Calculus Examples

$y = x^{2} + 2 x - 4$ , $y = x + 4$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{2} + 2 x - 4 = x + 4$

Step 1.2

Solve $x^{2} + 2 x - 4 = x + 4$ for $x$ .

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Step 1.2.1

Move all terms containing $x$ to the left side of the equation.

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Step 1.2.1.1

Subtract $x$ from both sides of the equation.

$x^{2} + 2 x - 4 - x = 4$

Step 1.2.1.2

Subtract $x$ from $2 x$ .

$x^{2} + x - 4 = 4$

Step 1.2.2

Move all terms to the left side of the equation and simplify.

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Step 1.2.2.1

Subtract $4$ from both sides of the equation.

$x^{2} + x - 4 - 4 = 0$

Step 1.2.2.2

Subtract $4$ from $- 4$ .

$x^{2} + x - 8 = 0$

Step 1.2.3

Use the quadratic formula to find the solutions.

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

Step 1.2.4

Substitute the values $a = 1$ , $b = 1$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (1 \cdot - 8)}}{2 \cdot 1} + y = x + 4$

Step 1.2.5

Simplify.

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 8}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 1 \pm \sqrt{1 + 32}}{2 \cdot 1}$

Step 1.2.5.1.3

Add $1$ and $32$ .

$x = \frac{- 1 \pm \sqrt{33}}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{33}}{2}$

Step 1.2.6

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

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 8}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 1 \pm \sqrt{1 + 32}}{2 \cdot 1}$

Step 1.2.6.1.3

Add $1$ and $32$ .

$x = \frac{- 1 \pm \sqrt{33}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{33}}{2}$

Step 1.2.6.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{33}}{2}$

Step 1.2.6.4

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

$x = \frac{- 1 \cdot 1 + \sqrt{33}}{2}$

Step 1.2.6.5

Factor $- 1$ out of $\sqrt{33}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{33})}{2}$

Step 1.2.6.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{33})$ .

$x = \frac{- 1 (1 - \sqrt{33})}{2}$

Step 1.2.6.7

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{33}}{2}$

Step 1.2.7

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

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 8}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 1 \pm \sqrt{1 + 32}}{2 \cdot 1}$

Step 1.2.7.1.3

Add $1$ and $32$ .

$x = \frac{- 1 \pm \sqrt{33}}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{33}}{2}$

Step 1.2.7.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{33}}{2}$

Step 1.2.7.4

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

$x = \frac{- 1 \cdot 1 - \sqrt{33}}{2}$

Step 1.2.7.5

Factor $- 1$ out of $- \sqrt{33}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{33})}{2}$

Step 1.2.7.6

Factor $- 1$ out of $- 1 (1) - (\sqrt{33})$ .

$x = \frac{- 1 (1 + \sqrt{33})}{2}$

Step 1.2.7.7

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{33}}{2}$

Step 1.2.8

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{33}}{2}, - \frac{1 + \sqrt{33}}{2}$

Step 1.3

Evaluate $y$ when $x = - \frac{1 - \sqrt{33}}{2}$ .

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Step 1.3.1

Substitute $- \frac{1 - \sqrt{33}}{2}$ for $x$ .

$y = (- \frac{1 - \sqrt{33}}{2}) + 4$

Step 1.3.2

Substitute $- \frac{1 - \sqrt{33}}{2}$ for $x$ in $y = (- \frac{1 - \sqrt{33}}{2}) + 4$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = - \frac{1 - \sqrt{33}}{2} + 4$

Step 1.3.2.2

Remove parentheses.

$y = (- \frac{1 - \sqrt{33}}{2}) + 4$

Step 1.3.2.3

Simplify $(- \frac{1 - \sqrt{33}}{2}) + 4$ .

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Step 1.3.2.3.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{1 - \sqrt{33}}{2} + 4 \cdot \frac{2}{2}$

Step 1.3.2.3.2

Combine fractions.

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Step 1.3.2.3.2.1

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

$y = - \frac{1 - \sqrt{33}}{2} + \frac{4 \cdot 2}{2}$

Step 1.3.2.3.2.2

Combine the numerators over the common denominator.

$y = \frac{- (1 - \sqrt{33}) + 4 \cdot 2}{2}$

Step 1.3.2.3.3

Simplify the numerator.

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Step 1.3.2.3.3.1

Apply the distributive property.

$y = \frac{- 1 \cdot 1 - - \sqrt{33} + 4 \cdot 2}{2}$

Step 1.3.2.3.3.2

Multiply $- 1$ by $1$ .

$y = \frac{- 1 - - \sqrt{33} + 4 \cdot 2}{2}$

Step 1.3.2.3.3.3

Multiply $- - \sqrt{33}$ .

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Step 1.3.2.3.3.3.1

Multiply $- 1$ by $- 1$ .

$y = \frac{- 1 + 1 \sqrt{33} + 4 \cdot 2}{2}$

Step 1.3.2.3.3.3.2

Multiply $\sqrt{33}$ by $1$ .

$y = \frac{- 1 + \sqrt{33} + 4 \cdot 2}{2}$

Step 1.3.2.3.3.4

Multiply $4$ by $2$ .

$y = \frac{- 1 + \sqrt{33} + 8}{2}$

Step 1.3.2.3.3.5

Add $- 1$ and $8$ .

