1

Factor $1745041$ out of $2305199161$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 - \sqrt{1745041 (1321)}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.15.2

Rewrite $1745041$ as $1321^{2}$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 - \sqrt{1321^{2} \cdot 1321}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.16

Pull terms out from under the radical.

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 - (1321 \sqrt{1321})}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.4.17

Multiply $1321$ by $- 1$ .

$\frac{\frac{15625 - 1875 \sqrt{1321} + 99075 - 1321 \sqrt{1321}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.5

Add $15625$ and $99075$ .

$\frac{\frac{114700 - 1875 \sqrt{1321} - 1321 \sqrt{1321}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.6

Subtract $1321 \sqrt{1321}$ from $- 1875 \sqrt{1321}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 {(\frac{25 - \sqrt{1321}}{3})}^{2} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.7

Apply the product rule to $\frac{25 - \sqrt{1321}}{3}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{{(25 - \sqrt{1321})}^{2}}{3^{2}} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.8

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

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{{(25 - \sqrt{1321})}^{2}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.9

Rewrite ${(25 - \sqrt{1321})}^{2}$ as $(25 - \sqrt{1321}) (25 - \sqrt{1321})$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{(25 - \sqrt{1321}) (25 - \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.10

Expand $(25 - \sqrt{1321}) (25 - \sqrt{1321})$ using the FOIL Method.

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Step 4.2.2.2.10.1

Apply the distributive property.

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{25 (25 - \sqrt{1321}) - \sqrt{1321} (25 - \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.10.2

Apply the distributive property.

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{25 \cdot 25 + 25 (- \sqrt{1321}) - \sqrt{1321} (25 - \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.10.3

Apply the distributive property.

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{25 \cdot 25 + 25 (- \sqrt{1321}) - \sqrt{1321} \cdot 25 - \sqrt{1321} (- \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11

Simplify and combine like terms.

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Step 4.2.2.2.11.1

Simplify each term.

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Step 4.2.2.2.11.1.1

Multiply $25$ by $25$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 + 25 (- \sqrt{1321}) - \sqrt{1321} \cdot 25 - \sqrt{1321} (- \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.2

Multiply $- 1$ by $25$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - \sqrt{1321} \cdot 25 - \sqrt{1321} (- \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.3

Multiply $25$ by $- 1$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} - \sqrt{1321} (- \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.4

Multiply $- \sqrt{1321} (- \sqrt{1321})$ .

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Step 4.2.2.2.11.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + 1 \sqrt{1321} \sqrt{1321}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.4.2

Multiply $\sqrt{1321}$ by $1$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + \sqrt{1321} \sqrt{1321}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.4.3

Raise $\sqrt{1321}$ to the power of $1$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + {\sqrt{1321}}^{1} \sqrt{1321}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.4.4

Raise $\sqrt{1321}$ to the power of $1$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + {\sqrt{1321}}^{1} {\sqrt{1321}}^{1}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.4.5

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

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + {\sqrt{1321}}^{1 + 1}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.4.6

Add $1$ and $1$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + {\sqrt{1321}}^{2}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.5

Rewrite ${\sqrt{1321}}^{2}$ as $1321$ .

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Step 4.2.2.2.11.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1321}$ as $1321^{\frac{1}{2}}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + {(1321^{\frac{1}{2}})}^{2}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.5.2

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

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + 1321^{\frac{1}{2} \cdot 2}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.5.3

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

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + 1321^{\frac{2}{2}}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.11.1.5.4.1

Cancel the common factor.

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + 1321^{\frac{2}{2}}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.5.4.2

Rewrite the expression.

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + 1321^{1}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.1.5.5

Evaluate the exponent.

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{625 - 25 \sqrt{1321} - 25 \sqrt{1321} + 1321}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.2

Add $625$ and $1321$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{1946 - 25 \sqrt{1321} - 25 \sqrt{1321}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.11.3

Subtract $25 \sqrt{1321}$ from $- 25 \sqrt{1321}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - 25 \frac{1946 - 50 \sqrt{1321}}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.12

Combine $- 25$ and $\frac{1946 - 50 \sqrt{1321}}{9}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} + \frac{- 25 (1946 - 50 \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.13

Move the negative in front of the fraction.

