Calculus Examples

$y = {(\frac{3 x - 1}{x^{2} + 3})}^{2}$

Step 1

Apply the product rule to $\frac{3 x - 1}{x^{2} + 3}$ .

$y = \frac{{(3 x - 1)}^{2}}{{(x^{2} + 3)}^{2}}$

Step 2

Set $y$ as a function of $x$ .

$f (x) = \frac{{(3 x - 1)}^{2}}{{(x^{2} + 3)}^{2}}$

Step 3

Find the derivative.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = {(3 x - 1)}^{2}$ and $g (x) = {(x^{2} + 3)}^{2}$ .

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

Step 3.2

Multiply the exponents in ${({(x^{2} + 3)}^{2})}^{2}$ .

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $2$ .

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

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 3 x - 1$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $3 x - 1$ .

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

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u_{1}$ with $3 x - 1$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

Move $2$ to the left of ${(x^{2} + 3)}^{2}$ .

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

Step 3.4.2

By the Sum Rule, the derivative of $3 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

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

Step 3.4.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 3.4.4

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

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

Step 3.4.5

Multiply $3$ by $1$ .

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

Step 3.4.6

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

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

Step 3.4.7

Simplify the expression.

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Step 3.4.7.1

Add $3$ and $0$ .

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

Step 3.4.7.2

Multiply $3$ by $2$ .

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) - {(3 x - 1)}^{2} \frac{d}{d x} [{(x^{2} + 3)}^{2}]}{{(x^{2} + 3)}^{4}}$

Step 3.5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 3$ .

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Step 3.5.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} + 3$ .

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) - {(3 x - 1)}^{2} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 3.5.2

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

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) - {(3 x - 1)}^{2} (2 u_{2} \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 3.5.3

Replace all occurrences of $u_{2}$ with $x^{2} + 3$ .

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) - {(3 x - 1)}^{2} (2 (x^{2} + 3) \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 3.6

Differentiate.

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Step 3.6.1

Multiply $2$ by $- 1$ .

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) - 2 {(3 x - 1)}^{2} ((x^{2} + 3) \frac{d}{d x} [x^{2} + 3])}{{(x^{2} + 3)}^{4}}$

Step 3.6.2

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

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) - 2 {(3 x - 1)}^{2} ((x^{2} + 3) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]))}{{(x^{2} + 3)}^{4}}$

Step 3.6.3

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

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) - 2 {(3 x - 1)}^{2} ((x^{2} + 3) (2 x + \frac{d}{d x} [3]))}{{(x^{2} + 3)}^{4}}$

Step 3.6.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{6 {(x^{2} + 3)}^{2} (3 x - 1) - 2 {(3 x - 1)}^{2} ((x^{2} + 3) (2 x + 0))}{{(x^{2} + 3)}^{4}}$

Step 3.6.5

Simplify the expression.

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Step 3.6.5.1

Add $2 x$ and $0$ .

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

Step 3.6.5.2

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

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

Step 3.6.5.3

Multiply $2$ by $- 2$ .

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

Step 3.7

Simplify.

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Step 3.7.1

Apply the distributive property.

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

Step 3.7.2

Simplify the numerator.

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Step 3.7.2.1

Factor $2 (3 x - 1)$ out of $6 {(x^{2} + 3)}^{2} (3 x - 1) - 4 {(3 x - 1)}^{2} (x^{2} x + 3 x)$ .

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Step 3.7.2.1.1

Factor $2 (3 x - 1)$ out of $6 {(x^{2} + 3)}^{2} (3 x - 1)$ .

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

Step 3.7.2.1.2

Factor $2 (3 x - 1)$ out of $- 4 {(3 x - 1)}^{2} (x^{2} x + 3 x)$ .

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

Step 3.7.2.1.3

Factor $2 (3 x - 1)$ out of $2 (3 x - 1) (3 {(x^{2} + 3)}^{2}) + 2 (3 x - 1) (- 2 (3 x - 1) (x^{2} x + 3 x))$ .

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

Step 3.7.2.2

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

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Step 3.7.2.2.1

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

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Step 3.7.2.2.1.1

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

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

Step 3.7.2.2.1.2

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

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

Step 3.7.2.2.2

Add $2$ and $1$ .