$y = \frac{7 + \sqrt{33}}{2}$

Step 1.4

Evaluate $y$ when $x = - \frac{1 + \sqrt{33}}{2}$ .

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Step 1.4.1

Substitute $- \frac{1 + \sqrt{33}}{2}$ for $x$ .

$y = (- \frac{1 + \sqrt{33}}{2}) + 4$

Step 1.4.2

Substitute $- \frac{1 + \sqrt{33}}{2}$ for $x$ in $y = (- \frac{1 + \sqrt{33}}{2}) + 4$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = - \frac{1 + \sqrt{33}}{2} + 4$

Step 1.4.2.2

Remove parentheses.

$y = (- \frac{1 + \sqrt{33}}{2}) + 4$

Step 1.4.2.3

Simplify $(- \frac{1 + \sqrt{33}}{2}) + 4$ .

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Step 1.4.2.3.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = - \frac{1 + \sqrt{33}}{2} + 4 \cdot \frac{2}{2}$

Step 1.4.2.3.2

Combine fractions.

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Step 1.4.2.3.2.1

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

$y = - \frac{1 + \sqrt{33}}{2} + \frac{4 \cdot 2}{2}$

Step 1.4.2.3.2.2

Combine the numerators over the common denominator.

$y = \frac{- (1 + \sqrt{33}) + 4 \cdot 2}{2}$

Step 1.4.2.3.3

Simplify the numerator.

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Step 1.4.2.3.3.1

Apply the distributive property.

$y = \frac{- 1 \cdot 1 - \sqrt{33} + 4 \cdot 2}{2}$

Step 1.4.2.3.3.2

Multiply $- 1$ by $1$ .

$y = \frac{- 1 - \sqrt{33} + 4 \cdot 2}{2}$

Step 1.4.2.3.3.3

Multiply $4$ by $2$ .

$y = \frac{- 1 - \sqrt{33} + 8}{2}$

Step 1.4.2.3.3.4

Add $- 1$ and $8$ .

$y = \frac{7 - \sqrt{33}}{2}$

Step 1.5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- \frac{1 - \sqrt{33}}{2}, \frac{7 + \sqrt{33}}{2})$

$(- \frac{1 + \sqrt{33}}{2}, \frac{7 - \sqrt{33}}{2})$

$(- \frac{1 - \sqrt{33}}{2}, \frac{7 + \sqrt{33}}{2})$

$(- \frac{1 + \sqrt{33}}{2}, \frac{7 - \sqrt{33}}{2})$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} x + 4 d x - \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} x^{2} + 2 x - 4 d x$

Step 3

Integrate to find the area between $- \frac{1 + \sqrt{33}}{2}$ and $- \frac{1 - \sqrt{33}}{2}$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} x + 4 - (x^{2} + 2 x - 4) d x$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

$x + 4 - x^{2} - (2 x) - - 4$

Step 3.2.2

Simplify.

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Step 3.2.2.1

Multiply $2$ by $- 1$ .

$x + 4 - x^{2} - 2 x - - 4$

Step 3.2.2.2

Multiply $- 1$ by $- 4$ .

$x + 4 - x^{2} - 2 x + 4$

$\int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} x + 4 - x^{2} - 2 x + 4 d x$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Subtract $2 x$ from $x$ .

$- x + 4 - x^{2} + 4$

Step 3.3.2

Add $4$ and $4$ .

$- x - x^{2} + 8$

$\int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} - x - x^{2} + 8 d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} - x d x + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} - x^{2} d x + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} 8 d x$

Step 3.5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} x d x + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} - x^{2} d x + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} 8 d x$

Step 3.6

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- (\frac{1}{2} x^{2}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} - x^{2} d x + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} 8 d x$

Step 3.7

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

$- (\frac{x^{2}}{2}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} - x^{2} d x + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} 8 d x$

Step 3.8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- (\frac{x^{2}}{2}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) - \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} x^{2} d x + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} 8 d x$

Step 3.9

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- (\frac{x^{2}}{2}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) - (\frac{1}{3} x^{3}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} 8 d x$

Step 3.10

Combine $\frac{1}{3}$ and $x^{3}$ .

$- (\frac{x^{2}}{2}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) - (\frac{x^{3}}{3}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) + \int_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}} 8 d x$

Step 3.11

Apply the constant rule.

$- (\frac{x^{2}}{2}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) - (\frac{x^{3}}{3}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) + 8 x]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}$

Step 3.12

Simplify the answer.

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Step 3.12.1

Substitute and simplify.

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Step 3.12.1.1

Evaluate $\frac{x^{2}}{2}$ at $- \frac{1 - \sqrt{33}}{2}$ and at $- \frac{1 + \sqrt{33}}{2}$ .

$- ((\frac{{(- \frac{1 - \sqrt{33}}{2})}^{2}}{2}) - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{x^{3}}{3}]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}) + 8 x]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}$

Step 3.12.1.2

Evaluate $\frac{x^{3}}{3}$ at $- \frac{1 - \sqrt{33}}{2}$ and at $- \frac{1 + \sqrt{33}}{2}$ .

$- (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + 8 x]_{- \frac{1 + \sqrt{33}}{2}}^{- \frac{1 - \sqrt{33}}{2}}$

Step 3.12.1.3

Evaluate $8 x$ at $- \frac{1 - \sqrt{33}}{2}$ and at $- \frac{1 + \sqrt{33}}{2}$ .

