Calculus Examples

$x {(x - 3)}^{2} (x + 1)$

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 ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

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

Step 1.1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 1.1.3.1.2

Move $- 3$ to the left of $x$ .

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

Step 1.1.3.1.3

Multiply $- 3$ by $- 3$ .

$\frac{d}{d x} [x (x^{2} - 3 x - 3 x + 9) (x + 1)]$

Step 1.1.3.2

Subtract $3 x$ from $- 3 x$ .

$\frac{d}{d x} [x (x^{2} - 6 x + 9) (x + 1)]$

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 $f (x) = x (x^{2} - 6 x + 9)$ and $g (x) = x + 1$ .

$x (x^{2} - 6 x + 9) \frac{d}{d x} [x + 1] + (x + 1) \frac{d}{d x} [x (x^{2} - 6 x + 9)]$

Step 1.1.5

Differentiate.

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Step 1.1.5.1

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$x (x^{2} - 6 x + 9) (\frac{d}{d x} [x] + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x (x^{2} - 6 x + 9)]$

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 $n = 1$ .

$x (x^{2} - 6 x + 9) (1 + \frac{d}{d x} [1]) + (x + 1) \frac{d}{d x} [x (x^{2} - 6 x + 9)]$

Step 1.1.5.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$x (x^{2} - 6 x + 9) (1 + 0) + (x + 1) \frac{d}{d x} [x (x^{2} - 6 x + 9)]$

Step 1.1.5.4

Simplify the expression.

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Step 1.1.5.4.1

Add $1$ and $0$ .

$x (x^{2} - 6 x + 9) \cdot 1 + (x + 1) \frac{d}{d x} [x (x^{2} - 6 x + 9)]$

Step 1.1.5.4.2

Multiply $x$ by $1$ .

$x (x^{2} - 6 x + 9) + (x + 1) \frac{d}{d x} [x (x^{2} - 6 x + 9)]$

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 $f (x) = x$ and $g (x) = x^{2} - 6 x + 9$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x \frac{d}{d x} [x^{2} - 6 x + 9] + (x^{2} - 6 x + 9) \frac{d}{d x} [x])$

Step 1.1.7

Differentiate.

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Step 1.1.7.1

By the Sum Rule, the derivative of $x^{2} - 6 x + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]) + (x^{2} - 6 x + 9) \frac{d}{d x} [x])$

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 $n = 2$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [9]) + (x^{2} - 6 x + 9) \frac{d}{d x} [x])$

Step 1.1.7.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [9]) + (x^{2} - 6 x + 9) \frac{d}{d x} [x])$

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 $n = 1$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (2 x - 6 \cdot 1 + \frac{d}{d x} [9]) + (x^{2} - 6 x + 9) \frac{d}{d x} [x])$

Step 1.1.7.5

Multiply $- 6$ by $1$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (2 x - 6 + \frac{d}{d x} [9]) + (x^{2} - 6 x + 9) \frac{d}{d x} [x])$

Step 1.1.7.6

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (2 x - 6 + 0) + (x^{2} - 6 x + 9) \frac{d}{d x} [x])$

Step 1.1.7.7

Add $2 x - 6$ and $0$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (2 x - 6) + (x^{2} - 6 x + 9) \frac{d}{d x} [x])$

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 $n = 1$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (2 x - 6) + (x^{2} - 6 x + 9) \cdot 1)$

Step 1.1.7.9

Multiply $x^{2} - 6 x + 9$ by $1$ .

$x (x^{2} - 6 x + 9) + (x + 1) (x (2 x - 6) + x^{2} - 6 x + 9)$

Step 1.1.8

Simplify.

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Step 1.1.8.1

Apply the distributive property.

$x \cdot x^{2} + x (- 6 x) + x \cdot 9 + (x + 1) (x (2 x - 6) + x^{2} - 6 x + 9)$

Step 1.1.8.2

Apply the distributive property.

$x \cdot x^{2} + x (- 6 x) + x \cdot 9 + (x + 1) (x (2 x) + x \cdot - 6 + x^{2} - 6 x + 9)$

Step 1.1.8.3

Apply the distributive property.

$x \cdot x^{2} + x (- 6 x) + x \cdot 9 + (x + 1) (x (2 x)) + (x + 1) (x \cdot - 6) + (x + 1) x^{2} + (x + 1) (- 6 x) + (x + 1) \cdot 9$

Step 1.1.8.4

Apply the distributive property.

$x \cdot x^{2} + x (- 6 x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + (x + 1) (x \cdot - 6) + (x + 1) x^{2} + (x + 1) (- 6 x) + (x + 1) \cdot 9$

Step 1.1.8.5

Apply the distributive property.

$x \cdot x^{2} + x (- 6 x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + (x + 1) x^{2} + (x + 1) (- 6 x) + (x + 1) \cdot 9$

Step 1.1.8.6

Apply the distributive property.

$x \cdot x^{2} + x (- 6 x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + (x + 1) (- 6 x) + (x + 1) \cdot 9$

Step 1.1.8.7

Apply the distributive property.

$x \cdot x^{2} + x (- 6 x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + (x + 1) \cdot 9$

Step 1.1.8.8

Apply the distributive property.

$x \cdot x^{2} + x (- 6 x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9

Combine terms.

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Step 1.1.8.9.1

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

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Step 1.1.8.9.1.1

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

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Step 1.1.8.9.1.1.1

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

$x^{1} x^{2} + x (- 6 x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.1.1.2

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

$x^{1 + 2} + x (- 6 x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.1.2

Add $1$ and $2$ .

$x^{3} + x (- 6 x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.2

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

$x^{3} - 6 (x^{1} x) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.3

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

$x^{3} - 6 (x^{1} x^{1}) + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.4

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

$x^{3} - 6 x^{1 + 1} + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.5

Add $1$ and $1$ .

$x^{3} - 6 x^{2} + x \cdot 9 + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.6

Move $9$ to the left of $x$ .

$x^{3} - 6 x^{2} + 9 \cdot x + x (x (2 x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.7

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

$x^{3} - 6 x^{2} + 9 x + x (2 (x^{1} x)) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.8

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

$x^{3} - 6 x^{2} + 9 x + x (2 (x^{1} x^{1})) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.9

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

$x^{3} - 6 x^{2} + 9 x + x (2 x^{1 + 1}) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.10

Add $1$ and $1$ .