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

Step 3.7.2.3

Simplify each term.

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Step 3.7.2.3.1

Rewrite ${(x^{2} + 3)}^{2}$ as $(x^{2} + 3) (x^{2} + 3)$ .

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

Step 3.7.2.3.2

Expand $(x^{2} + 3) (x^{2} + 3)$ using the FOIL Method.

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Step 3.7.2.3.2.1

Apply the distributive property.

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

Step 3.7.2.3.2.2

Apply the distributive property.

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

Step 3.7.2.3.2.3

Apply the distributive property.

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

Step 3.7.2.3.3

Simplify and combine like terms.

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Step 3.7.2.3.3.1

Simplify each term.

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Step 3.7.2.3.3.1.1

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

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Step 3.7.2.3.3.1.1.1

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

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

Step 3.7.2.3.3.1.1.2

Add $2$ and $2$ .

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

Step 3.7.2.3.3.1.2

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

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

Step 3.7.2.3.3.1.3

Multiply $3$ by $3$ .

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

Step 3.7.2.3.3.2

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

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

Step 3.7.2.3.4

Apply the distributive property.

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

Step 3.7.2.3.5

Simplify.

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Step 3.7.2.3.5.1

Multiply $6$ by $3$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 3 \cdot 9 - 2 (3 x - 1) (x^{3} + 3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.5.2

Multiply $3$ by $9$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 2 (3 x - 1) (x^{3} + 3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.6

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 + (- 2 (3 x) - 2 \cdot - 1) (x^{3} + 3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.7

Multiply $3$ by $- 2$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 + (- 6 x - 2 \cdot - 1) (x^{3} + 3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.8

Multiply $- 2$ by $- 1$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 + (- 6 x + 2) (x^{3} + 3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.9

Expand $(- 6 x + 2) (x^{3} + 3 x)$ using the FOIL Method.

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Step 3.7.2.3.9.1

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x (x^{3} + 3 x) + 2 (x^{3} + 3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.9.2

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x \cdot x^{3} - 6 x (3 x) + 2 (x^{3} + 3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.9.3

Apply the distributive property.

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x \cdot x^{3} - 6 x (3 x) + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10

Simplify each term.

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Step 3.7.2.3.10.1

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

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Step 3.7.2.3.10.1.1

Move $x^{3}$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 (x^{3} x) - 6 x (3 x) + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10.1.2

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

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Step 3.7.2.3.10.1.2.1

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

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 (x^{3} x^{1}) - 6 x (3 x) + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10.1.2.2

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

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x^{3 + 1} - 6 x (3 x) + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10.1.3

Add $3$ and $1$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x^{4} - 6 x (3 x) + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10.2

Rewrite using the commutative property of multiplication.

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x^{4} - 6 \cdot 3 x \cdot x + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.7.2.3.10.3.1

Move $x$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x^{4} - 6 \cdot 3 (x \cdot x) + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10.3.2

Multiply $x$ by $x$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x^{4} - 6 \cdot 3 x^{2} + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10.4

Multiply $- 6$ by $3$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x^{4} - 18 x^{2} + 2 x^{3} + 2 (3 x))}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.3.10.5

Multiply $3$ by $2$ .

$\frac{2 (3 x - 1) (3 x^{4} + 18 x^{2} + 27 - 6 x^{4} - 18 x^{2} + 2 x^{3} + 6 x)}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.4

Combine the opposite terms in $3 x^{4} + 18 x^{2} + 27 - 6 x^{4} - 18 x^{2} + 2 x^{3} + 6 x$ .

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Step 3.7.2.4.1

Subtract $18 x^{2}$ from $18 x^{2}$ .

$\frac{2 (3 x - 1) (3 x^{4} + 27 - 6 x^{4} + 0 + 2 x^{3} + 6 x)}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.4.2

Add $3 x^{4} + 27 - 6 x^{4}$ and $0$ .

$\frac{2 (3 x - 1) (3 x^{4} + 27 - 6 x^{4} + 2 x^{3} + 6 x)}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.5

Subtract $6 x^{4}$ from $3 x^{4}$ .