$- (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4

Simplify.

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Step 3.12.1.4.1

Factor $- 1$ out of $- \frac{1 - \sqrt{33}}{2}$ .

$- (\frac{{(- (\frac{1 - \sqrt{33}}{2}))}^{2}}{2} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.2

Apply the product rule to $- (\frac{1 - \sqrt{33}}{2})$ .

$- (\frac{{(- 1)}^{2} {(\frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.3

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

$- (\frac{1 {(\frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.4

Multiply ${(\frac{1 - \sqrt{33}}{2})}^{2}$ by $1$ .

$- (\frac{{(\frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.5

Factor $- 1$ out of $- \frac{1 + \sqrt{33}}{2}$ .

$- (\frac{{(\frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{{(- (\frac{1 + \sqrt{33}}{2}))}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.6

Apply the product rule to $- (\frac{1 + \sqrt{33}}{2})$ .

$- (\frac{{(\frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{{(- 1)}^{2} {(\frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.7

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

$- (\frac{{(\frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{1 {(\frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.8

Multiply ${(\frac{1 + \sqrt{33}}{2})}^{2}$ by $1$ .

$- (\frac{{(\frac{1 - \sqrt{33}}{2})}^{2}}{2} - \frac{{(\frac{1 + \sqrt{33}}{2})}^{2}}{2}) - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.9

Combine the numerators over the common denominator.

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (\frac{{(- \frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.10

Factor $- 1$ out of $- \frac{1 - \sqrt{33}}{2}$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (\frac{{(- (\frac{1 - \sqrt{33}}{2}))}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.11

Apply the product rule to $- (\frac{1 - \sqrt{33}}{2})$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (\frac{{(- 1)}^{3} {(\frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.12

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

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (\frac{- {(\frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.13

Move the negative in front of the fraction.

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (- \frac{{(\frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- \frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.14

Factor $- 1$ out of $- \frac{1 + \sqrt{33}}{2}$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (- \frac{{(\frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- (\frac{1 + \sqrt{33}}{2}))}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.15

Apply the product rule to $- (\frac{1 + \sqrt{33}}{2})$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (- \frac{{(\frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{{(- 1)}^{3} {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.16

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

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (- \frac{{(\frac{1 - \sqrt{33}}{2})}^{3}}{3} - \frac{- {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.17

Move the negative in front of the fraction.

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (- \frac{{(\frac{1 - \sqrt{33}}{2})}^{3}}{3} - - \frac{{(\frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.18

Multiply $- 1$ by $- 1$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (- \frac{{(\frac{1 - \sqrt{33}}{2})}^{3}}{3} + 1 \frac{{(\frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.19

Multiply $\frac{{(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$ by $1$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - (- \frac{{(\frac{1 - \sqrt{33}}{2})}^{3}}{3} + \frac{{(\frac{1 + \sqrt{33}}{2})}^{3}}{3}) + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.20

Combine the numerators over the common denominator.

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} + (8 (- \frac{1 - \sqrt{33}}{2})) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.21

Multiply $- 1$ by $8$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} - 8 \frac{1 - \sqrt{33}}{2} - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.22

Combine $- 8$ and $\frac{1 - \sqrt{33}}{2}$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} + \frac{- 8 (1 - \sqrt{33})}{2} - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.23

Cancel the common factor of $- 8$ and $2$ .

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Step 3.12.1.4.23.1

Factor $2$ out of $- 8 (1 - \sqrt{33})$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} + \frac{2 (- 4 (1 - \sqrt{33}))}{2} - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.23.2

Cancel the common factors.

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Step 3.12.1.4.23.2.1

Factor $2$ out of $2$ .

Step 3.12.1.4.23.2.2

Cancel the common factor.

Step 3.12.1.4.23.2.3

Rewrite the expression.

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} + \frac{- 4 (1 - \sqrt{33})}{1} - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.23.2.4

Divide $- 4 (1 - \sqrt{33})$ by $1$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} - 4 (1 - \sqrt{33}) - 8 (- \frac{1 + \sqrt{33}}{2})$

Step 3.12.1.4.24

Multiply $- 1$ by $- 8$ .

Step 3.12.1.4.25

Combine $8$ and $\frac{1 + \sqrt{33}}{2}$ .

Step 3.12.1.4.26

Cancel the common factor of $8$ and $2$ .

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Step 3.12.1.4.26.1

Factor $2$ out of $8 (1 + \sqrt{33})$ .

Step 3.12.1.4.26.2

Cancel the common factors.

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Step 3.12.1.4.26.2.1

Factor $2$ out of $2$ .

Step 3.12.1.4.26.2.2

Cancel the common factor.

Step 3.12.1.4.26.2.3

Rewrite the expression.

Step 3.12.1.4.26.2.4

Divide $4 (1 + \sqrt{33})$ by $1$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} - 4 (1 - \sqrt{33}) + 4 (1 + \sqrt{33})$

Step 3.12.1.4.27

To write $- 4 (1 - \sqrt{33})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} - 4 (1 - \sqrt{33}) \cdot \frac{2}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} + 4 (1 + \sqrt{33})$

Step 3.12.1.4.28

Combine $- 4 (1 - \sqrt{33})$ and $\frac{2}{2}$ .