$x^{3} - 6 x^{2} + 9 x + x (2 x^{2}) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.11

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

$x^{3} - 6 x^{2} + 9 x + 2 (x^{1} x^{2}) + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.12

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{1 + 2} + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.13

Add $1$ and $2$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 1 (x (2 x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.14

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 1 (2 (x^{1} x)) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.15

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 1 (2 (x^{1} x^{1})) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.16

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 1 (2 x^{1 + 1}) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.17

Add $1$ and $1$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 1 (2 x^{2}) + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.18

Multiply $2$ by $1$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 2 x^{2} + x (x \cdot - 6) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.19

Move $- 6$ to the left of $x$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 2 x^{2} + x (- 6 \cdot x) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.20

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 2 x^{2} - 6 (x^{1} x) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.21

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 2 x^{2} - 6 (x^{1} x^{1}) + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.22

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 2 x^{2} - 6 x^{1 + 1} + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.23

Add $1$ and $1$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 2 x^{2} - 6 x^{2} + 1 (x \cdot - 6) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.24

Move $- 6$ to the left of $x$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 2 x^{2} - 6 x^{2} + 1 (- 6 \cdot x) + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.25

Multiply $- 6$ by $1$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} + 2 x^{2} - 6 x^{2} - 6 x + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.26

Subtract $6 x^{2}$ from $2 x^{2}$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} - 4 x^{2} - 6 x + x \cdot x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.27

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

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Step 1.1.8.9.27.1

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

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Step 1.1.8.9.27.1.1

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} - 4 x^{2} - 6 x + x^{1} x^{2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.27.1.2

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

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} - 4 x^{2} - 6 x + x^{1 + 2} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.27.2

Add $1$ and $2$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} - 4 x^{2} - 6 x + x^{3} + 1 x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.28

Multiply $x^{2}$ by $1$ .

$x^{3} - 6 x^{2} + 9 x + 2 x^{3} - 4 x^{2} - 6 x + x^{3} + x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.29

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

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 4 x^{2} - 6 x + x^{2} + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.30

Add $- 4 x^{2}$ and $x^{2}$ .

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 3 x^{2} - 6 x + x (- 6 x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.31

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

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 3 x^{2} - 6 x - 6 (x^{1} x) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.32

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

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 3 x^{2} - 6 x - 6 (x^{1} x^{1}) + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.33

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

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 3 x^{2} - 6 x - 6 x^{1 + 1} + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.34

Add $1$ and $1$ .

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 3 x^{2} - 6 x - 6 x^{2} + 1 (- 6 x) + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.35

Multiply $- 6$ by $1$ .

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 3 x^{2} - 6 x - 6 x^{2} - 6 x + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.36

Subtract $6 x^{2}$ from $- 3 x^{2}$ .

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 9 x^{2} - 6 x - 6 x + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.37

Subtract $6 x$ from $- 6 x$ .

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 9 x^{2} - 12 x + x \cdot 9 + 1 \cdot 9$

Step 1.1.8.9.38

Move $9$ to the left of $x$ .

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 9 x^{2} - 12 x + 9 \cdot x + 1 \cdot 9$

Step 1.1.8.9.39

Multiply $9$ by $1$ .

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 9 x^{2} - 12 x + 9 x + 9$

Step 1.1.8.9.40

Add $- 12 x$ and $9 x$ .

$x^{3} - 6 x^{2} + 9 x + 3 x^{3} - 9 x^{2} - 3 x + 9$

Step 1.1.8.9.41

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

$4 x^{3} - 6 x^{2} + 9 x - 9 x^{2} - 3 x + 9$

Step 1.1.8.9.42

Subtract $9 x^{2}$ from $- 6 x^{2}$ .

$4 x^{3} - 15 x^{2} + 9 x - 3 x + 9$

Step 1.1.8.9.43

Subtract $3 x$ from $9 x$ .

$f' (x) = 4 x^{3} - 15 x^{2} + 6 x + 9$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x^{3} - 15 x^{2} + 6 x + 9$ .

$4 x^{3} - 15 x^{2} + 6 x + 9$

Step 2

Set the first derivative equal to $0$ then solve the equation $4 x^{3} - 15 x^{2} + 6 x + 9 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$4 x^{3} - 15 x^{2} + 6 x + 9 = 0$

Step 2.2

Factor $4 x^{3} - 15 x^{2} + 6 x + 9$ using the rational roots test.

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Step 2.2.1

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.

$p = \pm 1, \pm 9, \pm 3$

$q = \pm 1, \pm 4, \pm 2$

Step 2.2.2

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

$\pm 1, \pm 0.25, \pm 0.5, \pm 9, \pm 2.25, \pm 4.5, \pm 3, \pm 0.75, \pm 1.5$

Step 2.2.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 2.2.3.1

Substitute $3$ into the polynomial.

$4 \cdot 3^{3} - 15 \cdot 3^{2} + 6 \cdot 3 + 9$

Step 2.2.3.2

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

$4 \cdot 27 - 15 \cdot 3^{2} + 6 \cdot 3 + 9$

Step 2.2.3.3

Multiply $4$ by $27$ .

$108 - 15 \cdot 3^{2} + 6 \cdot 3 + 9$

Step 2.2.3.4

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

$108 - 15 \cdot 9 + 6 \cdot 3 + 9$

Step 2.2.3.5

Multiply $- 15$ by $9$ .

$108 - 135 + 6 \cdot 3 + 9$

Step 2.2.3.6

Subtract $135$ from $108$ .

$- 27 + 6 \cdot 3 + 9$

Step 2.2.3.7

Multiply $6$ by $3$ .

$- 27 + 18 + 9$

Step 2.2.3.8

Add $- 27$ and $18$ .

$- 9 + 9$

Step 2.2.3.9

Add $- 9$ and $9$ .

$0$

Step 2.2.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{4 x^{3} - 15 x^{2} + 6 x + 9}{x - 3}$

Step 2.2.5

Divide $4 x^{3} - 15 x^{2} + 6 x + 9$ by $x - 3$ .