$\frac{2 (3 x - 1) (- 3 x^{4} + 27 + 2 x^{3} + 6 x)}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.6

Reorder terms.

$\frac{2 (3 x - 1) (- 3 x^{4} + 2 x^{3} + 6 x + 27)}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.7

Rewrite $- 3 x^{4} + 2 x^{3} + 6 x + 27$ in a factored form.

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Step 3.7.2.7.1

Regroup terms.

$\frac{2 (3 x - 1) (- 3 x^{4} + 27 + 2 x^{3} + 6 x)}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.7.2

Factor $- 3$ out of $- 3 x^{4} + 27$ .

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Step 3.7.2.7.2.1

Factor $- 3$ out of $- 3 x^{4}$ .

$\frac{2 (3 x - 1) (- 3 (x^{4}) + 27 + 2 x^{3} + 6 x)}{{(x^{2} + 3)}^{4}}$

Step 3.7.2.7.2.2

Factor $- 3$ out of $27$ .

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

Step 3.7.2.7.2.3

Factor $- 3$ out of $- 3 (x^{4}) - 3 (- 9)$ .

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

Step 3.7.2.7.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 3.7.2.7.4

Rewrite $9$ as $3^{2}$ .

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

Step 3.7.2.7.5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 3$ .

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

Step 3.7.2.7.6

Factor $2 x$ out of $2 x^{3} + 6 x$ .

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Step 3.7.2.7.6.1

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

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

Step 3.7.2.7.6.2

Factor $2 x$ out of $6 x$ .

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

Step 3.7.2.7.6.3

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

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

Step 3.7.2.7.7

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

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Step 3.7.2.7.7.1

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

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

Step 3.7.2.7.7.2

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

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

Step 3.7.2.7.7.3

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

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

Step 3.7.2.7.8

Apply the distributive property.

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

Step 3.7.2.7.9

Multiply $- 3$ by $- 3$ .

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

Step 3.7.2.7.10

Reorder terms.

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

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

Step 3.7.3

Cancel the common factor of $x^{2} + 3$ and ${(x^{2} + 3)}^{4}$ .

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Step 3.7.3.1

Factor $x^{2} + 3$ out of $2 (3 x - 1) (x^{2} + 3) (- 3 x^{2} + 2 x + 9)$ .

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

Step 3.7.3.2

Cancel the common factors.

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Step 3.7.3.2.1

Factor $x^{2} + 3$ out of ${(x^{2} + 3)}^{4}$ .

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

Step 3.7.3.2.2

Cancel the common factor.

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

Step 3.7.3.2.3

Rewrite the expression.

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

Step 3.7.4

Reorder terms.

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

Step 3.7.5

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 3.7.6

Factor $- 1$ out of $2 x$ .

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

Step 3.7.7

Factor $- 1$ out of $- (3 x^{2}) - (- 2 x)$ .

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

Step 3.7.8

Rewrite $9$ as $- 1 (- 9)$ .

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

Step 3.7.9

Factor $- 1$ out of $- (3 x^{2} - 2 x) - 1 (- 9)$ .

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

Step 3.7.10

Rewrite $- (3 x^{2} - 2 x - 9)$ as $- 1 (3 x^{2} - 2 x - 9)$ .

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

Step 3.7.11

Move the negative in front of the fraction.

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

Step 3.7.12

Reorder factors in $- \frac{(2 (3 x^{2} - 2 x - 9)) (3 x - 1)}{{(x^{2} + 3)}^{3}}$ .

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

Step 4

Set the derivative equal to $0$ then solve the equation $- \frac{2 (3 x^{2} - 2 x - 9) (3 x - 1)}{{(x^{2} + 3)}^{3}} = 0$ .

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Step 4.1

Set the numerator equal to zero.

$2 (3 x^{2} - 2 x - 9) (3 x - 1) = 0$

Step 4.2

Solve the equation for $x$ .

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Step 4.2.1

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

$3 x^{2} - 2 x - 9 = 0$

$3 x - 1 = 0$

Step 4.2.2

Set $3 x^{2} - 2 x - 9$ equal to $0$ and solve for $x$ .

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Step 4.2.2.1

Set $3 x^{2} - 2 x - 9$ equal to $0$ .