$- \frac{{(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}}{2} + \frac{- 4 (1 - \sqrt{33}) \cdot 2}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} + 4 (1 + \sqrt{33})$

Step 3.12.1.4.29

Combine the numerators over the common denominator.

$\frac{- ({(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 4 (1 - \sqrt{33}) \cdot 2}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} + 4 (1 + \sqrt{33})$

Step 3.12.1.4.30

Multiply $2$ by $- 4$ .

$\frac{- ({(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3} + 4 (1 + \sqrt{33})$

Step 3.12.1.4.31

To write $4 (1 + \sqrt{33})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{- ({(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33})}{2} + 4 (1 + \sqrt{33}) \cdot \frac{2}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.1.4.32

Combine $4 (1 + \sqrt{33})$ and $\frac{2}{2}$ .

$\frac{- ({(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33})}{2} + \frac{4 (1 + \sqrt{33}) \cdot 2}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.1.4.33

Combine the numerators over the common denominator.

$\frac{- ({(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 4 (1 + \sqrt{33}) \cdot 2}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.1.4.34

Multiply $2$ by $4$ .

$\frac{- ({(\frac{1 - \sqrt{33}}{2})}^{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2

Simplify.

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Step 3.12.2.1

Simplify each term.

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Step 3.12.2.1.1

Simplify the numerator.

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Step 3.12.2.1.1.1

Simplify each term.

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Step 3.12.2.1.1.1.1

Apply the product rule to $\frac{1 - \sqrt{33}}{2}$ .

$\frac{- (\frac{{(1 - \sqrt{33})}^{2}}{2^{2}} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.2

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

$\frac{- (\frac{{(1 - \sqrt{33})}^{2}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.3

Rewrite ${(1 - \sqrt{33})}^{2}$ as $(1 - \sqrt{33}) (1 - \sqrt{33})$ .

$\frac{- (\frac{(1 - \sqrt{33}) (1 - \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.4

Expand $(1 - \sqrt{33}) (1 - \sqrt{33})$ using the FOIL Method.

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Step 3.12.2.1.1.1.4.1

Apply the distributive property.

$\frac{- (\frac{1 (1 - \sqrt{33}) - \sqrt{33} (1 - \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.4.2

Apply the distributive property.

$\frac{- (\frac{1 \cdot 1 + 1 (- \sqrt{33}) - \sqrt{33} (1 - \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.4.3

Apply the distributive property.

$\frac{- (\frac{1 \cdot 1 + 1 (- \sqrt{33}) - \sqrt{33} \cdot 1 - \sqrt{33} (- \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5

Simplify and combine like terms.

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Step 3.12.2.1.1.1.5.1

Simplify each term.

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Step 3.12.2.1.1.1.5.1.1

Multiply $1$ by $1$ .

$\frac{- (\frac{1 + 1 (- \sqrt{33}) - \sqrt{33} \cdot 1 - \sqrt{33} (- \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.2

Multiply $- \sqrt{33}$ by $1$ .

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} \cdot 1 - \sqrt{33} (- \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.3

Multiply $- 1$ by $1$ .

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} - \sqrt{33} (- \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.4

Multiply $- \sqrt{33} (- \sqrt{33})$ .

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Step 3.12.2.1.1.1.5.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + 1 \sqrt{33} \sqrt{33}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.4.2

Multiply $\sqrt{33}$ by $1$ .

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + \sqrt{33} \sqrt{33}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.4.3

Raise $\sqrt{33}$ to the power of $1$ .

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + {\sqrt{33}}^{1} \sqrt{33}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.4.4

Raise $\sqrt{33}$ to the power of $1$ .

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + {\sqrt{33}}^{1} {\sqrt{33}}^{1}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.4.5

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

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + {\sqrt{33}}^{1 + 1}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.4.6

Add $1$ and $1$ .

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + {\sqrt{33}}^{2}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.5

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

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Step 3.12.2.1.1.1.5.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + {(33^{\frac{1}{2}})}^{2}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.5.2

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

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + 33^{\frac{1}{2} \cdot 2}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.5.3

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

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + 33^{\frac{2}{2}}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.5.4

Cancel the common factor of $2$ .

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Step 3.12.2.1.1.1.5.1.5.4.1

Cancel the common factor.

Step 3.12.2.1.1.1.5.1.5.4.2

Rewrite the expression.

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + 33^{1}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.1.5.5

Evaluate the exponent.

$\frac{- (\frac{1 - \sqrt{33} - \sqrt{33} + 33}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.2

Add $1$ and $33$ .

$\frac{- (\frac{34 - \sqrt{33} - \sqrt{33}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.5.3

Subtract $\sqrt{33}$ from $- \sqrt{33}$ .

$\frac{- (\frac{34 - 2 \sqrt{33}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.6

Cancel the common factor of $34 - 2 \sqrt{33}$ and $4$ .

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Step 3.12.2.1.1.1.6.1

Factor $2$ out of $34$ .

$\frac{- (\frac{2 (17) - 2 \sqrt{33}}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.6.2

Factor $2$ out of $- 2 \sqrt{33}$ .

$\frac{- (\frac{2 (17) + 2 (- \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.6.3

Factor $2$ out of $2 (17) + 2 (- \sqrt{33})$ .