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Step 2.2.5.1

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

$x$

$3$

$4 x^{3}$

$15 x^{2}$

$6 x$

$9$

Step 2.2.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $x$ .

					$4 x^{2}$
	$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$

Step 2.2.5.3

Multiply the new quotient term by the divisor.

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			+	$4 x^{3}$	-	$12 x^{2}$

Step 2.2.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 12 x^{2}$

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$

Step 2.2.5.5

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

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$

Step 2.2.5.6

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

				$4 x^{2}$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$

Step 2.2.5.7

Divide the highest order term in the dividend $- 3 x^{2}$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$

Step 2.2.5.8

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					-	$3 x^{2}$	+	$9 x$

Step 2.2.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 9 x$

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$

Step 2.2.5.10

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

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$

Step 2.2.5.11

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

				$4 x^{2}$	-	$3 x$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$

Step 2.2.5.12

Divide the highest order term in the dividend $- 3 x$ by the highest order term in divisor $x$ .

				$4 x^{2}$	-	$3 x$	-	$3$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$

Step 2.2.5.13

Multiply the new quotient term by the divisor.

				$4 x^{2}$	-	$3 x$	-	$3$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$
							-	$3 x$	+	$9$

Step 2.2.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x + 9$

				$4 x^{2}$	-	$3 x$	-	$3$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$
							+	$3 x$	-	$9$

Step 2.2.5.15

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

				$4 x^{2}$	-	$3 x$	-	$3$
$x$	-	$3$		$4 x^{3}$	-	$15 x^{2}$	+	$6 x$	+	$9$
			-	$4 x^{3}$	+	$12 x^{2}$
					-	$3 x^{2}$	+	$6 x$
					+	$3 x^{2}$	-	$9 x$
							-	$3 x$	+	$9$
							+	$3 x$	-	$9$
										$0$

Step 2.2.5.16

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

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

Step 2.2.6

Write $4 x^{3} - 15 x^{2} + 6 x + 9$ as a set of factors.

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

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 $0$ .

$x - 3 = 0$

$4 x^{2} - 3 x - 3 = 0$

Step 2.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 2.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.5

Set $4 x^{2} - 3 x - 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $4 x^{2} - 3 x - 3$ equal to $0$ .

$4 x^{2} - 3 x - 3 = 0$

Step 2.5.2

Solve $4 x^{2} - 3 x - 3 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = 4$ , $b = - 3$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

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

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

Step 2.5.2.3.1.2

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

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 3}}{2 \cdot 4}$

Step 2.5.2.3.1.2.2

Multiply $- 16$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 48}}{2 \cdot 4}$

Step 2.5.2.3.1.3

Add $9$ and $48$ .

$x = \frac{3 \pm \sqrt{57}}{2 \cdot 4}$

Step 2.5.2.3.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{57}}{8}$

Step 2.5.2.4

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

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Step 2.5.2.4.1

Simplify the numerator.

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Step 2.5.2.4.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

Step 2.5.2.4.1.2

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

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Step 2.5.2.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 3}}{2 \cdot 4}$

Step 2.5.2.4.1.2.2

Multiply $- 16$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 48}}{2 \cdot 4}$

Step 2.5.2.4.1.3

Add $9$ and $48$ .

$x = \frac{3 \pm \sqrt{57}}{2 \cdot 4}$

Step 2.5.2.4.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{57}}{8}$

Step 2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{3 + \sqrt{57}}{8}$

Step 2.5.2.5

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

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Step 2.5.2.5.1

Simplify the numerator.

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Step 2.5.2.5.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 4 \cdot - 3}}{2 \cdot 4}$

Step 2.5.2.5.1.2

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

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Step 2.5.2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{3 \pm \sqrt{9 - 16 \cdot - 3}}{2 \cdot 4}$

Step 2.5.2.5.1.2.2

Multiply $- 16$ by $- 3$ .

$x = \frac{3 \pm \sqrt{9 + 48}}{2 \cdot 4}$

Step 2.5.2.5.1.3

Add $9$ and $48$ .

$x = \frac{3 \pm \sqrt{57}}{2 \cdot 4}$

Step 2.5.2.5.2

Multiply $2$ by $4$ .

$x = \frac{3 \pm \sqrt{57}}{8}$

Step 2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{3 - \sqrt{57}}{8}$

Step 2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{57}}{8}, \frac{3 - \sqrt{57}}{8}$

Step 2.6

The final solution is all the values that make $(x - 3) (4 x^{2} - 3 x - 3) = 0$ true.

$x = 3, \frac{3 + \sqrt{57}}{8}, \frac{3 - \sqrt{57}}{8}$

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 $x {(x - 3)}^{2} (x + 1)$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 3$ .

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Step 4.1.1

Substitute $3$ for $x$ .

$(3) {((3) - 3)}^{2} ((3) + 1)$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Subtract $3$ from $3$ .

$3 \cdot 0^{2} ((3) + 1)$

Step 4.1.2.2

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

$3 \cdot 0 ((3) + 1)$

Step 4.1.2.3

Multiply $3$ by $0$ .

$0 ((3) + 1)$

Step 4.1.2.4

Add $3$ and $1$ .

$0 \cdot 4$

Step 4.1.2.5

Multiply $0$ by $4$ .

$0$

Step 4.2

Evaluate at $x = \frac{3 + \sqrt{57}}{8}$ .

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Step 4.2.1

Substitute $\frac{3 + \sqrt{57}}{8}$ for $x$ .

$(\frac{3 + \sqrt{57}}{8}) {((\frac{3 + \sqrt{57}}{8}) - 3)}^{2} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2

Simplify.

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Step 4.2.2.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{3 + \sqrt{57}}{8} {(\frac{3 + \sqrt{57}}{8} - 3 \cdot \frac{8}{8})}^{2} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.2

Combine fractions.

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Step 4.2.2.2.1

Combine $- 3$ and $\frac{8}{8}$ .

$\frac{3 + \sqrt{57}}{8} {(\frac{3 + \sqrt{57}}{8} + \frac{- 3 \cdot 8}{8})}^{2} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.2.2

Combine the numerators over the common denominator.

$\frac{3 + \sqrt{57}}{8} {(\frac{3 + \sqrt{57} - 3 \cdot 8}{8})}^{2} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.3

Simplify the numerator.