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - \frac{25 (1946 - 50 \sqrt{1321})}{9} - 232 \frac{25 - \sqrt{1321}}{3} + 57700}{8000}$

Step 4.2.2.2.14

Combine $- 232$ and $\frac{25 - \sqrt{1321}}{3}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - \frac{25 (1946 - 50 \sqrt{1321})}{9} + \frac{- 232 (25 - \sqrt{1321})}{3} + 57700}{8000}$

Step 4.2.2.2.15

Move the negative in front of the fraction.

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - \frac{25 (1946 - 50 \sqrt{1321})}{9} - \frac{232 (25 - \sqrt{1321})}{3} + 57700}{8000}$

Step 4.2.2.2.16

To write $- \frac{25 (1946 - 50 \sqrt{1321})}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - \frac{25 (1946 - 50 \sqrt{1321})}{9} \cdot \frac{3}{3} - \frac{232 (25 - \sqrt{1321})}{3} + 57700}{8000}$

Step 4.2.2.2.17

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

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Step 4.2.2.2.17.1

Multiply $\frac{25 (1946 - 50 \sqrt{1321})}{9}$ by $\frac{3}{3}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - \frac{25 (1946 - 50 \sqrt{1321}) \cdot 3}{9 \cdot 3} - \frac{232 (25 - \sqrt{1321})}{3} + 57700}{8000}$

Step 4.2.2.2.17.2

Multiply $9$ by $3$ .

$\frac{\frac{114700 - 3196 \sqrt{1321}}{27} - \frac{25 (1946 - 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 - \sqrt{1321})}{3} + 57700}{8000}$

Step 4.2.2.2.18

Combine the numerators over the common denominator.

$\frac{\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 - \sqrt{1321})}{3} + 57700}{8000}$

Step 4.2.2.2.19

To write $- \frac{232 (25 - \sqrt{1321})}{3}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 - \sqrt{1321})}{3} \cdot \frac{9}{9} + 57700}{8000}$

Step 4.2.2.2.20

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

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Step 4.2.2.2.20.1

Multiply $\frac{232 (25 - \sqrt{1321})}{3}$ by $\frac{9}{9}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 - \sqrt{1321}) \cdot 9}{3 \cdot 9} + 57700}{8000}$

Step 4.2.2.2.20.2

Multiply $3$ by $9$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3}{27} - \frac{232 (25 - \sqrt{1321}) \cdot 9}{27} + 57700}{8000}$

Step 4.2.2.2.21

Combine the numerators over the common denominator.

$\frac{\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9}{27} + 57700}{8000}$

Step 4.2.2.2.22

To write $57700$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9}{27} + 57700 \cdot \frac{27}{27}}{8000}$

Step 4.2.2.2.23

Combine $57700$ and $\frac{27}{27}$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9}{27} + \frac{57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.24

Combine the numerators over the common denominator.

$\frac{\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25

Rewrite $\frac{114700 - 3196 \sqrt{1321} - 25 (1946 - 50 \sqrt{1321}) \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}$ in a factored form.

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Step 4.2.2.2.25.1

Apply the distributive property.

$\frac{\frac{114700 - 3196 \sqrt{1321} + (- 25 \cdot 1946 - 25 (- 50 \sqrt{1321})) \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.2

Multiply $- 25$ by $1946$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} + (- 48650 - 25 (- 50 \sqrt{1321})) \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.3

Multiply $- 50$ by $- 25$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} + (- 48650 + 1250 \sqrt{1321}) \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.4

Apply the distributive property.

$\frac{\frac{114700 - 3196 \sqrt{1321} - 48650 \cdot 3 + 1250 \sqrt{1321} \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.5

Multiply $- 48650$ by $3$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 1250 \sqrt{1321} \cdot 3 - 232 (25 - \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.6

Multiply $3$ by $1250$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 3750 \sqrt{1321} - 232 (25 - \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.7

Apply the distributive property.

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 3750 \sqrt{1321} + (- 232 \cdot 25 - 232 (- \sqrt{1321})) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.8

Multiply $- 232$ by $25$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 3750 \sqrt{1321} + (- 5800 - 232 (- \sqrt{1321})) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.9

Multiply $- 1$ by $- 232$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 3750 \sqrt{1321} + (- 5800 + 232 \sqrt{1321}) \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.10

Apply the distributive property.