$3 x^{2} - 2 x - 9 = 0$

Step 4.2.2.2

Solve $3 x^{2} - 2 x - 9 = 0$ for $x$ .

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Step 4.2.2.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2.2.2

Substitute the values $a = 3$ , $b = - 2$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot - 9)}}{2 \cdot 3}$

Step 4.2.2.2.3

Simplify.

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Step 4.2.2.2.3.1

Simplify the numerator.

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Step 4.2.2.2.3.1.1

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

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

Step 4.2.2.2.3.1.2

Multiply $- 4 \cdot 3 \cdot - 9$ .

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Step 4.2.2.2.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 9}}{2 \cdot 3}$

Step 4.2.2.2.3.1.2.2

Multiply $- 12$ by $- 9$ .

$x = \frac{2 \pm \sqrt{4 + 108}}{2 \cdot 3}$

Step 4.2.2.2.3.1.3

Add $4$ and $108$ .

$x = \frac{2 \pm \sqrt{112}}{2 \cdot 3}$

Step 4.2.2.2.3.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 4.2.2.2.3.1.4.1

Factor $16$ out of $112$ .

$x = \frac{2 \pm \sqrt{16 (7)}}{2 \cdot 3}$

Step 4.2.2.2.3.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{2 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 3}$

Step 4.2.2.2.3.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 4 \sqrt{7}}{2 \cdot 3}$

Step 4.2.2.2.3.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 4 \sqrt{7}}{6}$

Step 4.2.2.2.3.3

Simplify $\frac{2 \pm 4 \sqrt{7}}{6}$ .

$x = \frac{1 \pm 2 \sqrt{7}}{3}$

Step 4.2.2.2.4

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

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Step 4.2.2.2.4.1

Simplify the numerator.

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Step 4.2.2.2.4.1.1

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

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

Step 4.2.2.2.4.1.2

Multiply $- 4 \cdot 3 \cdot - 9$ .

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Step 4.2.2.2.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 9}}{2 \cdot 3}$

Step 4.2.2.2.4.1.2.2

Multiply $- 12$ by $- 9$ .

$x = \frac{2 \pm \sqrt{4 + 108}}{2 \cdot 3}$

Step 4.2.2.2.4.1.3

Add $4$ and $108$ .

$x = \frac{2 \pm \sqrt{112}}{2 \cdot 3}$

Step 4.2.2.2.4.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 4.2.2.2.4.1.4.1

Factor $16$ out of $112$ .

$x = \frac{2 \pm \sqrt{16 (7)}}{2 \cdot 3}$

Step 4.2.2.2.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{2 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 3}$

Step 4.2.2.2.4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 4 \sqrt{7}}{2 \cdot 3}$

Step 4.2.2.2.4.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 4 \sqrt{7}}{6}$

Step 4.2.2.2.4.3

Simplify $\frac{2 \pm 4 \sqrt{7}}{6}$ .

$x = \frac{1 \pm 2 \sqrt{7}}{3}$

Step 4.2.2.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + 2 \sqrt{7}}{3}$

Step 4.2.2.2.5

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

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Step 4.2.2.2.5.1

Simplify the numerator.

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Step 4.2.2.2.5.1.1

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

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

Step 4.2.2.2.5.1.2

Multiply $- 4 \cdot 3 \cdot - 9$ .

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Step 4.2.2.2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 9}}{2 \cdot 3}$

Step 4.2.2.2.5.1.2.2

Multiply $- 12$ by $- 9$ .

$x = \frac{2 \pm \sqrt{4 + 108}}{2 \cdot 3}$

Step 4.2.2.2.5.1.3

Add $4$ and $108$ .

$x = \frac{2 \pm \sqrt{112}}{2 \cdot 3}$

Step 4.2.2.2.5.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 4.2.2.2.5.1.4.1

Factor $16$ out of $112$ .

$x = \frac{2 \pm \sqrt{16 (7)}}{2 \cdot 3}$

Step 4.2.2.2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{2 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 3}$

Step 4.2.2.2.5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 4 \sqrt{7}}{2 \cdot 3}$

Step 4.2.2.2.5.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 4 \sqrt{7}}{6}$

Step 4.2.2.2.5.3

Simplify $\frac{2 \pm 4 \sqrt{7}}{6}$ .

