Calculus Examples

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

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

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Step 1.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

Move $- 2$ to the left of $x$ .

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

Step 1.1.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.1.3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 1.1.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3} (x^{2} - 4 x + 4)$ and $g (x) = x - 4$ .

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

Step 1.1.5

Differentiate.

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Step 1.1.5.1

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

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

Step 1.1.5.2

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

$x^{3} (x^{2} - 4 x + 4) (1 + \frac{d}{d x} [- 4]) + (x - 4) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.1.5.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$x^{3} (x^{2} - 4 x + 4) (1 + 0) + (x - 4) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.1.5.4

Simplify the expression.

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Step 1.1.5.4.1

Add $1$ and $0$ .

$x^{3} (x^{2} - 4 x + 4) \cdot 1 + (x - 4) \frac{d}{d x} [x^{3} (x^{2} - 4 x + 4)]$

Step 1.1.5.4.2

Multiply $x^{3}$ by $1$ .

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

Step 1.1.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3}$ and $g (x) = x^{2} - 4 x + 4$ .

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

Step 1.1.7

Differentiate.

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Step 1.1.7.1

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

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

Step 1.1.7.2

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

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

Step 1.1.7.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

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

Step 1.1.7.4

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

$x^{3} (x^{2} - 4 x + 4) + (x - 4) (x^{3} (2 x - 4 \cdot 1 + \frac{d}{d x} [4]) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.1.7.5

Multiply $- 4$ by $1$ .

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

Step 1.1.7.6

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$x^{3} (x^{2} - 4 x + 4) + (x - 4) (x^{3} (2 x - 4 + 0) + (x^{2} - 4 x + 4) \frac{d}{d x} [x^{3}])$

Step 1.1.7.7

Add $2 x - 4$ and $0$ .

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

Step 1.1.7.8

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

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

Step 1.1.7.9

Move $3$ to the left of $x^{2} - 4 x + 4$ .

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

Step 1.1.8

Simplify.

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Step 1.1.8.1

Apply the distributive property.

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

Step 1.1.8.2

Apply the distributive property.

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

Step 1.1.8.3

Apply the distributive property.

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

Step 1.1.8.4

Apply the distributive property.

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

Step 1.1.8.5

Apply the distributive property.

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

Step 1.1.8.6

Apply the distributive property.

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

Step 1.1.8.7

Apply the distributive property.

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

Step 1.1.8.8

Apply the distributive property.

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

Step 1.1.8.9

Apply the distributive property.

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

Step 1.1.8.10

Apply the distributive property.

Step 1.1.8.11

Combine terms.

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Step 1.1.8.11.1

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

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Step 1.1.8.11.1.1

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

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

Step 1.1.8.11.1.2

Add $3$ and $2$ .

$x^{5} + x^{3} (- 4 x) + x^{3} \cdot 4 + x (x^{3} (2 x)) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.2

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

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Step 1.1.8.11.2.1

Move $x$ .

$x^{5} + x \cdot x^{3} \cdot - 4 + x^{3} \cdot 4 + x (x^{3} (2 x)) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.2.2

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

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Step 1.1.8.11.2.2.1

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

$x^{5} + x^{1} x^{3} \cdot - 4 + x^{3} \cdot 4 + x (x^{3} (2 x)) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.2.2.2

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

$x^{5} + x^{1 + 3} \cdot - 4 + x^{3} \cdot 4 + x (x^{3} (2 x)) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.2.3

Add $1$ and $3$ .

$x^{5} + x^{4} \cdot - 4 + x^{3} \cdot 4 + x (x^{3} (2 x)) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.3

Move $- 4$ to the left of $x^{4}$ .

$x^{5} - 4 \cdot x^{4} + x^{3} \cdot 4 + x (x^{3} (2 x)) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.4

Move $4$ to the left of $x^{3}$ .

$x^{5} - 4 x^{4} + 4 \cdot x^{3} + x (x^{3} (2 x)) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.5

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

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Step 1.1.8.11.5.1

Move $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + x (x \cdot x^{3} \cdot 2) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.5.2

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

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Step 1.1.8.11.5.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + x (x^{1} x^{3} \cdot 2) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.5.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + x (x^{1 + 3} \cdot 2) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.5.3

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + x (x^{4} \cdot 2) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.6

Move $2$ to the left of $x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + x (2 \cdot x^{4}) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.7

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 (x^{1} x^{4}) - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.8

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{1 + 4} - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.9

Add $1$ and $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 4 (x^{3} (2 x)) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.10

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

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Step 1.1.8.11.10.1

Move $x$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 4 (x \cdot x^{3} \cdot 2) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.10.2

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

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Step 1.1.8.11.10.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 4 (x^{1} x^{3} \cdot 2) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.10.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 4 (x^{1 + 3} \cdot 2) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.10.3

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 4 (x^{4} \cdot 2) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.11

Move $2$ to the left of $x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 4 (2 \cdot x^{4}) + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.12

Multiply $2$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 8 x^{4} + x (x^{3} \cdot - 4) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.13

Move $- 4$ to the left of $x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 8 x^{4} + x (- 4 \cdot x^{3}) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.14

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 8 x^{4} - 4 (x^{1} x^{3}) - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.15

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 8 x^{4} - 4 x^{1 + 3} - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.16

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 8 x^{4} - 4 x^{4} - 4 (x^{3} \cdot - 4) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.17

Move $- 4$ to the left of $x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 8 x^{4} - 4 x^{4} - 4 (- 4 \cdot x^{3}) + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.18

Multiply $- 4$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 8 x^{4} - 4 x^{4} + 16 x^{3} + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.19

Subtract $4 x^{4}$ from $- 8 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + x (3 x^{2} x^{2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.20

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

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Step 1.1.8.11.20.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + x (3 (x^{2} x^{2})) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.20.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + x (3 x^{2 + 2}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.20.3

Add $2$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + x (3 x^{4}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.21

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 (x^{1} x^{4}) - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.22

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{1 + 4} - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.23

Add $1$ and $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 4 (3 x^{2} x^{2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.24

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

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Step 1.1.8.11.24.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 4 (3 (x^{2} x^{2})) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.24.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 4 (3 x^{2 + 2}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.24.3

Add $2$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 4 (3 x^{4}) + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.25

Multiply $3$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} + x (3 (- 4 x) x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.26

Multiply $- 4$ by $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} + x (- 12 x \cdot x^{2}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.27

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

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Step 1.1.8.11.27.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} + x (- 12 (x^{2} x)) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.27.2

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

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Step 1.1.8.11.27.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} + x (- 12 (x^{2} x^{1})) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.27.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} + x (- 12 x^{2 + 1}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.27.3

Add $2$ and $1$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} + x (- 12 x^{3}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.28

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 (x^{1} x^{3}) - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.29

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 x^{1 + 3} - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.30

Add $1$ and $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 x^{4} - 4 (3 (- 4 x) x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.31

Multiply $- 4$ by $3$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 x^{4} - 4 (- 12 x \cdot x^{2}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.32

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

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Step 1.1.8.11.32.1

Move $x^{2}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 x^{4} - 4 (- 12 (x^{2} x)) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.32.2

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

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Step 1.1.8.11.32.2.1

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 x^{4} - 4 (- 12 (x^{2} x^{1})) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.32.2.2