$\frac{- (\frac{2 (17 - \sqrt{33})}{4} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.6.4

Cancel the common factors.

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Step 3.12.2.1.1.1.6.4.1

Factor $2$ out of $4$ .

$\frac{- (\frac{2 (17 - \sqrt{33})}{2 \cdot 2} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.6.4.2

Cancel the common factor.

Step 3.12.2.1.1.1.6.4.3

Rewrite the expression.

$\frac{- (\frac{17 - \sqrt{33}}{2} - {(\frac{1 + \sqrt{33}}{2})}^{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.7

Apply the product rule to $\frac{1 + \sqrt{33}}{2}$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{{(1 + \sqrt{33})}^{2}}{2^{2}}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.8

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

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{{(1 + \sqrt{33})}^{2}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.9

Rewrite ${(1 + \sqrt{33})}^{2}$ as $(1 + \sqrt{33}) (1 + \sqrt{33})$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{(1 + \sqrt{33}) (1 + \sqrt{33})}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.10

Expand $(1 + \sqrt{33}) (1 + \sqrt{33})$ using the FOIL Method.

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Step 3.12.2.1.1.1.10.1

Apply the distributive property.

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 (1 + \sqrt{33}) + \sqrt{33} (1 + \sqrt{33})}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.10.2

Apply the distributive property.

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 \cdot 1 + 1 \sqrt{33} + \sqrt{33} (1 + \sqrt{33})}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.10.3

Apply the distributive property.

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 \cdot 1 + 1 \sqrt{33} + \sqrt{33} \cdot 1 + \sqrt{33} \sqrt{33}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11

Simplify and combine like terms.

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Step 3.12.2.1.1.1.11.1

Simplify each term.

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Step 3.12.2.1.1.1.11.1.1

Multiply $1$ by $1$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 + 1 \sqrt{33} + \sqrt{33} \cdot 1 + \sqrt{33} \sqrt{33}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11.1.2

Multiply $\sqrt{33}$ by $1$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 + \sqrt{33} + \sqrt{33} \cdot 1 + \sqrt{33} \sqrt{33}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11.1.3

Multiply $\sqrt{33}$ by $1$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 + \sqrt{33} + \sqrt{33} + \sqrt{33} \sqrt{33}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11.1.4

Combine using the product rule for radicals.

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 + \sqrt{33} + \sqrt{33} + \sqrt{33 \cdot 33}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11.1.5

Multiply $33$ by $33$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 + \sqrt{33} + \sqrt{33} + \sqrt{1089}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11.1.6

Rewrite $1089$ as $33^{2}$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 + \sqrt{33} + \sqrt{33} + \sqrt{33^{2}}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11.1.7

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

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{1 + \sqrt{33} + \sqrt{33} + 33}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11.2

Add $1$ and $33$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{34 + \sqrt{33} + \sqrt{33}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.11.3

Add $\sqrt{33}$ and $\sqrt{33}$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{34 + 2 \sqrt{33}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.12

Cancel the common factor of $34 + 2 \sqrt{33}$ and $4$ .

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Step 3.12.2.1.1.1.12.1

Factor $2$ out of $34$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{2 \cdot 17 + 2 \sqrt{33}}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.12.2

Factor $2$ out of $2 \sqrt{33}$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{2 \cdot 17 + 2 (\sqrt{33})}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.12.3

Factor $2$ out of $2 \cdot 17 + 2 (\sqrt{33})$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{2 \cdot (17 + \sqrt{33})}{4}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.12.4

Cancel the common factors.

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Step 3.12.2.1.1.1.12.4.1

Factor $2$ out of $4$ .

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{2 \cdot (17 + \sqrt{33})}{2 (2)}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.12.4.2

Cancel the common factor.

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{2 \cdot (17 + \sqrt{33})}{2 \cdot 2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.1.12.4.3

Rewrite the expression.

$\frac{- (\frac{17 - \sqrt{33}}{2} - \frac{17 + \sqrt{33}}{2}) - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.2

Combine the numerators over the common denominator.

$\frac{- \frac{17 - \sqrt{33} - (17 + \sqrt{33})}{2} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.3

Simplify each term.

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Step 3.12.2.1.1.3.1

Apply the distributive property.

$\frac{- \frac{17 - \sqrt{33} - 1 \cdot 17 - \sqrt{33}}{2} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.3.2

Multiply $- 1$ by $17$ .

$\frac{- \frac{17 - \sqrt{33} - 17 - \sqrt{33}}{2} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.4

Subtract $17$ from $17$ .

$\frac{- \frac{0 - \sqrt{33} - \sqrt{33}}{2} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.5

Subtract $\sqrt{33}$ from $0$ .

$\frac{- \frac{- \sqrt{33} - \sqrt{33}}{2} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.6

Subtract $\sqrt{33}$ from $- \sqrt{33}$ .

$\frac{- \frac{- 2 \sqrt{33}}{2} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.7

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

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Step 3.12.2.1.1.7.1

Factor $2$ out of $- 2 \sqrt{33}$ .

$\frac{- \frac{2 (- \sqrt{33})}{2} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.7.2

Cancel the common factors.

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Step 3.12.2.1.1.7.2.1

Factor $2$ out of $2$ .