$x = \frac{1 \pm 2 \sqrt{7}}{3}$

Step 4.2.2.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - 2 \sqrt{7}}{3}$

Step 4.2.2.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + 2 \sqrt{7}}{3}, \frac{1 - 2 \sqrt{7}}{3}$

Step 4.2.3

Set $3 x - 1$ equal to $0$ and solve for $x$ .

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Step 4.2.3.1

Set $3 x - 1$ equal to $0$ .

$3 x - 1 = 0$

Step 4.2.3.2

Solve $3 x - 1 = 0$ for $x$ .

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Step 4.2.3.2.1

Add $1$ to both sides of the equation.

$3 x = 1$

Step 4.2.3.2.2

Divide each term in $3 x = 1$ by $3$ and simplify.

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Step 4.2.3.2.2.1

Divide each term in $3 x = 1$ by $3$ .

$\frac{3 x}{3} = \frac{1}{3}$

Step 4.2.3.2.2.2

Simplify the left side.

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Step 4.2.3.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.3.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{1}{3}$

Step 4.2.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{3}$

Step 4.2.4

The final solution is all the values that make $2 (3 x^{2} - 2 x - 9) (3 x - 1) = 0$ true.

$x = \frac{1 + 2 \sqrt{7}}{3}, \frac{1 - 2 \sqrt{7}}{3}, \frac{1}{3}$

Step 5

Solve the original function $f (x) = \frac{{(3 x - 1)}^{2}}{{(x^{2} + 3)}^{2}}$ at $x = \frac{1 + 2 \sqrt{7}}{3}$ .

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Step 5.1

Replace the variable $x$ with $\frac{1 + 2 \sqrt{7}}{3}$ in the expression.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{{(3 (\frac{1 + 2 \sqrt{7}}{3}) - 1)}^{2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Cancel the common factor of $3$ .

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Step 5.2.1.1.1

Cancel the common factor.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{{(3 (\frac{1 + 2 \sqrt{7}}{3}) - 1)}^{2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.1.2

Rewrite the expression.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{{(1 + 2 \sqrt{7} - 1)}^{2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.2

Subtract $1$ from $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{{(0 + 2 \sqrt{7})}^{2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.3

Add $0$ and $2 \sqrt{7}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{{(2 \sqrt{7})}^{2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.4

Apply the product rule to $2 \sqrt{7}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{2^{2} {\sqrt{7}}^{2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.5

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 {\sqrt{7}}^{2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.2.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 {(7^{\frac{1}{2}})}^{2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.6.2

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7^{\frac{1}{2} \cdot 2}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.6.3

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7^{\frac{2}{2}}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.6.4

Cancel the common factor of $2$ .

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Step 5.2.1.6.4.1

Cancel the common factor.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7^{\frac{2}{2}}}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.6.4.2

Rewrite the expression.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.1.6.5

Evaluate the exponent.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{({(\frac{1 + 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 5.2.2

Simplify the denominator.

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Step 5.2.2.1

Apply the product rule to $\frac{1 + 2 \sqrt{7}}{3}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{{(1 + 2 \sqrt{7})}^{2}}{3^{2}} + 3)}^{2}}$

Step 5.2.2.2

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{{(1 + 2 \sqrt{7})}^{2}}{9} + 3)}^{2}}$

Step 5.2.2.3

Rewrite ${(1 + 2 \sqrt{7})}^{2}$ as $(1 + 2 \sqrt{7}) (1 + 2 \sqrt{7})$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{(1 + 2 \sqrt{7}) (1 + 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.4

Expand $(1 + 2 \sqrt{7}) (1 + 2 \sqrt{7})$ using the FOIL Method.

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Step 5.2.2.4.1

Apply the distributive property.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 (1 + 2 \sqrt{7}) + 2 \sqrt{7} (1 + 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.4.2

Apply the distributive property.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 \cdot 1 + 1 (2 \sqrt{7}) + 2 \sqrt{7} (1 + 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.4.3

Apply the distributive property.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 \cdot 1 + 1 (2 \sqrt{7}) + 2 \sqrt{7} \cdot 1 + 2 \sqrt{7} (2 \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.5

Simplify and combine like terms.