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 x^{4} - 4 (- 12 x^{2 + 1}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.32.3

Add $2$ and $1$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 x^{4} - 4 (- 12 x^{3}) + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.33

Multiply $- 12$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 12 x^{4} - 12 x^{4} + 48 x^{3} + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.34

Subtract $12 x^{4}$ from $- 12 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 24 x^{4} + 48 x^{3} + x (3 \cdot 4 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.35

Multiply $3$ by $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 24 x^{4} + 48 x^{3} + x (12 x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.36

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 24 x^{4} + 48 x^{3} + 12 (x^{1} x^{2}) - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.37

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

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 24 x^{4} + 48 x^{3} + 12 x^{1 + 2} - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.38

Add $1$ and $2$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 24 x^{4} + 48 x^{3} + 12 x^{3} - 4 (3 \cdot 4 x^{2})$

Step 1.1.8.11.39

Multiply $3$ by $4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 24 x^{4} + 48 x^{3} + 12 x^{3} - 4 (12 x^{2})$

Step 1.1.8.11.40

Multiply $12$ by $- 4$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 24 x^{4} + 48 x^{3} + 12 x^{3} - 48 x^{2}$

Step 1.1.8.11.41

Add $48 x^{3}$ and $12 x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 2 x^{5} - 12 x^{4} + 16 x^{3} + 3 x^{5} - 24 x^{4} + 60 x^{3} - 48 x^{2}$

Step 1.1.8.11.42

Add $2 x^{5}$ and $3 x^{5}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} - 12 x^{4} + 16 x^{3} - 24 x^{4} + 60 x^{3} - 48 x^{2}$

Step 1.1.8.11.43

Subtract $24 x^{4}$ from $- 12 x^{4}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} - 36 x^{4} + 16 x^{3} + 60 x^{3} - 48 x^{2}$

Step 1.1.8.11.44

Add $16 x^{3}$ and $60 x^{3}$ .

$x^{5} - 4 x^{4} + 4 x^{3} + 5 x^{5} - 36 x^{4} + 76 x^{3} - 48 x^{2}$

Step 1.1.8.11.45

Add $x^{5}$ and $5 x^{5}$ .

$6 x^{5} - 4 x^{4} + 4 x^{3} - 36 x^{4} + 76 x^{3} - 48 x^{2}$

Step 1.1.8.11.46

Subtract $36 x^{4}$ from $- 4 x^{4}$ .

$6 x^{5} - 40 x^{4} + 4 x^{3} + 76 x^{3} - 48 x^{2}$

Step 1.1.8.11.47

Add $4 x^{3}$ and $76 x^{3}$ .

$f' (x) = 6 x^{5} - 40 x^{4} + 80 x^{3} - 48 x^{2}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $6 x^{5} - 40 x^{4} + 80 x^{3} - 48 x^{2}$ .

$6 x^{5} - 40 x^{4} + 80 x^{3} - 48 x^{2}$

Step 2

Set the first derivative equal to $0$ then solve the equation $6 x^{5} - 40 x^{4} + 80 x^{3} - 48 x^{2} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$6 x^{5} - 40 x^{4} + 80 x^{3} - 48 x^{2} = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $2 x^{2}$ out of $6 x^{5} - 40 x^{4} + 80 x^{3} - 48 x^{2}$ .

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Step 2.2.1.1

Factor $2 x^{2}$ out of $6 x^{5}$ .

$2 x^{2} (3 x^{3}) - 40 x^{4} + 80 x^{3} - 48 x^{2} = 0$

Step 2.2.1.2

Factor $2 x^{2}$ out of $- 40 x^{4}$ .

$2 x^{2} (3 x^{3}) + 2 x^{2} (- 20 x^{2}) + 80 x^{3} - 48 x^{2} = 0$

Step 2.2.1.3

Factor $2 x^{2}$ out of $80 x^{3}$ .

$2 x^{2} (3 x^{3}) + 2 x^{2} (- 20 x^{2}) + 2 x^{2} (40 x) - 48 x^{2} = 0$

Step 2.2.1.4

Factor $2 x^{2}$ out of $- 48 x^{2}$ .

$2 x^{2} (3 x^{3}) + 2 x^{2} (- 20 x^{2}) + 2 x^{2} (40 x) + 2 x^{2} (- 24) = 0$

Step 2.2.1.5

Factor $2 x^{2}$ out of $2 x^{2} (3 x^{3}) + 2 x^{2} (- 20 x^{2})$ .

$2 x^{2} (3 x^{3} - 20 x^{2}) + 2 x^{2} (40 x) + 2 x^{2} (- 24) = 0$

Step 2.2.1.6

Factor $2 x^{2}$ out of $2 x^{2} (3 x^{3} - 20 x^{2}) + 2 x^{2} (40 x)$ .

$2 x^{2} (3 x^{3} - 20 x^{2} + 40 x) + 2 x^{2} (- 24) = 0$

Step 2.2.1.7

Factor $2 x^{2}$ out of $2 x^{2} (3 x^{3} - 20 x^{2} + 40 x) + 2 x^{2} (- 24)$ .

$2 x^{2} (3 x^{3} - 20 x^{2} + 40 x - 24) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $3 x^{3} - 20 x^{2} + 40 x - 24$ using the rational roots test.

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Step 2.2.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 24, \pm 2, \pm 12, \pm 3, \pm 8, \pm 4, \pm 6$

$q = \pm 1, \pm 3$

Step 2.2.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.\overline{3}, \pm 24, \pm 8, \pm 2, \pm 0.\overline{6}, \pm 12, \pm 4, \pm 3, \pm 2.\overline{6}, \pm 1.\overline{3}, \pm 6$

Step 2.2.2.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 2.2.2.1.3.1

Substitute $2$ into the polynomial.

$3 \cdot 2^{3} - 20 \cdot 2^{2} + 40 \cdot 2 - 24$

Step 2.2.2.1.3.2

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

$3 \cdot 8 - 20 \cdot 2^{2} + 40 \cdot 2 - 24$

Step 2.2.2.1.3.3

Multiply $3$ by $8$ .

$24 - 20 \cdot 2^{2} + 40 \cdot 2 - 24$

Step 2.2.2.1.3.4

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

$24 - 20 \cdot 4 + 40 \cdot 2 - 24$

Step 2.2.2.1.3.5

Multiply $- 20$ by $4$ .

$24 - 80 + 40 \cdot 2 - 24$

Step 2.2.2.1.3.6

Subtract $80$ from $24$ .

$- 56 + 40 \cdot 2 - 24$

Step 2.2.2.1.3.7

Multiply $40$ by $2$ .

$- 56 + 80 - 24$

Step 2.2.2.1.3.8

Add $- 56$ and $80$ .

$24 - 24$

Step 2.2.2.1.3.9

Subtract $24$ from $24$ .

$0$

Step 2.2.2.1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{3 x^{3} - 20 x^{2} + 40 x - 24}{x - 2}$

Step 2.2.2.1.5

Divide $3 x^{3} - 20 x^{2} + 40 x - 24$ by $x - 2$ .

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Step 2.2.2.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$2$

$3 x^{3}$

$20 x^{2}$

$40 x$

$24$

Step 2.2.2.1.5.2

Divide the highest order term in the dividend $3 x^{3}$ by the highest order term in divisor $x$ .