$\frac{- \frac{2 (- \sqrt{33})}{2 (1)} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.7.2.2

Cancel the common factor.

$\frac{- \frac{2 (- \sqrt{33})}{2 \cdot 1} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.7.2.3

Rewrite the expression.

$\frac{- \frac{- \sqrt{33}}{1} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.7.2.4

Divide $- \sqrt{33}$ by $1$ .

$\frac{- - \sqrt{33} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.8

Multiply $- - \sqrt{33}$ .

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Step 3.12.2.1.1.8.1

Multiply $- 1$ by $- 1$ .

$\frac{1 \sqrt{33} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.8.2

Multiply $\sqrt{33}$ by $1$ .

$\frac{\sqrt{33} - 8 (1 - \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.9

Apply the distributive property.

$\frac{\sqrt{33} - 8 \cdot 1 - 8 (- \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.10

Multiply $- 8$ by $1$ .

$\frac{\sqrt{33} - 8 - 8 (- \sqrt{33}) + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.11

Multiply $- 1$ by $- 8$ .

$\frac{\sqrt{33} - 8 + 8 \sqrt{33} + 8 (1 + \sqrt{33})}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.12

Apply the distributive property.

$\frac{\sqrt{33} - 8 + 8 \sqrt{33} + 8 \cdot 1 + 8 \sqrt{33}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.13

Multiply $8$ by $1$ .

$\frac{\sqrt{33} - 8 + 8 \sqrt{33} + 8 + 8 \sqrt{33}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.14

Add $\sqrt{33}$ and $8 \sqrt{33}$ .

$\frac{- 8 + 9 \sqrt{33} + 8 + 8 \sqrt{33}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.15

Add $- 8$ and $8$ .

$\frac{0 + 9 \sqrt{33} + 8 \sqrt{33}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.16

Add $0$ and $9 \sqrt{33}$ .

$\frac{9 \sqrt{33} + 8 \sqrt{33}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.1.17

Add $9 \sqrt{33}$ and $8 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- {(\frac{1 - \sqrt{33}}{2})}^{3} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2

Simplify the numerator.

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Step 3.12.2.1.2.1

Apply the product rule to $\frac{1 - \sqrt{33}}{2}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{{(1 - \sqrt{33})}^{3}}{2^{3}} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.2

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

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{{(1 - \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.3

Use the Binomial Theorem.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1^{3} + 3 \cdot 1^{2} (- \sqrt{33}) + 3 \cdot 1 {(- \sqrt{33})}^{2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4

Simplify each term.

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Step 3.12.2.1.2.4.1

One to any power is one.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 + 3 \cdot 1^{2} (- \sqrt{33}) + 3 \cdot 1 {(- \sqrt{33})}^{2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.2

One to any power is one.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 + 3 \cdot 1 (- \sqrt{33}) + 3 \cdot 1 {(- \sqrt{33})}^{2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.3

Multiply $3$ by $1$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 + 3 (- \sqrt{33}) + 3 \cdot 1 {(- \sqrt{33})}^{2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.4

Multiply $- 1$ by $3$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 \cdot 1 {(- \sqrt{33})}^{2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.5

Multiply $3$ by $1$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 {(- \sqrt{33})}^{2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.6

Apply the product rule to $- \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 ({(- 1)}^{2} {\sqrt{33}}^{2}) + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.7

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

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 (1 {\sqrt{33}}^{2}) + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.8

Multiply ${\sqrt{33}}^{2}$ by $1$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 {\sqrt{33}}^{2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.9

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

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Step 3.12.2.1.2.4.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 {(33^{\frac{1}{2}})}^{2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.9.2

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

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 \cdot 33^{\frac{1}{2} \cdot 2} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.9.3

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

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 \cdot 33^{\frac{2}{2}} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.9.4

Cancel the common factor of $2$ .

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Step 3.12.2.1.2.4.9.4.1

Cancel the common factor.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 \cdot 33^{\frac{2}{2}} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.9.4.2

Rewrite the expression.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 \cdot 33^{1} + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.9.5

Evaluate the exponent.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 3 \cdot 33 + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.10

Multiply $3$ by $33$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 + {(- \sqrt{33})}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.11

Apply the product rule to $- \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 + {(- 1)}^{3} {\sqrt{33}}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.12

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

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 - {\sqrt{33}}^{3}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.13

Rewrite ${\sqrt{33}}^{3}$ as $\sqrt{33^{3}}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 - \sqrt{33^{3}}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.14

Raise $33$ to the power of $3$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 - \sqrt{35937}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.15

Rewrite $35937$ as $33^{2} \cdot 33$ .

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Step 3.12.2.1.2.4.15.1

Factor $1089$ out of $35937$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 - \sqrt{1089 (33)}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.15.2

Rewrite $1089$ as $33^{2}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 - \sqrt{33^{2} \cdot 33}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.16

Pull terms out from under the radical.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 - (33 \sqrt{33})}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.4.17

Multiply $33$ by $- 1$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{1 - 3 \sqrt{33} + 99 - 33 \sqrt{33}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.5

Add $1$ and $99$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{100 - 3 \sqrt{33} - 33 \sqrt{33}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.6

Subtract $33 \sqrt{33}$ from $- 3 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{100 - 36 \sqrt{33}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.7

Cancel the common factor of $100 - 36 \sqrt{33}$ and $8$ .