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Step 5.2.2.5.1

Simplify each term.

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Step 5.2.2.5.1.1

Multiply $1$ by $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 1 (2 \sqrt{7}) + 2 \sqrt{7} \cdot 1 + 2 \sqrt{7} (2 \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.5.1.2

Multiply $2 \sqrt{7}$ by $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} \cdot 1 + 2 \sqrt{7} (2 \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.5.1.3

Multiply $2$ by $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 2 \sqrt{7} (2 \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.5.1.4

Multiply $2 \sqrt{7} (2 \sqrt{7})$ .

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Step 5.2.2.5.1.4.1

Multiply $2$ by $2$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 \sqrt{7} \sqrt{7}}{9} + 3)}^{2}}$

Step 5.2.2.5.1.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 (\sqrt{7} \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.5.1.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 (\sqrt{7} \sqrt{7})}{9} + 3)}^{2}}$

Step 5.2.2.5.1.4.4

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 {\sqrt{7}}^{1 + 1}}{9} + 3)}^{2}}$

Step 5.2.2.5.1.4.5

Add $1$ and $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 {\sqrt{7}}^{2}}{9} + 3)}^{2}}$

Step 5.2.2.5.1.5

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.2.2.5.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 {(7^{\frac{1}{2}})}^{2}}{9} + 3)}^{2}}$

Step 5.2.2.5.1.5.2

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 \cdot 7^{\frac{1}{2} \cdot 2}}{9} + 3)}^{2}}$

Step 5.2.2.5.1.5.3

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 \cdot 7^{\frac{2}{2}}}{9} + 3)}^{2}}$

Step 5.2.2.5.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.2.5.1.5.4.1

Cancel the common factor.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 \cdot 7^{\frac{2}{2}}}{9} + 3)}^{2}}$

Step 5.2.2.5.1.5.4.2

Rewrite the expression.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 \cdot 7}{9} + 3)}^{2}}$

Step 5.2.2.5.1.5.5

Evaluate the exponent.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 4 \cdot 7}{9} + 3)}^{2}}$

Step 5.2.2.5.1.6

Multiply $4$ by $7$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 2 \sqrt{7} + 2 \sqrt{7} + 28}{9} + 3)}^{2}}$

Step 5.2.2.5.2

Add $1$ and $28$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 + 2 \sqrt{7} + 2 \sqrt{7}}{9} + 3)}^{2}}$

Step 5.2.2.5.3

Add $2 \sqrt{7}$ and $2 \sqrt{7}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 + 4 \sqrt{7}}{9} + 3)}^{2}}$

Step 5.2.2.6

To write $3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 + 4 \sqrt{7}}{9} + 3 \cdot \frac{9}{9})}^{2}}$

Step 5.2.2.7

Combine $3$ and $\frac{9}{9}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 + 4 \sqrt{7}}{9} + \frac{3 \cdot 9}{9})}^{2}}$

Step 5.2.2.8

Combine the numerators over the common denominator.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 + 4 \sqrt{7} + 3 \cdot 9}{9})}^{2}}$

Step 5.2.2.9

Rewrite $\frac{29 + 4 \sqrt{7} + 3 \cdot 9}{9}$ in a factored form.

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Step 5.2.2.9.1

Multiply $3$ by $9$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 + 4 \sqrt{7} + 27}{9})}^{2}}$

Step 5.2.2.9.2

Add $29$ and $27$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{56 + 4 \sqrt{7}}{9})}^{2}}$

Step 5.2.2.10

Apply the product rule to $\frac{56 + 4 \sqrt{7}}{9}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{{(56 + 4 \sqrt{7})}^{2}}{9^{2}}}$

Step 5.2.2.11

Raise $9$ to the power of $2$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{{(56 + 4 \sqrt{7})}^{2}}{81}}$

Step 5.2.2.12

Rewrite ${(56 + 4 \sqrt{7})}^{2}$ as $(56 + 4 \sqrt{7}) (56 + 4 \sqrt{7})$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{(56 + 4 \sqrt{7}) (56 + 4 \sqrt{7})}{81}}$

Step 5.2.2.13

Expand $(56 + 4 \sqrt{7}) (56 + 4 \sqrt{7})$ using the FOIL Method.