					$3 x^{2}$
	$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$

Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$3 x^{2}$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			+	$3 x^{3}$	-	$6 x^{2}$

Step 2.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{3} - 6 x^{2}$

				$3 x^{2}$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$

Step 2.2.2.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$3 x^{2}$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$

Step 2.2.2.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$3 x^{2}$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$

Step 2.2.2.1.5.7

Divide the highest order term in the dividend $- 14 x^{2}$ by the highest order term in divisor $x$ .

				$3 x^{2}$	-	$14 x$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$

Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$14 x$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$
					-	$14 x^{2}$	+	$28 x$

Step 2.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{2} + 28 x$

				$3 x^{2}$	-	$14 x$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$
					+	$14 x^{2}$	-	$28 x$

Step 2.2.2.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$3 x^{2}$	-	$14 x$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$
					+	$14 x^{2}$	-	$28 x$
							+	$12 x$

Step 2.2.2.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$3 x^{2}$	-	$14 x$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$
					+	$14 x^{2}$	-	$28 x$
							+	$12 x$	-	$24$

Step 2.2.2.1.5.12

Divide the highest order term in the dividend $12 x$ by the highest order term in divisor $x$ .

				$3 x^{2}$	-	$14 x$	+	$12$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$
					+	$14 x^{2}$	-	$28 x$
							+	$12 x$	-	$24$

Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$3 x^{2}$	-	$14 x$	+	$12$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$
					+	$14 x^{2}$	-	$28 x$
							+	$12 x$	-	$24$
							+	$12 x$	-	$24$

Step 2.2.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $12 x - 24$

				$3 x^{2}$	-	$14 x$	+	$12$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$
					+	$14 x^{2}$	-	$28 x$
							+	$12 x$	-	$24$
							-	$12 x$	+	$24$

Step 2.2.2.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$3 x^{2}$	-	$14 x$	+	$12$
$x$	-	$2$		$3 x^{3}$	-	$20 x^{2}$	+	$40 x$	-	$24$
			-	$3 x^{3}$	+	$6 x^{2}$
					-	$14 x^{2}$	+	$40 x$
					+	$14 x^{2}$	-	$28 x$
							+	$12 x$	-	$24$
							-	$12 x$	+	$24$
										$0$

Step 2.2.2.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$3 x^{2} - 14 x + 12$

Step 2.2.2.1.6

Write $3 x^{3} - 20 x^{2} + 40 x - 24$ as a set of factors.

$2 x^{2} ((x - 2) (3 x^{2} - 14 x + 12)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$2 x^{2} (x - 2) (3 x^{2} - 14 x + 12) = 0$

Step 2.3

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

$x^{2} = 0$

$x - 2 = 0$

$3 x^{2} - 14 x + 12 = 0$

Step 2.4

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 2.4.2

Solve $x^{2} = 0$ for $x$ .

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Step 2.4.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.6

Set $3 x^{2} - 14 x + 12$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $3 x^{2} - 14 x + 12$ equal to $0$ .

$3 x^{2} - 14 x + 12 = 0$

Step 2.6.2

Solve $3 x^{2} - 14 x + 12 = 0$ for $x$ .

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Step 2.6.2.1

Use the quadratic formula to find the solutions.

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

Step 2.6.2.2

Substitute the values $a = 3$ , $b = - 14$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (3 \cdot 12)}}{2 \cdot 3}$

Step 2.6.2.3

Simplify.

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Step 2.6.2.3.1

Simplify the numerator.

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Step 2.6.2.3.1.1

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 12}}{2 \cdot 3}$

Step 2.6.2.3.1.2

Multiply $- 4 \cdot 3 \cdot 12$ .

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Step 2.6.2.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 12}}{2 \cdot 3}$

Step 2.6.2.3.1.2.2

Multiply $- 12$ by $12$ .

$x = \frac{14 \pm \sqrt{196 - 144}}{2 \cdot 3}$

Step 2.6.2.3.1.3

Subtract $144$ from $196$ .

$x = \frac{14 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.6.2.3.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 2.6.2.3.1.4.1

Factor $4$ out of $52$ .

$x = \frac{14 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.6.2.3.1.4.2

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.6.2.3.1.5

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.6.2.3.2

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{13}}{6}$

Step 2.6.2.3.3

Simplify $\frac{14 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{7 \pm \sqrt{13}}{3}$

Step 2.6.2.4

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

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Step 2.6.2.4.1

Simplify the numerator.

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Step 2.6.2.4.1.1

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 12}}{2 \cdot 3}$

Step 2.6.2.4.1.2

Multiply $- 4 \cdot 3 \cdot 12$ .

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Step 2.6.2.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 12}}{2 \cdot 3}$

Step 2.6.2.4.1.2.2

Multiply $- 12$ by $12$ .

$x = \frac{14 \pm \sqrt{196 - 144}}{2 \cdot 3}$

Step 2.6.2.4.1.3

Subtract $144$ from $196$ .

$x = \frac{14 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.6.2.4.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 2.6.2.4.1.4.1

Factor $4$ out of $52$ .

$x = \frac{14 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.6.2.4.1.4.2

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.6.2.4.1.5

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.6.2.4.2

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{13}}{6}$

Step 2.6.2.4.3

Simplify $\frac{14 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{7 \pm \sqrt{13}}{3}$

Step 2.6.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{7 + \sqrt{13}}{3}$

Step 2.6.2.5

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

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Step 2.6.2.5.1

Simplify the numerator.

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Step 2.6.2.5.1.1

Raise $- 14$ to the power of $2$ .

$x = \frac{14 \pm \sqrt{196 - 4 \cdot 3 \cdot 12}}{2 \cdot 3}$

Step 2.6.2.5.1.2

Multiply $- 4 \cdot 3 \cdot 12$ .

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Step 2.6.2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{14 \pm \sqrt{196 - 12 \cdot 12}}{2 \cdot 3}$

Step 2.6.2.5.1.2.2

Multiply $- 12$ by $12$ .

$x = \frac{14 \pm \sqrt{196 - 144}}{2 \cdot 3}$

Step 2.6.2.5.1.3

Subtract $144$ from $196$ .

$x = \frac{14 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.6.2.5.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 2.6.2.5.1.4.1

Factor $4$ out of $52$ .

$x = \frac{14 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.6.2.5.1.4.2

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

$x = \frac{14 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.6.2.5.1.5

Pull terms out from under the radical.

$x = \frac{14 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.6.2.5.2

Multiply $2$ by $3$ .

$x = \frac{14 \pm 2 \sqrt{13}}{6}$

Step 2.6.2.5.3

Simplify $\frac{14 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{7 \pm \sqrt{13}}{3}$

Step 2.6.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{7 - \sqrt{13}}{3}$

Step 2.6.2.6

The final answer is the combination of both solutions.

$x = \frac{7 + \sqrt{13}}{3}, \frac{7 - \sqrt{13}}{3}$

Step 2.7

The final solution is all the values that make $2 x^{2} (x - 2) (3 x^{2} - 14 x + 12) = 0$ true.

$x = 0, 2, \frac{7 + \sqrt{13}}{3}, \frac{7 - \sqrt{13}}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

Step 4

Evaluate $x^{3} {(x - 2)}^{2} (x - 4)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

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

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

Step 4.1.2.2

Subtract $2$ from $0$ .