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 3750 \sqrt{1321} - 5800 \cdot 9 + 232 \sqrt{1321} \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.11

Multiply $- 5800$ by $9$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 3750 \sqrt{1321} - 52200 + 232 \sqrt{1321} \cdot 9 + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.12

Multiply $9$ by $232$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 3750 \sqrt{1321} - 52200 + 2088 \sqrt{1321} + 57700 \cdot 27}{27}}{8000}$

Step 4.2.2.2.25.13

Multiply $57700$ by $27$ .

$\frac{\frac{114700 - 3196 \sqrt{1321} - 145950 + 3750 \sqrt{1321} - 52200 + 2088 \sqrt{1321} + 1557900}{27}}{8000}$

Step 4.2.2.2.25.14

Subtract $145950$ from $114700$ .

$\frac{\frac{- 31250 - 3196 \sqrt{1321} + 3750 \sqrt{1321} - 52200 + 2088 \sqrt{1321} + 1557900}{27}}{8000}$

Step 4.2.2.2.25.15

Subtract $52200$ from $- 31250$ .

$\frac{\frac{- 83450 - 3196 \sqrt{1321} + 3750 \sqrt{1321} + 2088 \sqrt{1321} + 1557900}{27}}{8000}$

Step 4.2.2.2.25.16

Add $- 83450$ and $1557900$ .

$\frac{\frac{1474450 - 3196 \sqrt{1321} + 3750 \sqrt{1321} + 2088 \sqrt{1321}}{27}}{8000}$

Step 4.2.2.2.25.17

Add $- 3196 \sqrt{1321}$ and $3750 \sqrt{1321}$ .

$\frac{\frac{1474450 + 554 \sqrt{1321} + 2088 \sqrt{1321}}{27}}{8000}$

Step 4.2.2.2.25.18

Add $554 \sqrt{1321}$ and $2088 \sqrt{1321}$ .

$\frac{\frac{1474450 + 2642 \sqrt{1321}}{27}}{8000}$

Step 4.2.2.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1474450 + 2642 \sqrt{1321}}{27} \cdot \frac{1}{8000}$

Step 4.2.2.4

Multiply $\frac{1474450 + 2642 \sqrt{1321}}{27} \cdot \frac{1}{8000}$ .

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Step 4.2.2.4.1

Multiply $\frac{1474450 + 2642 \sqrt{1321}}{27}$ by $\frac{1}{8000}$ .

$\frac{1474450 + 2642 \sqrt{1321}}{27 \cdot 8000}$

Step 4.2.2.4.2

Multiply $27$ by $8000$ .

$\frac{1474450 + 2642 \sqrt{1321}}{216000}$

Step 4.2.2.5

Cancel the common factor of $1474450 + 2642 \sqrt{1321}$ and $216000$ .

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Step 4.2.2.5.1

Factor $2$ out of $1474450$ .

$\frac{2 (737225) + 2642 \sqrt{1321}}{216000}$

Step 4.2.2.5.2

Factor $2$ out of $2642 \sqrt{1321}$ .

$\frac{2 (737225) + 2 (1321 \sqrt{1321})}{216000}$

Step 4.2.2.5.3

Factor $2$ out of $2 (737225) + 2 (1321 \sqrt{1321})$ .

$\frac{2 (737225 + 1321 \sqrt{1321})}{216000}$

Step 4.2.2.5.4

Cancel the common factors.

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Step 4.2.2.5.4.1

Factor $2$ out of $216000$ .

$\frac{2 (737225 + 1321 \sqrt{1321})}{2 \cdot 108000}$

Step 4.2.2.5.4.2

Cancel the common factor.

$\frac{2 (737225 + 1321 \sqrt{1321})}{2 \cdot 108000}$

Step 4.2.2.5.4.3

Rewrite the expression.

$\frac{737225 + 1321 \sqrt{1321}}{108000}$

Step 4.3

List all of the points.

$(\frac{25 + \sqrt{1321}}{3}, \frac{737225 - 1321 \sqrt{1321}}{108000}), (\frac{25 - \sqrt{1321}}{3}, \frac{737225 + 1321 \sqrt{1321}}{108000})$

Step 5