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Step 4.2.2.3.1

Multiply $- 3$ by $8$ .

$\frac{3 + \sqrt{57}}{8} {(\frac{3 + \sqrt{57} - 24}{8})}^{2} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.3.2

Subtract $24$ from $3$ .

$\frac{3 + \sqrt{57}}{8} {(\frac{- 21 + \sqrt{57}}{8})}^{2} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.4

Simplify the expression.

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Step 4.2.2.4.1

Apply the product rule to $\frac{- 21 + \sqrt{57}}{8}$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{{(- 21 + \sqrt{57})}^{2}}{8^{2}} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.4.2

Raise $8$ to the power of $2$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{{(- 21 + \sqrt{57})}^{2}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.4.3

Rewrite ${(- 21 + \sqrt{57})}^{2}$ as $(- 21 + \sqrt{57}) (- 21 + \sqrt{57})$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{(- 21 + \sqrt{57}) (- 21 + \sqrt{57})}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.5

Expand $(- 21 + \sqrt{57}) (- 21 + \sqrt{57})$ using the FOIL Method.

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Step 4.2.2.5.1

Apply the distributive property.

$\frac{3 + \sqrt{57}}{8} \cdot \frac{- 21 (- 21 + \sqrt{57}) + \sqrt{57} (- 21 + \sqrt{57})}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.5.2

Apply the distributive property.

$\frac{3 + \sqrt{57}}{8} \cdot \frac{- 21 \cdot - 21 - 21 \sqrt{57} + \sqrt{57} (- 21 + \sqrt{57})}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.5.3

Apply the distributive property.

$\frac{3 + \sqrt{57}}{8} \cdot \frac{- 21 \cdot - 21 - 21 \sqrt{57} + \sqrt{57} \cdot - 21 + \sqrt{57} \sqrt{57}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.6

Simplify and combine like terms.

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Step 4.2.2.6.1

Simplify each term.

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Step 4.2.2.6.1.1

Multiply $- 21$ by $- 21$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{441 - 21 \sqrt{57} + \sqrt{57} \cdot - 21 + \sqrt{57} \sqrt{57}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.6.1.2

Move $- 21$ to the left of $\sqrt{57}$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{441 - 21 \sqrt{57} - 21 \cdot \sqrt{57} + \sqrt{57} \sqrt{57}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.6.1.3

Combine using the product rule for radicals.

$\frac{3 + \sqrt{57}}{8} \cdot \frac{441 - 21 \sqrt{57} - 21 \sqrt{57} + \sqrt{57 \cdot 57}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.6.1.4

Multiply $57$ by $57$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{441 - 21 \sqrt{57} - 21 \sqrt{57} + \sqrt{3249}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.6.1.5

Rewrite $3249$ as $57^{2}$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{441 - 21 \sqrt{57} - 21 \sqrt{57} + \sqrt{57^{2}}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.6.1.6

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

$\frac{3 + \sqrt{57}}{8} \cdot \frac{441 - 21 \sqrt{57} - 21 \sqrt{57} + 57}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.6.2

Add $441$ and $57$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{498 - 21 \sqrt{57} - 21 \sqrt{57}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.6.3

Subtract $21 \sqrt{57}$ from $- 21 \sqrt{57}$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{498 - 42 \sqrt{57}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.7

Cancel the common factor of $498 - 42 \sqrt{57}$ and $64$ .

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Step 4.2.2.7.1

Factor $2$ out of $498$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{2 (249) - 42 \sqrt{57}}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.7.2

Factor $2$ out of $- 42 \sqrt{57}$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{2 (249) + 2 (- 21 \sqrt{57})}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.7.3

Factor $2$ out of $2 (249) + 2 (- 21 \sqrt{57})$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{2 (249 - 21 \sqrt{57})}{64} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.7.4

Cancel the common factors.

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Step 4.2.2.7.4.1

Factor $2$ out of $64$ .

$\frac{3 + \sqrt{57}}{8} \cdot \frac{2 (249 - 21 \sqrt{57})}{2 \cdot 32} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.7.4.2

Cancel the common factor.

$\frac{3 + \sqrt{57}}{8} \cdot \frac{2 (249 - 21 \sqrt{57})}{2 \cdot 32} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.7.4.3

Rewrite the expression.

$\frac{3 + \sqrt{57}}{8} \cdot \frac{249 - 21 \sqrt{57}}{32} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.8

Multiply $\frac{3 + \sqrt{57}}{8} \cdot \frac{249 - 21 \sqrt{57}}{32}$ .

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Step 4.2.2.8.1

Multiply $\frac{3 + \sqrt{57}}{8}$ by $\frac{249 - 21 \sqrt{57}}{32}$ .

$\frac{(3 + \sqrt{57}) (249 - 21 \sqrt{57})}{8 \cdot 32} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.8.2

Multiply $8$ by $32$ .

$\frac{(3 + \sqrt{57}) (249 - 21 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.9

Expand $(3 + \sqrt{57}) (249 - 21 \sqrt{57})$ using the FOIL Method.

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Step 4.2.2.9.1

Apply the distributive property.

$\frac{3 (249 - 21 \sqrt{57}) + \sqrt{57} (249 - 21 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.9.2

Apply the distributive property.

$\frac{3 \cdot 249 + 3 (- 21 \sqrt{57}) + \sqrt{57} (249 - 21 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.9.3

Apply the distributive property.

$\frac{3 \cdot 249 + 3 (- 21 \sqrt{57}) + \sqrt{57} \cdot 249 + \sqrt{57} (- 21 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10

Simplify and combine like terms.

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Step 4.2.2.10.1

Simplify each term.

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Step 4.2.2.10.1.1

Multiply $3$ by $249$ .

$\frac{747 + 3 (- 21 \sqrt{57}) + \sqrt{57} \cdot 249 + \sqrt{57} (- 21 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.2

Multiply $- 21$ by $3$ .

$\frac{747 - 63 \sqrt{57} + \sqrt{57} \cdot 249 + \sqrt{57} (- 21 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.3

Move $249$ to the left of $\sqrt{57}$ .