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

Step 4.1.2.3

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

$0 \cdot 4 ((0) - 4)$

Step 4.1.2.4

Multiply $0$ by $4$ .

$0 ((0) - 4)$

Step 4.1.2.5

Subtract $4$ from $0$ .

$0 \cdot - 4$

Step 4.1.2.6

Multiply $0$ by $- 4$ .

$0$

Step 4.2

Evaluate at $x = 2$ .

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Step 4.2.1

Substitute $2$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$8 {((2) - 2)}^{2} ((2) - 4)$

Step 4.2.2.2

Subtract $2$ from $2$ .

$8 \cdot 0^{2} ((2) - 4)$

Step 4.2.2.3

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

$8 \cdot 0 ((2) - 4)$

Step 4.2.2.4

Multiply $8$ by $0$ .

$0 ((2) - 4)$

Step 4.2.2.5

Subtract $4$ from $2$ .

$0 \cdot - 2$

Step 4.2.2.6

Multiply $0$ by $- 2$ .

$0$

Step 4.3

Evaluate at $x = \frac{7 + \sqrt{13}}{3}$ .

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Step 4.3.1

Substitute $\frac{7 + \sqrt{13}}{3}$ for $x$ .

${(\frac{7 + \sqrt{13}}{3})}^{3} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify the expression.

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Step 4.3.2.1.1

Apply the product rule to $\frac{7 + \sqrt{13}}{3}$ .

$\frac{{(7 + \sqrt{13})}^{3}}{3^{3}} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.1.2

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

$\frac{{(7 + \sqrt{13})}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.2

Use the Binomial Theorem.

$\frac{7^{3} + 3 \cdot 7^{2} \sqrt{13} + 3 \cdot 7 {\sqrt{13}}^{2} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3

Simplify terms.

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Step 4.3.2.3.1

Simplify each term.

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Step 4.3.2.3.1.1

Raise $7$ to the power of $3$ .

$\frac{343 + 3 \cdot 7^{2} \sqrt{13} + 3 \cdot 7 {\sqrt{13}}^{2} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.2

Raise $7$ to the power of $2$ .

$\frac{343 + 3 \cdot 49 \sqrt{13} + 3 \cdot 7 {\sqrt{13}}^{2} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.3

Multiply $3$ by $49$ .

$\frac{343 + 147 \sqrt{13} + 3 \cdot 7 {\sqrt{13}}^{2} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.4

Multiply $3$ by $7$ .

$\frac{343 + 147 \sqrt{13} + 21 {\sqrt{13}}^{2} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.5

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.3.2.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{343 + 147 \sqrt{13} + 21 {(13^{\frac{1}{2}})}^{2} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.5.2

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

$\frac{343 + 147 \sqrt{13} + 21 \cdot 13^{\frac{1}{2} \cdot 2} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.5.3

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

$\frac{343 + 147 \sqrt{13} + 21 \cdot 13^{\frac{2}{2}} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.3.1.5.4.1

Cancel the common factor.

$\frac{343 + 147 \sqrt{13} + 21 \cdot 13^{\frac{2}{2}} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.5.4.2

Rewrite the expression.

$\frac{343 + 147 \sqrt{13} + 21 \cdot 13^{1} + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.5.5

Evaluate the exponent.

$\frac{343 + 147 \sqrt{13} + 21 \cdot 13 + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.6

Multiply $21$ by $13$ .

$\frac{343 + 147 \sqrt{13} + 273 + {\sqrt{13}}^{3}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.7

Rewrite ${\sqrt{13}}^{3}$ as $\sqrt{13^{3}}$ .

$\frac{343 + 147 \sqrt{13} + 273 + \sqrt{13^{3}}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.8

Raise $13$ to the power of $3$ .

$\frac{343 + 147 \sqrt{13} + 273 + \sqrt{2197}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.9

Rewrite $2197$ as $13^{2} \cdot 13$ .

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Step 4.3.2.3.1.9.1

Factor $169$ out of $2197$ .

$\frac{343 + 147 \sqrt{13} + 273 + \sqrt{169 (13)}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.9.2

Rewrite $169$ as $13^{2}$ .

$\frac{343 + 147 \sqrt{13} + 273 + \sqrt{13^{2} \cdot 13}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.1.10

Pull terms out from under the radical.

$\frac{343 + 147 \sqrt{13} + 273 + 13 \sqrt{13}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.2

Simplify by adding terms.

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Step 4.3.2.3.2.1

Add $343$ and $273$ .

$\frac{616 + 147 \sqrt{13} + 13 \sqrt{13}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.3.2.2

Add $147 \sqrt{13}$ and $13 \sqrt{13}$ .

$\frac{616 + 160 \sqrt{13}}{27} {((\frac{7 + \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.4

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

$\frac{616 + 160 \sqrt{13}}{27} {(\frac{7 + \sqrt{13}}{3} - 2 \cdot \frac{3}{3})}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.5

Combine fractions.

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Step 4.3.2.5.1

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

$\frac{616 + 160 \sqrt{13}}{27} {(\frac{7 + \sqrt{13}}{3} + \frac{- 2 \cdot 3}{3})}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.5.2

Combine the numerators over the common denominator.

$\frac{616 + 160 \sqrt{13}}{27} {(\frac{7 + \sqrt{13} - 2 \cdot 3}{3})}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.6

Simplify the numerator.

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Step 4.3.2.6.1

Multiply $- 2$ by $3$ .

$\frac{616 + 160 \sqrt{13}}{27} {(\frac{7 + \sqrt{13} - 6}{3})}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.6.2

Subtract $6$ from $7$ .

$\frac{616 + 160 \sqrt{13}}{27} {(\frac{1 + \sqrt{13}}{3})}^{2} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.7

Simplify the expression.

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Step 4.3.2.7.1

Apply the product rule to $\frac{1 + \sqrt{13}}{3}$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{{(1 + \sqrt{13})}^{2}}{3^{2}} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.7.2

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

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{{(1 + \sqrt{13})}^{2}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.7.3

Rewrite ${(1 + \sqrt{13})}^{2}$ as $(1 + \sqrt{13}) (1 + \sqrt{13})$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{(1 + \sqrt{13}) (1 + \sqrt{13})}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.8

Expand $(1 + \sqrt{13}) (1 + \sqrt{13})$ using the FOIL Method.

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Step 4.3.2.8.1

Apply the distributive property.

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 (1 + \sqrt{13}) + \sqrt{13} (1 + \sqrt{13})}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.8.2

Apply the distributive property.

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 \cdot 1 + 1 \sqrt{13} + \sqrt{13} (1 + \sqrt{13})}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.8.3

Apply the distributive property.

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 \cdot 1 + 1 \sqrt{13} + \sqrt{13} \cdot 1 + \sqrt{13} \sqrt{13}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9

Simplify and combine like terms.

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Step 4.3.2.9.1

Simplify each term.

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Step 4.3.2.9.1.1

Multiply $1$ by $1$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 + 1 \sqrt{13} + \sqrt{13} \cdot 1 + \sqrt{13} \sqrt{13}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9.1.2

Multiply $\sqrt{13}$ by $1$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 + \sqrt{13} + \sqrt{13} \cdot 1 + \sqrt{13} \sqrt{13}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9.1.3

Multiply $\sqrt{13}$ by $1$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 + \sqrt{13} + \sqrt{13} + \sqrt{13} \sqrt{13}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9.1.4

Combine using the product rule for radicals.