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Step 3.12.2.1.2.7.1

Factor $4$ out of $100$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{4 (25) - 36 \sqrt{33}}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.7.2

Factor $4$ out of $- 36 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{4 (25) + 4 (- 9 \sqrt{33})}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.7.3

Factor $4$ out of $4 (25) + 4 (- 9 \sqrt{33})$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{4 (25 - 9 \sqrt{33})}{8} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.7.4

Cancel the common factors.

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Step 3.12.2.1.2.7.4.1

Factor $4$ out of $8$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{4 (25 - 9 \sqrt{33})}{4 \cdot 2} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.7.4.2

Cancel the common factor.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{4 (25 - 9 \sqrt{33})}{4 \cdot 2} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.7.4.3

Rewrite the expression.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + {(\frac{1 + \sqrt{33}}{2})}^{3}}{3}$

Step 3.12.2.1.2.8

Apply the product rule to $\frac{1 + \sqrt{33}}{2}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{{(1 + \sqrt{33})}^{3}}{2^{3}}}{3}$

Step 3.12.2.1.2.9

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

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{{(1 + \sqrt{33})}^{3}}{8}}{3}$

Step 3.12.2.1.2.10

Use the Binomial Theorem.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1^{3} + 3 \cdot 1^{2} \sqrt{33} + 3 \cdot 1 {\sqrt{33}}^{2} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11

Simplify each term.

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Step 3.12.2.1.2.11.1

One to any power is one.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \cdot 1^{2} \sqrt{33} + 3 \cdot 1 {\sqrt{33}}^{2} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.2

One to any power is one.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \cdot 1 \sqrt{33} + 3 \cdot 1 {\sqrt{33}}^{2} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.3

Multiply $3$ by $1$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 3 \cdot 1 {\sqrt{33}}^{2} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.4

Multiply $3$ by $1$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 3 {\sqrt{33}}^{2} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.5

Rewrite ${\sqrt{33}}^{2}$ as $33$ .

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Step 3.12.2.1.2.11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{33}$ as $33^{\frac{1}{2}}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 3 {(33^{\frac{1}{2}})}^{2} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.5.2

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

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 3 \cdot 33^{\frac{1}{2} \cdot 2} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.5.3

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

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 3 \cdot 33^{\frac{2}{2}} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.5.4

Cancel the common factor of $2$ .

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Step 3.12.2.1.2.11.5.4.1

Cancel the common factor.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 3 \cdot 33^{\frac{2}{2}} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.5.4.2

Rewrite the expression.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 3 \cdot 33^{1} + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.5.5

Evaluate the exponent.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 3 \cdot 33 + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.6

Multiply $3$ by $33$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 99 + {\sqrt{33}}^{3}}{8}}{3}$

Step 3.12.2.1.2.11.7

Rewrite ${\sqrt{33}}^{3}$ as $\sqrt{33^{3}}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 99 + \sqrt{33^{3}}}{8}}{3}$

Step 3.12.2.1.2.11.8

Raise $33$ to the power of $3$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 99 + \sqrt{35937}}{8}}{3}$

Step 3.12.2.1.2.11.9

Rewrite $35937$ as $33^{2} \cdot 33$ .

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Step 3.12.2.1.2.11.9.1

Factor $1089$ out of $35937$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 99 + \sqrt{1089 (33)}}{8}}{3}$

Step 3.12.2.1.2.11.9.2

Rewrite $1089$ as $33^{2}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 99 + \sqrt{33^{2} \cdot 33}}{8}}{3}$

Step 3.12.2.1.2.11.10

Pull terms out from under the radical.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{1 + 3 \sqrt{33} + 99 + 33 \sqrt{33}}{8}}{3}$

Step 3.12.2.1.2.12

Add $1$ and $99$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{100 + 3 \sqrt{33} + 33 \sqrt{33}}{8}}{3}$

Step 3.12.2.1.2.13

Add $3 \sqrt{33}$ and $33 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{100 + 36 \sqrt{33}}{8}}{3}$

Step 3.12.2.1.2.14

Cancel the common factor of $100 + 36 \sqrt{33}$ and $8$ .

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Step 3.12.2.1.2.14.1

Factor $4$ out of $100$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{4 (25) + 36 \sqrt{33}}{8}}{3}$

Step 3.12.2.1.2.14.2

Factor $4$ out of $36 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{4 (25) + 4 (9 \sqrt{33})}{8}}{3}$

Step 3.12.2.1.2.14.3

Factor $4$ out of $4 (25) + 4 (9 \sqrt{33})$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{4 (25 + 9 \sqrt{33})}{8}}{3}$

Step 3.12.2.1.2.14.4

Cancel the common factors.

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Step 3.12.2.1.2.14.4.1

Factor $4$ out of $8$ .

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{4 (25 + 9 \sqrt{33})}{4 \cdot 2}}{3}$

Step 3.12.2.1.2.14.4.2

Cancel the common factor.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{4 (25 + 9 \sqrt{33})}{4 \cdot 2}}{3}$

Step 3.12.2.1.2.14.4.3

Rewrite the expression.

$\frac{17 \sqrt{33}}{2} - \frac{- \frac{25 - 9 \sqrt{33}}{2} + \frac{25 + 9 \sqrt{33}}{2}}{3}$

Step 3.12.2.1.2.15

Combine the numerators over the common denominator.