$\frac{747 - 63 \sqrt{57} + 249 \cdot \sqrt{57} + \sqrt{57} (- 21 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.4

Multiply $\sqrt{57} (- 21 \sqrt{57})$ .

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Step 4.2.2.10.1.4.1

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 ({\sqrt{57}}^{1} \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.4.2

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 ({\sqrt{57}}^{1} {\sqrt{57}}^{1})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.4.3

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

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 {\sqrt{57}}^{1 + 1}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.4.4

Add $1$ and $1$ .

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 {\sqrt{57}}^{2}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.5

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 4.2.2.10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 {(57^{\frac{1}{2}})}^{2}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.5.2

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

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 \cdot 57^{\frac{1}{2} \cdot 2}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.5.3

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

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 \cdot 57^{\frac{2}{2}}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.10.1.5.4.1

Cancel the common factor.

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 \cdot 57^{\frac{2}{2}}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.5.4.2

Rewrite the expression.

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 \cdot 57^{1}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.5.5

Evaluate the exponent.

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 21 \cdot 57}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.1.6

Multiply $- 21$ by $57$ .

$\frac{747 - 63 \sqrt{57} + 249 \sqrt{57} - 1197}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.2

Subtract $1197$ from $747$ .

$\frac{- 450 - 63 \sqrt{57} + 249 \sqrt{57}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.10.3

Add $- 63 \sqrt{57}$ and $249 \sqrt{57}$ .

$\frac{- 450 + 186 \sqrt{57}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.11

Cancel the common factor of $- 450 + 186 \sqrt{57}$ and $256$ .

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Step 4.2.2.11.1

Factor $2$ out of $- 450$ .

$\frac{2 (- 225) + 186 \sqrt{57}}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.11.2

Factor $2$ out of $186 \sqrt{57}$ .

$\frac{2 (- 225) + 2 (93 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.11.3

Factor $2$ out of $2 (- 225) + 2 (93 \sqrt{57})$ .

$\frac{2 (- 225 + 93 \sqrt{57})}{256} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.11.4

Cancel the common factors.

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Step 4.2.2.11.4.1

Factor $2$ out of $256$ .

$\frac{2 (- 225 + 93 \sqrt{57})}{2 \cdot 128} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.11.4.2

Cancel the common factor.

$\frac{2 (- 225 + 93 \sqrt{57})}{2 \cdot 128} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.11.4.3

Rewrite the expression.

$\frac{- 225 + 93 \sqrt{57}}{128} ((\frac{3 + \sqrt{57}}{8}) + 1)$

Step 4.2.2.12

Simplify the expression.

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Step 4.2.2.12.1

Write $1$ as a fraction with a common denominator.

$\frac{- 225 + 93 \sqrt{57}}{128} (\frac{3 + \sqrt{57}}{8} + \frac{8}{8})$

Step 4.2.2.12.2

Combine the numerators over the common denominator.

$\frac{- 225 + 93 \sqrt{57}}{128} \cdot \frac{3 + \sqrt{57} + 8}{8}$

Step 4.2.2.12.3

Add $3$ and $8$ .

$\frac{- 225 + 93 \sqrt{57}}{128} \cdot \frac{11 + \sqrt{57}}{8}$

Step 4.2.2.13

Multiply $\frac{- 225 + 93 \sqrt{57}}{128} \cdot \frac{11 + \sqrt{57}}{8}$ .

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Step 4.2.2.13.1

Multiply $\frac{- 225 + 93 \sqrt{57}}{128}$ by $\frac{11 + \sqrt{57}}{8}$ .

$\frac{(- 225 + 93 \sqrt{57}) (11 + \sqrt{57})}{128 \cdot 8}$

Step 4.2.2.13.2

Multiply $128$ by $8$ .

$\frac{(- 225 + 93 \sqrt{57}) (11 + \sqrt{57})}{1024}$

Step 4.2.2.14

Expand $(- 225 + 93 \sqrt{57}) (11 + \sqrt{57})$ using the FOIL Method.

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Step 4.2.2.14.1

Apply the distributive property.

$\frac{- 225 (11 + \sqrt{57}) + 93 \sqrt{57} (11 + \sqrt{57})}{1024}$

Step 4.2.2.14.2

Apply the distributive property.

$\frac{- 225 \cdot 11 - 225 \sqrt{57} + 93 \sqrt{57} (11 + \sqrt{57})}{1024}$

Step 4.2.2.14.3

Apply the distributive property.

$\frac{- 225 \cdot 11 - 225 \sqrt{57} + 93 \sqrt{57} \cdot 11 + 93 \sqrt{57} \sqrt{57}}{1024}$

Step 4.2.2.15

Simplify and combine like terms.

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Step 4.2.2.15.1

Simplify each term.

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Step 4.2.2.15.1.1

Multiply $- 225$ by $11$ .

$\frac{- 2475 - 225 \sqrt{57} + 93 \sqrt{57} \cdot 11 + 93 \sqrt{57} \sqrt{57}}{1024}$

Step 4.2.2.15.1.2

Multiply $11$ by $93$ .

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 \sqrt{57} \sqrt{57}}{1024}$

Step 4.2.2.15.1.3

Multiply $93 \sqrt{57} \sqrt{57}$ .

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Step 4.2.2.15.1.3.1

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 ({\sqrt{57}}^{1} \sqrt{57})}{1024}$

Step 4.2.2.15.1.3.2

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 ({\sqrt{57}}^{1} {\sqrt{57}}^{1})}{1024}$

Step 4.2.2.15.1.3.3

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

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 {\sqrt{57}}^{1 + 1}}{1024}$

Step 4.2.2.15.1.3.4

Add $1$ and $1$ .

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 {\sqrt{57}}^{2}}{1024}$

Step 4.2.2.15.1.4

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 4.2.2.15.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 {(57^{\frac{1}{2}})}^{2}}{1024}$

Step 4.2.2.15.1.4.2

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

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 \cdot 57^{\frac{1}{2} \cdot 2}}{1024}$

Step 4.2.2.15.1.4.3

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

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 \cdot 57^{\frac{2}{2}}}{1024}$

Step 4.2.2.15.1.4.4

Cancel the common factor of $2$ .

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Step 4.2.2.15.1.4.4.1

Cancel the common factor.