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 + \sqrt{13} + \sqrt{13} + \sqrt{13 \cdot 13}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9.1.5

Multiply $13$ by $13$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 + \sqrt{13} + \sqrt{13} + \sqrt{169}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9.1.6

Rewrite $169$ as $13^{2}$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 + \sqrt{13} + \sqrt{13} + \sqrt{13^{2}}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9.1.7

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

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{1 + \sqrt{13} + \sqrt{13} + 13}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9.2

Add $1$ and $13$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{14 + \sqrt{13} + \sqrt{13}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.9.3

Add $\sqrt{13}$ and $\sqrt{13}$ .

$\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{14 + 2 \sqrt{13}}{9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.10

Multiply $\frac{616 + 160 \sqrt{13}}{27} \cdot \frac{14 + 2 \sqrt{13}}{9}$ .

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Step 4.3.2.10.1

Multiply $\frac{616 + 160 \sqrt{13}}{27}$ by $\frac{14 + 2 \sqrt{13}}{9}$ .

$\frac{(616 + 160 \sqrt{13}) (14 + 2 \sqrt{13})}{27 \cdot 9} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.10.2

Multiply $27$ by $9$ .

$\frac{(616 + 160 \sqrt{13}) (14 + 2 \sqrt{13})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.11

Expand $(616 + 160 \sqrt{13}) (14 + 2 \sqrt{13})$ using the FOIL Method.

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Step 4.3.2.11.1

Apply the distributive property.

$\frac{616 (14 + 2 \sqrt{13}) + 160 \sqrt{13} (14 + 2 \sqrt{13})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.11.2

Apply the distributive property.

$\frac{616 \cdot 14 + 616 (2 \sqrt{13}) + 160 \sqrt{13} (14 + 2 \sqrt{13})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.11.3

Apply the distributive property.

$\frac{616 \cdot 14 + 616 (2 \sqrt{13}) + 160 \sqrt{13} \cdot 14 + 160 \sqrt{13} (2 \sqrt{13})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12

Simplify and combine like terms.

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Step 4.3.2.12.1

Simplify each term.

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Step 4.3.2.12.1.1

Multiply $616$ by $14$ .

$\frac{8624 + 616 (2 \sqrt{13}) + 160 \sqrt{13} \cdot 14 + 160 \sqrt{13} (2 \sqrt{13})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.2

Multiply $2$ by $616$ .

$\frac{8624 + 1232 \sqrt{13} + 160 \sqrt{13} \cdot 14 + 160 \sqrt{13} (2 \sqrt{13})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.3

Multiply $14$ by $160$ .

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 160 \sqrt{13} (2 \sqrt{13})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.4

Multiply $160 \sqrt{13} (2 \sqrt{13})$ .

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Step 4.3.2.12.1.4.1

Multiply $2$ by $160$ .

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 \sqrt{13} \sqrt{13}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.4.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 ({\sqrt{13}}^{1} \sqrt{13})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.4.4

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

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 {\sqrt{13}}^{1 + 1}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.4.5

Add $1$ and $1$ .

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 {\sqrt{13}}^{2}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.5

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.3.2.12.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 {(13^{\frac{1}{2}})}^{2}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.5.2

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

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 \cdot 13^{\frac{1}{2} \cdot 2}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.5.3

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

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 \cdot 13^{\frac{2}{2}}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.12.1.5.4.1

Cancel the common factor.

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 \cdot 13^{\frac{2}{2}}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.5.4.2

Rewrite the expression.

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 \cdot 13^{1}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.5.5

Evaluate the exponent.

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 320 \cdot 13}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.1.6

Multiply $320$ by $13$ .

$\frac{8624 + 1232 \sqrt{13} + 2240 \sqrt{13} + 4160}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.2

Add $8624$ and $4160$ .

$\frac{12784 + 1232 \sqrt{13} + 2240 \sqrt{13}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.12.3

Add $1232 \sqrt{13}$ and $2240 \sqrt{13}$ .

$\frac{12784 + 3472 \sqrt{13}}{243} ((\frac{7 + \sqrt{13}}{3}) - 4)$

Step 4.3.2.13

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{12784 + 3472 \sqrt{13}}{243} (\frac{7 + \sqrt{13}}{3} - 4 \cdot \frac{3}{3})$

Step 4.3.2.14

Combine fractions.

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Step 4.3.2.14.1

Combine $- 4$ and $\frac{3}{3}$ .

$\frac{12784 + 3472 \sqrt{13}}{243} (\frac{7 + \sqrt{13}}{3} + \frac{- 4 \cdot 3}{3})$

Step 4.3.2.14.2

Combine the numerators over the common denominator.

$\frac{12784 + 3472 \sqrt{13}}{243} \cdot \frac{7 + \sqrt{13} - 4 \cdot 3}{3}$

Step 4.3.2.15

Simplify the numerator.

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Step 4.3.2.15.1

Multiply $- 4$ by $3$ .

$\frac{12784 + 3472 \sqrt{13}}{243} \cdot \frac{7 + \sqrt{13} - 12}{3}$

Step 4.3.2.15.2

Subtract $12$ from $7$ .

$\frac{12784 + 3472 \sqrt{13}}{243} \cdot \frac{- 5 + \sqrt{13}}{3}$

Step 4.3.2.16

Multiply $\frac{12784 + 3472 \sqrt{13}}{243} \cdot \frac{- 5 + \sqrt{13}}{3}$ .

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Step 4.3.2.16.1

Multiply $\frac{12784 + 3472 \sqrt{13}}{243}$ by $\frac{- 5 + \sqrt{13}}{3}$ .

$\frac{(12784 + 3472 \sqrt{13}) (- 5 + \sqrt{13})}{243 \cdot 3}$

Step 4.3.2.16.2

Multiply $243$ by $3$ .

$\frac{(12784 + 3472 \sqrt{13}) (- 5 + \sqrt{13})}{729}$

Step 4.3.2.17

Expand $(12784 + 3472 \sqrt{13}) (- 5 + \sqrt{13})$ using the FOIL Method.

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Step 4.3.2.17.1

Apply the distributive property.

$\frac{12784 (- 5 + \sqrt{13}) + 3472 \sqrt{13} (- 5 + \sqrt{13})}{729}$

Step 4.3.2.17.2

Apply the distributive property.

$\frac{12784 \cdot - 5 + 12784 \sqrt{13} + 3472 \sqrt{13} (- 5 + \sqrt{13})}{729}$

Step 4.3.2.17.3

Apply the distributive property.

$\frac{12784 \cdot - 5 + 12784 \sqrt{13} + 3472 \sqrt{13} \cdot - 5 + 3472 \sqrt{13} \sqrt{13}}{729}$

Step 4.3.2.18

Simplify and combine like terms.

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Step 4.3.2.18.1

Simplify each term.

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Step 4.3.2.18.1.1

Multiply $12784$ by $- 5$ .

$\frac{- 63920 + 12784 \sqrt{13} + 3472 \sqrt{13} \cdot - 5 + 3472 \sqrt{13} \sqrt{13}}{729}$

Step 4.3.2.18.1.2

Multiply $- 5$ by $3472$ .