$\frac{17 \sqrt{33}}{2} - \frac{\frac{- (25 - 9 \sqrt{33}) + 25 + 9 \sqrt{33}}{2}}{3}$

Step 3.12.2.1.2.16

Rewrite $\frac{- (25 - 9 \sqrt{33}) + 25 + 9 \sqrt{33}}{2}$ in a factored form.

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Step 3.12.2.1.2.16.1

Apply the distributive property.

$\frac{17 \sqrt{33}}{2} - \frac{\frac{- 1 \cdot 25 - (- 9 \sqrt{33}) + 25 + 9 \sqrt{33}}{2}}{3}$

Step 3.12.2.1.2.16.2

Multiply $- 1$ by $25$ .

$\frac{17 \sqrt{33}}{2} - \frac{\frac{- 25 - (- 9 \sqrt{33}) + 25 + 9 \sqrt{33}}{2}}{3}$

Step 3.12.2.1.2.16.3

Multiply $- 9$ by $- 1$ .

$\frac{17 \sqrt{33}}{2} - \frac{\frac{- 25 + 9 \sqrt{33} + 25 + 9 \sqrt{33}}{2}}{3}$

Step 3.12.2.1.2.16.4

Add $- 25$ and $25$ .

$\frac{17 \sqrt{33}}{2} - \frac{\frac{0 + 9 \sqrt{33} + 9 \sqrt{33}}{2}}{3}$

Step 3.12.2.1.2.16.5

Add $0$ and $9 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{\frac{9 \sqrt{33} + 9 \sqrt{33}}{2}}{3}$

Step 3.12.2.1.2.16.6

Add $9 \sqrt{33}$ and $9 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{\frac{18 \sqrt{33}}{2}}{3}$

Step 3.12.2.1.2.16.7

Reduce the expression by cancelling the common factors.

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Step 3.12.2.1.2.16.7.1

Reduce the expression $\frac{18 \sqrt{33}}{2}$ by cancelling the common factors.

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Step 3.12.2.1.2.16.7.1.1

Factor $2$ out of $18 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{\frac{2 (9 \sqrt{33})}{2}}{3}$

Step 3.12.2.1.2.16.7.1.2

Factor $2$ out of $2$ .

$\frac{17 \sqrt{33}}{2} - \frac{\frac{2 (9 \sqrt{33})}{2 (1)}}{3}$

Step 3.12.2.1.2.16.7.1.3

Cancel the common factor.

$\frac{17 \sqrt{33}}{2} - \frac{\frac{2 (9 \sqrt{33})}{2 \cdot 1}}{3}$

Step 3.12.2.1.2.16.7.1.4

Rewrite the expression.

$\frac{17 \sqrt{33}}{2} - \frac{\frac{9 \sqrt{33}}{1}}{3}$

Step 3.12.2.1.2.16.7.2

Divide $9 \sqrt{33}$ by $1$ .

$\frac{17 \sqrt{33}}{2} - \frac{9 \sqrt{33}}{3}$

Step 3.12.2.1.3

Cancel the common factor of $9$ and $3$ .

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Step 3.12.2.1.3.1

Factor $3$ out of $9 \sqrt{33}$ .

$\frac{17 \sqrt{33}}{2} - \frac{3 (3 \sqrt{33})}{3}$

Step 3.12.2.1.3.2

Cancel the common factors.

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Step 3.12.2.1.3.2.1

Factor $3$ out of $3$ .

$\frac{17 \sqrt{33}}{2} - \frac{3 (3 \sqrt{33})}{3 (1)}$

Step 3.12.2.1.3.2.2

Cancel the common factor.

$\frac{17 \sqrt{33}}{2} - \frac{3 (3 \sqrt{33})}{3 \cdot 1}$

Step 3.12.2.1.3.2.3

Rewrite the expression.

$\frac{17 \sqrt{33}}{2} - \frac{3 \sqrt{33}}{1}$

Step 3.12.2.1.3.2.4

Divide $3 \sqrt{33}$ by $1$ .

$\frac{17 \sqrt{33}}{2} - (3 \sqrt{33})$

Step 3.12.2.1.4

Multiply $3$ by $- 1$ .

$\frac{17 \sqrt{33}}{2} - 3 \sqrt{33}$

Step 3.12.2.2

To write $- 3 \sqrt{33}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{17 \sqrt{33}}{2} - 3 \sqrt{33} \cdot \frac{2}{2}$

Step 3.12.2.3

Combine $- 3 \sqrt{33}$ and $\frac{2}{2}$ .

$\frac{17 \sqrt{33}}{2} + \frac{- 3 \sqrt{33} \cdot 2}{2}$

Step 3.12.2.4

Combine the numerators over the common denominator.

$\frac{17 \sqrt{33} - 3 \sqrt{33} \cdot 2}{2}$

Step 3.12.2.5

Simplify the numerator.

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Step 3.12.2.5.1

Multiply $2$ by $- 3$ .

$\frac{17 \sqrt{33} - 6 \sqrt{33}}{2}$

Step 3.12.2.5.2

Subtract $6 \sqrt{33}$ from $17 \sqrt{33}$ .

$\frac{11 \sqrt{33}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{11 \sqrt{33}}{2}$

Decimal Form:

$31.59509455 \dots$

Step 5