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 \cdot 57^{\frac{2}{2}}}{1024}$

Step 4.2.2.15.1.4.4.2

Rewrite the expression.

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 \cdot 57^{1}}{1024}$

Step 4.2.2.15.1.4.5

Evaluate the exponent.

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 93 \cdot 57}{1024}$

Step 4.2.2.15.1.5

Multiply $93$ by $57$ .

$\frac{- 2475 - 225 \sqrt{57} + 1023 \sqrt{57} + 5301}{1024}$

Step 4.2.2.15.2

Add $- 2475$ and $5301$ .

$\frac{2826 - 225 \sqrt{57} + 1023 \sqrt{57}}{1024}$

Step 4.2.2.15.3

Add $- 225 \sqrt{57}$ and $1023 \sqrt{57}$ .

$\frac{2826 + 798 \sqrt{57}}{1024}$

Step 4.2.2.16

Cancel the common factor of $2826 + 798 \sqrt{57}$ and $1024$ .

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Step 4.2.2.16.1

Factor $2$ out of $2826$ .

$\frac{2 (1413) + 798 \sqrt{57}}{1024}$

Step 4.2.2.16.2

Factor $2$ out of $798 \sqrt{57}$ .

$\frac{2 (1413) + 2 (399 \sqrt{57})}{1024}$

Step 4.2.2.16.3

Factor $2$ out of $2 (1413) + 2 (399 \sqrt{57})$ .

$\frac{2 (1413 + 399 \sqrt{57})}{1024}$

Step 4.2.2.16.4

Cancel the common factors.

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Step 4.2.2.16.4.1

Factor $2$ out of $1024$ .

$\frac{2 (1413 + 399 \sqrt{57})}{2 \cdot 512}$

Step 4.2.2.16.4.2

Cancel the common factor.

$\frac{2 (1413 + 399 \sqrt{57})}{2 \cdot 512}$

Step 4.2.2.16.4.3

Rewrite the expression.

$\frac{1413 + 399 \sqrt{57}}{512}$

Step 4.3

Evaluate at $x = \frac{3 - \sqrt{57}}{8}$ .

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Step 4.3.1

Substitute $\frac{3 - \sqrt{57}}{8}$ for $x$ .

$(\frac{3 - \sqrt{57}}{8}) {((\frac{3 - \sqrt{57}}{8}) - 3)}^{2} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2

Simplify.

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Step 4.3.2.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{3 - \sqrt{57}}{8} {(\frac{3 - \sqrt{57}}{8} - 3 \cdot \frac{8}{8})}^{2} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.2

Combine fractions.

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Step 4.3.2.2.1

Combine $- 3$ and $\frac{8}{8}$ .

$\frac{3 - \sqrt{57}}{8} {(\frac{3 - \sqrt{57}}{8} + \frac{- 3 \cdot 8}{8})}^{2} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.2.2

Combine the numerators over the common denominator.

$\frac{3 - \sqrt{57}}{8} {(\frac{3 - \sqrt{57} - 3 \cdot 8}{8})}^{2} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.3

Simplify the numerator.

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Step 4.3.2.3.1

Multiply $- 3$ by $8$ .

$\frac{3 - \sqrt{57}}{8} {(\frac{3 - \sqrt{57} - 24}{8})}^{2} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.3.2

Subtract $24$ from $3$ .

$\frac{3 - \sqrt{57}}{8} {(\frac{- 21 - \sqrt{57}}{8})}^{2} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.4

Simplify the expression.

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Step 4.3.2.4.1

Apply the product rule to $\frac{- 21 - \sqrt{57}}{8}$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{{(- 21 - \sqrt{57})}^{2}}{8^{2}} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.4.2

Raise $8$ to the power of $2$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{{(- 21 - \sqrt{57})}^{2}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.4.3

Rewrite ${(- 21 - \sqrt{57})}^{2}$ as $(- 21 - \sqrt{57}) (- 21 - \sqrt{57})$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{(- 21 - \sqrt{57}) (- 21 - \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.5

Expand $(- 21 - \sqrt{57}) (- 21 - \sqrt{57})$ using the FOIL Method.

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Step 4.3.2.5.1

Apply the distributive property.

$\frac{3 - \sqrt{57}}{8} \cdot \frac{- 21 (- 21 - \sqrt{57}) - \sqrt{57} (- 21 - \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.5.2

Apply the distributive property.

$\frac{3 - \sqrt{57}}{8} \cdot \frac{- 21 \cdot - 21 - 21 (- \sqrt{57}) - \sqrt{57} (- 21 - \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.5.3

Apply the distributive property.

$\frac{3 - \sqrt{57}}{8} \cdot \frac{- 21 \cdot - 21 - 21 (- \sqrt{57}) - \sqrt{57} \cdot - 21 - \sqrt{57} (- \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6

Simplify and combine like terms.

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Step 4.3.2.6.1

Simplify each term.

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Step 4.3.2.6.1.1

Multiply $- 21$ by $- 21$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 - 21 (- \sqrt{57}) - \sqrt{57} \cdot - 21 - \sqrt{57} (- \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.2

Multiply $- 1$ by $- 21$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} - \sqrt{57} \cdot - 21 - \sqrt{57} (- \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.3

Multiply $- 21$ by $- 1$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} - \sqrt{57} (- \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.4

Multiply $- \sqrt{57} (- \sqrt{57})$ .

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Step 4.3.2.6.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + 1 \sqrt{57} \sqrt{57}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.4.2

Multiply $\sqrt{57}$ by $1$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + \sqrt{57} \sqrt{57}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.4.3

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + {\sqrt{57}}^{1} \sqrt{57}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.4.4

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + {\sqrt{57}}^{1} {\sqrt{57}}^{1}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.4.5

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

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + {\sqrt{57}}^{1 + 1}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.4.6

Add $1$ and $1$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + {\sqrt{57}}^{2}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.5

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 4.3.2.6.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + {(57^{\frac{1}{2}})}^{2}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.5.2

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

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + 57^{\frac{1}{2} \cdot 2}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.5.3

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

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + 57^{\frac{2}{2}}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.6.1.5.4.1

Cancel the common factor.

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + 57^{\frac{2}{2}}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.5.4.2

Rewrite the expression.