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 \sqrt{13} \sqrt{13}}{729}$

Step 4.3.2.18.1.3

Multiply $3472 \sqrt{13} \sqrt{13}$ .

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Step 4.3.2.18.1.3.1

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 ({\sqrt{13}}^{1} \sqrt{13})}{729}$

Step 4.3.2.18.1.3.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{729}$

Step 4.3.2.18.1.3.3

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

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 {\sqrt{13}}^{1 + 1}}{729}$

Step 4.3.2.18.1.3.4

Add $1$ and $1$ .

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 {\sqrt{13}}^{2}}{729}$

Step 4.3.2.18.1.4

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.3.2.18.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 {(13^{\frac{1}{2}})}^{2}}{729}$

Step 4.3.2.18.1.4.2

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

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 \cdot 13^{\frac{1}{2} \cdot 2}}{729}$

Step 4.3.2.18.1.4.3

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

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 \cdot 13^{\frac{2}{2}}}{729}$

Step 4.3.2.18.1.4.4

Cancel the common factor of $2$ .

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Step 4.3.2.18.1.4.4.1

Cancel the common factor.

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 \cdot 13^{\frac{2}{2}}}{729}$

Step 4.3.2.18.1.4.4.2

Rewrite the expression.

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 \cdot 13^{1}}{729}$

Step 4.3.2.18.1.4.5

Evaluate the exponent.

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 3472 \cdot 13}{729}$

Step 4.3.2.18.1.5

Multiply $3472$ by $13$ .

$\frac{- 63920 + 12784 \sqrt{13} - 17360 \sqrt{13} + 45136}{729}$

Step 4.3.2.18.2

Add $- 63920$ and $45136$ .

$\frac{- 18784 + 12784 \sqrt{13} - 17360 \sqrt{13}}{729}$

Step 4.3.2.18.3

Subtract $17360 \sqrt{13}$ from $12784 \sqrt{13}$ .

$\frac{- 18784 - 4576 \sqrt{13}}{729}$

Step 4.3.2.19

Rewrite $- 18784$ as $- 1 (18784)$ .

$\frac{- 1 (18784) - 4576 \sqrt{13}}{729}$

Step 4.3.2.20

Factor $- 1$ out of $- 4576 \sqrt{13}$ .

$\frac{- 1 (18784) - (4576 \sqrt{13})}{729}$

Step 4.3.2.21

Factor $- 1$ out of $- 1 (18784) - (4576 \sqrt{13})$ .

$\frac{- 1 (18784 + 4576 \sqrt{13})}{729}$

Step 4.3.2.22

Move the negative in front of the fraction.

$- \frac{18784 + 4576 \sqrt{13}}{729}$

Step 4.4

Evaluate at $x = \frac{7 - \sqrt{13}}{3}$ .

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Step 4.4.1

Substitute $\frac{7 - \sqrt{13}}{3}$ for $x$ .

${(\frac{7 - \sqrt{13}}{3})}^{3} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2

Simplify.

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Step 4.4.2.1

Simplify the expression.

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Step 4.4.2.1.1

Apply the product rule to $\frac{7 - \sqrt{13}}{3}$ .

$\frac{{(7 - \sqrt{13})}^{3}}{3^{3}} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.1.2

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

$\frac{{(7 - \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.2

Use the Binomial Theorem.

$\frac{7^{3} + 3 \cdot 7^{2} (- \sqrt{13}) + 3 \cdot 7 {(- \sqrt{13})}^{2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3

Simplify terms.

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Step 4.4.2.3.1

Simplify each term.

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Step 4.4.2.3.1.1

Raise $7$ to the power of $3$ .

$\frac{343 + 3 \cdot 7^{2} (- \sqrt{13}) + 3 \cdot 7 {(- \sqrt{13})}^{2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.2

Raise $7$ to the power of $2$ .

$\frac{343 + 3 \cdot 49 (- \sqrt{13}) + 3 \cdot 7 {(- \sqrt{13})}^{2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.3

Multiply $3$ by $49$ .

$\frac{343 + 147 (- \sqrt{13}) + 3 \cdot 7 {(- \sqrt{13})}^{2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.4

Multiply $- 1$ by $147$ .

$\frac{343 - 147 \sqrt{13} + 3 \cdot 7 {(- \sqrt{13})}^{2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.5

Multiply $3$ by $7$ .

$\frac{343 - 147 \sqrt{13} + 21 {(- \sqrt{13})}^{2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.6

Apply the product rule to $- \sqrt{13}$ .

$\frac{343 - 147 \sqrt{13} + 21 ({(- 1)}^{2} {\sqrt{13}}^{2}) + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.7

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

$\frac{343 - 147 \sqrt{13} + 21 (1 {\sqrt{13}}^{2}) + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.8

Multiply ${\sqrt{13}}^{2}$ by $1$ .

$\frac{343 - 147 \sqrt{13} + 21 {\sqrt{13}}^{2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.9

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.4.2.3.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{343 - 147 \sqrt{13} + 21 {(13^{\frac{1}{2}})}^{2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.9.2

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

$\frac{343 - 147 \sqrt{13} + 21 \cdot 13^{\frac{1}{2} \cdot 2} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.9.3

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

$\frac{343 - 147 \sqrt{13} + 21 \cdot 13^{\frac{2}{2}} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.9.4

Cancel the common factor of $2$ .

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Step 4.4.2.3.1.9.4.1

Cancel the common factor.

$\frac{343 - 147 \sqrt{13} + 21 \cdot 13^{\frac{2}{2}} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.9.4.2

Rewrite the expression.

$\frac{343 - 147 \sqrt{13} + 21 \cdot 13^{1} + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.9.5

Evaluate the exponent.

$\frac{343 - 147 \sqrt{13} + 21 \cdot 13 + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.10

Multiply $21$ by $13$ .

$\frac{343 - 147 \sqrt{13} + 273 + {(- \sqrt{13})}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.11

Apply the product rule to $- \sqrt{13}$ .

$\frac{343 - 147 \sqrt{13} + 273 + {(- 1)}^{3} {\sqrt{13}}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.12

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

$\frac{343 - 147 \sqrt{13} + 273 - {\sqrt{13}}^{3}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.13

Rewrite ${\sqrt{13}}^{3}$ as $\sqrt{13^{3}}$ .

$\frac{343 - 147 \sqrt{13} + 273 - \sqrt{13^{3}}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.14

Raise $13$ to the power of $3$ .

$\frac{343 - 147 \sqrt{13} + 273 - \sqrt{2197}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.15

Rewrite $2197$ as $13^{2} \cdot 13$ .

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Step 4.4.2.3.1.15.1

Factor $169$ out of $2197$ .

$\frac{343 - 147 \sqrt{13} + 273 - \sqrt{169 (13)}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.15.2

Rewrite $169$ as $13^{2}$ .

$\frac{343 - 147 \sqrt{13} + 273 - \sqrt{13^{2} \cdot 13}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.16

Pull terms out from under the radical.

$\frac{343 - 147 \sqrt{13} + 273 - (13 \sqrt{13})}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.1.17

Multiply $13$ by $- 1$ .

$\frac{343 - 147 \sqrt{13} + 273 - 13 \sqrt{13}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.2

Simplify by adding terms.