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + 57^{1}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.1.5.5

Evaluate the exponent.

$\frac{3 - \sqrt{57}}{8} \cdot \frac{441 + 21 \sqrt{57} + 21 \sqrt{57} + 57}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.2

Add $441$ and $57$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{498 + 21 \sqrt{57} + 21 \sqrt{57}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.6.3

Add $21 \sqrt{57}$ and $21 \sqrt{57}$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{498 + 42 \sqrt{57}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.7

Cancel the common factor of $498 + 42 \sqrt{57}$ and $64$ .

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Step 4.3.2.7.1

Factor $2$ out of $498$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{2 (249) + 42 \sqrt{57}}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.7.2

Factor $2$ out of $42 \sqrt{57}$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{2 (249) + 2 (21 \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.7.3

Factor $2$ out of $2 (249) + 2 (21 \sqrt{57})$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{2 (249 + 21 \sqrt{57})}{64} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.7.4

Cancel the common factors.

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Step 4.3.2.7.4.1

Factor $2$ out of $64$ .

$\frac{3 - \sqrt{57}}{8} \cdot \frac{2 (249 + 21 \sqrt{57})}{2 \cdot 32} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.7.4.2

Cancel the common factor.

$\frac{3 - \sqrt{57}}{8} \cdot \frac{2 (249 + 21 \sqrt{57})}{2 \cdot 32} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.7.4.3

Rewrite the expression.

$\frac{3 - \sqrt{57}}{8} \cdot \frac{249 + 21 \sqrt{57}}{32} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.8

Multiply $\frac{3 - \sqrt{57}}{8} \cdot \frac{249 + 21 \sqrt{57}}{32}$ .

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Step 4.3.2.8.1

Multiply $\frac{3 - \sqrt{57}}{8}$ by $\frac{249 + 21 \sqrt{57}}{32}$ .

$\frac{(3 - \sqrt{57}) (249 + 21 \sqrt{57})}{8 \cdot 32} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.8.2

Multiply $8$ by $32$ .

$\frac{(3 - \sqrt{57}) (249 + 21 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.9

Expand $(3 - \sqrt{57}) (249 + 21 \sqrt{57})$ using the FOIL Method.

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Step 4.3.2.9.1

Apply the distributive property.

$\frac{3 (249 + 21 \sqrt{57}) - \sqrt{57} (249 + 21 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.9.2

Apply the distributive property.

$\frac{3 \cdot 249 + 3 (21 \sqrt{57}) - \sqrt{57} (249 + 21 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.9.3

Apply the distributive property.

$\frac{3 \cdot 249 + 3 (21 \sqrt{57}) - \sqrt{57} \cdot 249 - \sqrt{57} (21 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10

Simplify and combine like terms.

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Step 4.3.2.10.1

Simplify each term.

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Step 4.3.2.10.1.1

Multiply $3$ by $249$ .

$\frac{747 + 3 (21 \sqrt{57}) - \sqrt{57} \cdot 249 - \sqrt{57} (21 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.2

Multiply $21$ by $3$ .

$\frac{747 + 63 \sqrt{57} - \sqrt{57} \cdot 249 - \sqrt{57} (21 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.3

Multiply $249$ by $- 1$ .

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - \sqrt{57} (21 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.4

Multiply $- \sqrt{57} (21 \sqrt{57})$ .

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Step 4.3.2.10.1.4.1

Multiply $21$ by $- 1$ .

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 \sqrt{57} \sqrt{57}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.4.2

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 ({\sqrt{57}}^{1} \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.4.3

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 ({\sqrt{57}}^{1} {\sqrt{57}}^{1})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.4.4

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

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 {\sqrt{57}}^{1 + 1}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.4.5

Add $1$ and $1$ .

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 {\sqrt{57}}^{2}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.5

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 4.3.2.10.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 {(57^{\frac{1}{2}})}^{2}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.5.2

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

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 \cdot 57^{\frac{1}{2} \cdot 2}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.5.3

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

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 \cdot 57^{\frac{2}{2}}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.10.1.5.4.1

Cancel the common factor.

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 \cdot 57^{\frac{2}{2}}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.5.4.2

Rewrite the expression.

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 \cdot 57^{1}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.5.5

Evaluate the exponent.

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 21 \cdot 57}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.1.6

Multiply $- 21$ by $57$ .

$\frac{747 + 63 \sqrt{57} - 249 \sqrt{57} - 1197}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.2

Subtract $1197$ from $747$ .

$\frac{- 450 + 63 \sqrt{57} - 249 \sqrt{57}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.10.3

Subtract $249 \sqrt{57}$ from $63 \sqrt{57}$ .

$\frac{- 450 - 186 \sqrt{57}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.11

Cancel the common factor of $- 450 - 186 \sqrt{57}$ and $256$ .

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Step 4.3.2.11.1

Factor $2$ out of $- 450$ .

$\frac{2 (- 225) - 186 \sqrt{57}}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.11.2

Factor $2$ out of $- 186 \sqrt{57}$ .

$\frac{2 (- 225) + 2 (- 93 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.11.3

Factor $2$ out of $2 (- 225) + 2 (- 93 \sqrt{57})$ .

$\frac{2 (- 225 - 93 \sqrt{57})}{256} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.11.4

Cancel the common factors.

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Step 4.3.2.11.4.1

Factor $2$ out of $256$ .

$\frac{2 (- 225 - 93 \sqrt{57})}{2 \cdot 128} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.11.4.2

Cancel the common factor.

$\frac{2 (- 225 - 93 \sqrt{57})}{2 \cdot 128} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.11.4.3

Rewrite the expression.

$\frac{- 225 - 93 \sqrt{57}}{128} ((\frac{3 - \sqrt{57}}{8}) + 1)$

Step 4.3.2.12

Simplify the expression.

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Step 4.3.2.12.1

Write $1$ as a fraction with a common denominator.

$\frac{- 225 - 93 \sqrt{57}}{128} (\frac{3 - \sqrt{57}}{8} + \frac{8}{8})$

Step 4.3.2.12.2

Combine the numerators over the common denominator.

$\frac{- 225 - 93 \sqrt{57}}{128} \cdot \frac{3 - \sqrt{57} + 8}{8}$

Step 4.3.2.12.3

Add $3$ and $8$ .

$\frac{- 225 - 93 \sqrt{57}}{128} \cdot \frac{11 - \sqrt{57}}{8}$

Step 4.3.2.13

Multiply $\frac{- 225 - 93 \sqrt{57}}{128} \cdot \frac{11 - \sqrt{57}}{8}$ .

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Step 4.3.2.13.1

Multiply $\frac{- 225 - 93 \sqrt{57}}{128}$ by $\frac{11 - \sqrt{57}}{8}$ .

$\frac{(- 225 - 93 \sqrt{57}) (11 - \sqrt{57})}{128 \cdot 8}$

Step 4.3.2.13.2

Multiply $128$ by $8$ .

$\frac{(- 225 - 93 \sqrt{57}) (11 - \sqrt{57})}{1024}$

Step 4.3.2.14

Expand $(- 225 - 93 \sqrt{57}) (11 - \sqrt{57})$ using the FOIL Method.

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Step 4.3.2.14.1

Apply the distributive property.

$\frac{- 225 (11 - \sqrt{57}) - 93 \sqrt{57} (11 - \sqrt{57})}{1024}$

Step 4.3.2.14.2

Apply the distributive property.

$\frac{- 225 \cdot 11 - 225 (- \sqrt{57}) - 93 \sqrt{57} (11 - \sqrt{57})}{1024}$

Step 4.3.2.14.3

Apply the distributive property.

$\frac{- 225 \cdot 11 - 225 (- \sqrt{57}) - 93 \sqrt{57} \cdot 11 - 93 \sqrt{57} (- \sqrt{57})}{1024}$

Step 4.3.2.15

Simplify and combine like terms.

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Step 4.3.2.15.1

Simplify each term.

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Step 4.3.2.15.1.1

Multiply $- 225$ by $11$ .

$\frac{- 2475 - 225 (- \sqrt{57}) - 93 \sqrt{57} \cdot 11 - 93 \sqrt{57} (- \sqrt{57})}{1024}$

Step 4.3.2.15.1.2

Multiply $- 1$ by $- 225$ .

$\frac{- 2475 + 225 \sqrt{57} - 93 \sqrt{57} \cdot 11 - 93 \sqrt{57} (- \sqrt{57})}{1024}$

Step 4.3.2.15.1.3

Multiply $11$ by $- 93$ .

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} - 93 \sqrt{57} (- \sqrt{57})}{1024}$

Step 4.3.2.15.1.4

Multiply $- 93 \sqrt{57} (- \sqrt{57})$ .

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Step 4.3.2.15.1.4.1

Multiply $- 1$ by $- 93$ .

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 \sqrt{57} \sqrt{57}}{1024}$

Step 4.3.2.15.1.4.2

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 ({\sqrt{57}}^{1} \sqrt{57})}{1024}$

Step 4.3.2.15.1.4.3

Raise $\sqrt{57}$ to the power of $1$ .

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 ({\sqrt{57}}^{1} {\sqrt{57}}^{1})}{1024}$

Step 4.3.2.15.1.4.4

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

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 {\sqrt{57}}^{1 + 1}}{1024}$

Step 4.3.2.15.1.4.5

Add $1$ and $1$ .

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 {\sqrt{57}}^{2}}{1024}$

Step 4.3.2.15.1.5

Rewrite ${\sqrt{57}}^{2}$ as $57$ .

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Step 4.3.2.15.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{57}$ as $57^{\frac{1}{2}}$ .

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 {(57^{\frac{1}{2}})}^{2}}{1024}$

Step 4.3.2.15.1.5.2

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

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 \cdot 57^{\frac{1}{2} \cdot 2}}{1024}$

Step 4.3.2.15.1.5.3

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

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 \cdot 57^{\frac{2}{2}}}{1024}$

Step 4.3.2.15.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.15.1.5.4.1

Cancel the common factor.

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 \cdot 57^{\frac{2}{2}}}{1024}$

Step 4.3.2.15.1.5.4.2

Rewrite the expression.

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 \cdot 57^{1}}{1024}$

Step 4.3.2.15.1.5.5

Evaluate the exponent.

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 93 \cdot 57}{1024}$

Step 4.3.2.15.1.6

Multiply $93$ by $57$ .

$\frac{- 2475 + 225 \sqrt{57} - 1023 \sqrt{57} + 5301}{1024}$

Step 4.3.2.15.2

Add $- 2475$ and $5301$ .

$\frac{2826 + 225 \sqrt{57} - 1023 \sqrt{57}}{1024}$

Step 4.3.2.15.3

Subtract $1023 \sqrt{57}$ from $225 \sqrt{57}$ .

$\frac{2826 - 798 \sqrt{57}}{1024}$

Step 4.3.2.16

Cancel the common factor of $2826 - 798 \sqrt{57}$ and $1024$ .

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Step 4.3.2.16.1

Factor $2$ out of $2826$ .

$\frac{2 (1413) - 798 \sqrt{57}}{1024}$

Step 4.3.2.16.2

Factor $2$ out of $- 798 \sqrt{57}$ .

$\frac{2 (1413) + 2 (- 399 \sqrt{57})}{1024}$

Step 4.3.2.16.3

Factor $2$ out of $2 (1413) + 2 (- 399 \sqrt{57})$ .

$\frac{2 (1413 - 399 \sqrt{57})}{1024}$

Step 4.3.2.16.4

Cancel the common factors.

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Step 4.3.2.16.4.1

Factor $2$ out of $1024$ .

$\frac{2 (1413 - 399 \sqrt{57})}{2 \cdot 512}$

Step 4.3.2.16.4.2

Cancel the common factor.

$\frac{2 (1413 - 399 \sqrt{57})}{2 \cdot 512}$

Step 4.3.2.16.4.3

Rewrite the expression.

$\frac{1413 - 399 \sqrt{57}}{512}$

Step 4.4

List all of the points.

$(3, 0), (\frac{3 + \sqrt{57}}{8}, \frac{1413 + 399 \sqrt{57}}{512}), (\frac{3 - \sqrt{57}}{8}, \frac{1413 - 399 \sqrt{57}}{512})$

Step 5