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Step 4.4.2.3.2.1

Add $343$ and $273$ .

$\frac{616 - 147 \sqrt{13} - 13 \sqrt{13}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.3.2.2

Subtract $13 \sqrt{13}$ from $- 147 \sqrt{13}$ .

$\frac{616 - 160 \sqrt{13}}{27} {((\frac{7 - \sqrt{13}}{3}) - 2)}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.4

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

$\frac{616 - 160 \sqrt{13}}{27} {(\frac{7 - \sqrt{13}}{3} - 2 \cdot \frac{3}{3})}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.5

Combine fractions.

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Step 4.4.2.5.1

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

$\frac{616 - 160 \sqrt{13}}{27} {(\frac{7 - \sqrt{13}}{3} + \frac{- 2 \cdot 3}{3})}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.5.2

Combine the numerators over the common denominator.

$\frac{616 - 160 \sqrt{13}}{27} {(\frac{7 - \sqrt{13} - 2 \cdot 3}{3})}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.6

Simplify the numerator.

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Step 4.4.2.6.1

Multiply $- 2$ by $3$ .

$\frac{616 - 160 \sqrt{13}}{27} {(\frac{7 - \sqrt{13} - 6}{3})}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.6.2

Subtract $6$ from $7$ .

$\frac{616 - 160 \sqrt{13}}{27} {(\frac{1 - \sqrt{13}}{3})}^{2} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.7

Simplify the expression.

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Step 4.4.2.7.1

Apply the product rule to $\frac{1 - \sqrt{13}}{3}$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{{(1 - \sqrt{13})}^{2}}{3^{2}} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.7.2

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

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{{(1 - \sqrt{13})}^{2}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.7.3

Rewrite ${(1 - \sqrt{13})}^{2}$ as $(1 - \sqrt{13}) (1 - \sqrt{13})$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{(1 - \sqrt{13}) (1 - \sqrt{13})}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.8

Expand $(1 - \sqrt{13}) (1 - \sqrt{13})$ using the FOIL Method.

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Step 4.4.2.8.1

Apply the distributive property.

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 (1 - \sqrt{13}) - \sqrt{13} (1 - \sqrt{13})}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.8.2

Apply the distributive property.

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 \cdot 1 + 1 (- \sqrt{13}) - \sqrt{13} (1 - \sqrt{13})}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.8.3

Apply the distributive property.

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 \cdot 1 + 1 (- \sqrt{13}) - \sqrt{13} \cdot 1 - \sqrt{13} (- \sqrt{13})}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9

Simplify and combine like terms.

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Step 4.4.2.9.1

Simplify each term.

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Step 4.4.2.9.1.1

Multiply $1$ by $1$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 + 1 (- \sqrt{13}) - \sqrt{13} \cdot 1 - \sqrt{13} (- \sqrt{13})}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.2

Multiply $- \sqrt{13}$ by $1$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} \cdot 1 - \sqrt{13} (- \sqrt{13})}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.3

Multiply $- 1$ by $1$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} - \sqrt{13} (- \sqrt{13})}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.4

Multiply $- \sqrt{13} (- \sqrt{13})$ .

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Step 4.4.2.9.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + 1 \sqrt{13} \sqrt{13}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.4.2

Multiply $\sqrt{13}$ by $1$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + \sqrt{13} \sqrt{13}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + {\sqrt{13}}^{1} \sqrt{13}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.4.4

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + {\sqrt{13}}^{1} {\sqrt{13}}^{1}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.4.5

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

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + {\sqrt{13}}^{1 + 1}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.4.6

Add $1$ and $1$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + {\sqrt{13}}^{2}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.5

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.4.2.9.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + {(13^{\frac{1}{2}})}^{2}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.5.2

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

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + 13^{\frac{1}{2} \cdot 2}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.5.3

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

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + 13^{\frac{2}{2}}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.5.4

Cancel the common factor of $2$ .

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Step 4.4.2.9.1.5.4.1

Cancel the common factor.

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + 13^{\frac{2}{2}}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.5.4.2

Rewrite the expression.

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + 13^{1}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.1.5.5

Evaluate the exponent.

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{1 - \sqrt{13} - \sqrt{13} + 13}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.2

Add $1$ and $13$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{14 - \sqrt{13} - \sqrt{13}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.9.3

Subtract $\sqrt{13}$ from $- \sqrt{13}$ .

$\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{14 - 2 \sqrt{13}}{9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.10

Multiply $\frac{616 - 160 \sqrt{13}}{27} \cdot \frac{14 - 2 \sqrt{13}}{9}$ .

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Step 4.4.2.10.1

Multiply $\frac{616 - 160 \sqrt{13}}{27}$ by $\frac{14 - 2 \sqrt{13}}{9}$ .

$\frac{(616 - 160 \sqrt{13}) (14 - 2 \sqrt{13})}{27 \cdot 9} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.10.2

Multiply $27$ by $9$ .

$\frac{(616 - 160 \sqrt{13}) (14 - 2 \sqrt{13})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.11

Expand $(616 - 160 \sqrt{13}) (14 - 2 \sqrt{13})$ using the FOIL Method.

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Step 4.4.2.11.1

Apply the distributive property.

$\frac{616 (14 - 2 \sqrt{13}) - 160 \sqrt{13} (14 - 2 \sqrt{13})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.11.2

Apply the distributive property.

$\frac{616 \cdot 14 + 616 (- 2 \sqrt{13}) - 160 \sqrt{13} (14 - 2 \sqrt{13})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.11.3

Apply the distributive property.

$\frac{616 \cdot 14 + 616 (- 2 \sqrt{13}) - 160 \sqrt{13} \cdot 14 - 160 \sqrt{13} (- 2 \sqrt{13})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12

Simplify and combine like terms.

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Step 4.4.2.12.1

Simplify each term.

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Step 4.4.2.12.1.1

Multiply $616$ by $14$ .

$\frac{8624 + 616 (- 2 \sqrt{13}) - 160 \sqrt{13} \cdot 14 - 160 \sqrt{13} (- 2 \sqrt{13})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.2

Multiply $- 2$ by $616$ .

$\frac{8624 - 1232 \sqrt{13} - 160 \sqrt{13} \cdot 14 - 160 \sqrt{13} (- 2 \sqrt{13})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.3

Multiply $14$ by $- 160$ .

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} - 160 \sqrt{13} (- 2 \sqrt{13})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.4

Multiply $- 160 \sqrt{13} (- 2 \sqrt{13})$ .

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Step 4.4.2.12.1.4.1

Multiply $- 2$ by $- 160$ .

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 \sqrt{13} \sqrt{13}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.4.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 ({\sqrt{13}}^{1} \sqrt{13})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.4.4

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

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 {\sqrt{13}}^{1 + 1}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.4.5

Add $1$ and $1$ .

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 {\sqrt{13}}^{2}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.5

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.4.2.12.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 {(13^{\frac{1}{2}})}^{2}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.5.2

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

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 \cdot 13^{\frac{1}{2} \cdot 2}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.5.3

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

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 \cdot 13^{\frac{2}{2}}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.5.4

Cancel the common factor of $2$ .

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Step 4.4.2.12.1.5.4.1

Cancel the common factor.

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 \cdot 13^{\frac{2}{2}}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.5.4.2

Rewrite the expression.

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 \cdot 13^{1}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.5.5

Evaluate the exponent.

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 320 \cdot 13}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.1.6

Multiply $320$ by $13$ .

$\frac{8624 - 1232 \sqrt{13} - 2240 \sqrt{13} + 4160}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.2

Add $8624$ and $4160$ .

$\frac{12784 - 1232 \sqrt{13} - 2240 \sqrt{13}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.12.3

Subtract $2240 \sqrt{13}$ from $- 1232 \sqrt{13}$ .

$\frac{12784 - 3472 \sqrt{13}}{243} ((\frac{7 - \sqrt{13}}{3}) - 4)$

Step 4.4.2.13

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{12784 - 3472 \sqrt{13}}{243} (\frac{7 - \sqrt{13}}{3} - 4 \cdot \frac{3}{3})$

Step 4.4.2.14

Combine fractions.

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Step 4.4.2.14.1

Combine $- 4$ and $\frac{3}{3}$ .

$\frac{12784 - 3472 \sqrt{13}}{243} (\frac{7 - \sqrt{13}}{3} + \frac{- 4 \cdot 3}{3})$

Step 4.4.2.14.2

Combine the numerators over the common denominator.

$\frac{12784 - 3472 \sqrt{13}}{243} \cdot \frac{7 - \sqrt{13} - 4 \cdot 3}{3}$

Step 4.4.2.15

Simplify the numerator.

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Step 4.4.2.15.1

Multiply $- 4$ by $3$ .

$\frac{12784 - 3472 \sqrt{13}}{243} \cdot \frac{7 - \sqrt{13} - 12}{3}$

Step 4.4.2.15.2

Subtract $12$ from $7$ .

$\frac{12784 - 3472 \sqrt{13}}{243} \cdot \frac{- 5 - \sqrt{13}}{3}$

Step 4.4.2.16

Multiply $\frac{12784 - 3472 \sqrt{13}}{243} \cdot \frac{- 5 - \sqrt{13}}{3}$ .

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Step 4.4.2.16.1

Multiply $\frac{12784 - 3472 \sqrt{13}}{243}$ by $\frac{- 5 - \sqrt{13}}{3}$ .

$\frac{(12784 - 3472 \sqrt{13}) (- 5 - \sqrt{13})}{243 \cdot 3}$

Step 4.4.2.16.2

Multiply $243$ by $3$ .

$\frac{(12784 - 3472 \sqrt{13}) (- 5 - \sqrt{13})}{729}$

Step 4.4.2.17

Expand $(12784 - 3472 \sqrt{13}) (- 5 - \sqrt{13})$ using the FOIL Method.

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Step 4.4.2.17.1

Apply the distributive property.

$\frac{12784 (- 5 - \sqrt{13}) - 3472 \sqrt{13} (- 5 - \sqrt{13})}{729}$

Step 4.4.2.17.2

Apply the distributive property.

$\frac{12784 \cdot - 5 + 12784 (- \sqrt{13}) - 3472 \sqrt{13} (- 5 - \sqrt{13})}{729}$

Step 4.4.2.17.3

Apply the distributive property.

$\frac{12784 \cdot - 5 + 12784 (- \sqrt{13}) - 3472 \sqrt{13} \cdot - 5 - 3472 \sqrt{13} (- \sqrt{13})}{729}$

Step 4.4.2.18

Simplify and combine like terms.

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Step 4.4.2.18.1

Simplify each term.

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Step 4.4.2.18.1.1

Multiply $12784$ by $- 5$ .

$\frac{- 63920 + 12784 (- \sqrt{13}) - 3472 \sqrt{13} \cdot - 5 - 3472 \sqrt{13} (- \sqrt{13})}{729}$

Step 4.4.2.18.1.2

Multiply $- 1$ by $12784$ .

$\frac{- 63920 - 12784 \sqrt{13} - 3472 \sqrt{13} \cdot - 5 - 3472 \sqrt{13} (- \sqrt{13})}{729}$

Step 4.4.2.18.1.3

Multiply $- 5$ by $- 3472$ .

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} - 3472 \sqrt{13} (- \sqrt{13})}{729}$

Step 4.4.2.18.1.4

Multiply $- 3472 \sqrt{13} (- \sqrt{13})$ .

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Step 4.4.2.18.1.4.1

Multiply $- 1$ by $- 3472$ .

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 \sqrt{13} \sqrt{13}}{729}$

Step 4.4.2.18.1.4.2

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 ({\sqrt{13}}^{1} \sqrt{13})}{729}$

Step 4.4.2.18.1.4.3

Raise $\sqrt{13}$ to the power of $1$ .

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{729}$

Step 4.4.2.18.1.4.4

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

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 {\sqrt{13}}^{1 + 1}}{729}$

Step 4.4.2.18.1.4.5

Add $1$ and $1$ .

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 {\sqrt{13}}^{2}}{729}$

Step 4.4.2.18.1.5

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 4.4.2.18.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 {(13^{\frac{1}{2}})}^{2}}{729}$

Step 4.4.2.18.1.5.2

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

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 \cdot 13^{\frac{1}{2} \cdot 2}}{729}$

Step 4.4.2.18.1.5.3

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

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 \cdot 13^{\frac{2}{2}}}{729}$

Step 4.4.2.18.1.5.4

Cancel the common factor of $2$ .

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Step 4.4.2.18.1.5.4.1

Cancel the common factor.

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 \cdot 13^{\frac{2}{2}}}{729}$

Step 4.4.2.18.1.5.4.2

Rewrite the expression.

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 \cdot 13^{1}}{729}$

Step 4.4.2.18.1.5.5

Evaluate the exponent.

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 3472 \cdot 13}{729}$

Step 4.4.2.18.1.6

Multiply $3472$ by $13$ .

$\frac{- 63920 - 12784 \sqrt{13} + 17360 \sqrt{13} + 45136}{729}$

Step 4.4.2.18.2

Add $- 63920$ and $45136$ .

$\frac{- 18784 - 12784 \sqrt{13} + 17360 \sqrt{13}}{729}$

Step 4.4.2.18.3

Add $- 12784 \sqrt{13}$ and $17360 \sqrt{13}$ .

$\frac{- 18784 + 4576 \sqrt{13}}{729}$

Step 4.4.2.19

Rewrite $- 18784$ as $- 1 (18784)$ .

$\frac{- 1 (18784) + 4576 \sqrt{13}}{729}$

Step 4.4.2.20

Factor $- 1$ out of $4576 \sqrt{13}$ .

$\frac{- 1 (18784) - (- 4576 \sqrt{13})}{729}$

Step 4.4.2.21

Factor $- 1$ out of $- 1 (18784) - (- 4576 \sqrt{13})$ .

$\frac{- 1 (18784 - 4576 \sqrt{13})}{729}$

Step 4.4.2.22

Move the negative in front of the fraction.

$- \frac{18784 - 4576 \sqrt{13}}{729}$

Step 4.5

List all of the points.

$(0, 0), (2, 0), (\frac{7 + \sqrt{13}}{3}, - \frac{18784 + 4576 \sqrt{13}}{729}), (\frac{7 - \sqrt{13}}{3}, - \frac{18784 - 4576 \sqrt{13}}{729})$

Step 5