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Step 5.2.2.13.1

Apply the distributive property.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{56 (56 + 4 \sqrt{7}) + 4 \sqrt{7} (56 + 4 \sqrt{7})}{81}}$

Step 5.2.2.13.2

Apply the distributive property.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{56 \cdot 56 + 56 (4 \sqrt{7}) + 4 \sqrt{7} (56 + 4 \sqrt{7})}{81}}$

Step 5.2.2.13.3

Apply the distributive property.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{56 \cdot 56 + 56 (4 \sqrt{7}) + 4 \sqrt{7} \cdot 56 + 4 \sqrt{7} (4 \sqrt{7})}{81}}$

Step 5.2.2.14

Simplify and combine like terms.

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Step 5.2.2.14.1

Simplify each term.

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Step 5.2.2.14.1.1

Multiply $56$ by $56$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 56 (4 \sqrt{7}) + 4 \sqrt{7} \cdot 56 + 4 \sqrt{7} (4 \sqrt{7})}{81}}$

Step 5.2.2.14.1.2

Multiply $4$ by $56$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 4 \sqrt{7} \cdot 56 + 4 \sqrt{7} (4 \sqrt{7})}{81}}$

Step 5.2.2.14.1.3

Multiply $56$ by $4$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 4 \sqrt{7} (4 \sqrt{7})}{81}}$

Step 5.2.2.14.1.4

Multiply $4 \sqrt{7} (4 \sqrt{7})$ .

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Step 5.2.2.14.1.4.1

Multiply $4$ by $4$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 \sqrt{7} \sqrt{7}}{81}}$

Step 5.2.2.14.1.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 (\sqrt{7} \sqrt{7})}{81}}$

Step 5.2.2.14.1.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 (\sqrt{7} \sqrt{7})}{81}}$

Step 5.2.2.14.1.4.4

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 {\sqrt{7}}^{1 + 1}}{81}}$

Step 5.2.2.14.1.4.5

Add $1$ and $1$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 {\sqrt{7}}^{2}}{81}}$

Step 5.2.2.14.1.5

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 5.2.2.14.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 {(7^{\frac{1}{2}})}^{2}}{81}}$

Step 5.2.2.14.1.5.2

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 \cdot 7^{\frac{1}{2} \cdot 2}}{81}}$

Step 5.2.2.14.1.5.3

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

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 \cdot 7^{\frac{2}{2}}}{81}}$

Step 5.2.2.14.1.5.4

Cancel the common factor of $2$ .

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Step 5.2.2.14.1.5.4.1

Cancel the common factor.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 \cdot 7^{\frac{2}{2}}}{81}}$

Step 5.2.2.14.1.5.4.2

Rewrite the expression.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 \cdot 7}{81}}$

Step 5.2.2.14.1.5.5

Evaluate the exponent.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 16 \cdot 7}{81}}$

Step 5.2.2.14.1.6

Multiply $16$ by $7$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 224 \sqrt{7} + 224 \sqrt{7} + 112}{81}}$

Step 5.2.2.14.2

Add $3136$ and $112$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3248 + 224 \sqrt{7} + 224 \sqrt{7}}{81}}$

Step 5.2.2.14.3

Add $224 \sqrt{7}$ and $224 \sqrt{7}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3248 + 448 \sqrt{7}}{81}}$

Step 5.2.3

Multiply $4$ by $7$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{28}{\frac{3248 + 448 \sqrt{7}}{81}}$

Step 5.2.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{1 + 2 \sqrt{7}}{3}) = 28 (\frac{81}{3248 + 448 \sqrt{7}})$

Step 5.2.5

Multiply $\frac{81}{3248 + 448 \sqrt{7}}$ by $\frac{3248 - 448 \sqrt{7}}{3248 - 448 \sqrt{7}}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = 28 (\frac{81}{3248 + 448 \sqrt{7}} \cdot \frac{3248 - 448 \sqrt{7}}{3248 - 448 \sqrt{7}})$

Step 5.2.6

Multiply $\frac{81}{3248 + 448 \sqrt{7}}$ by $\frac{3248 - 448 \sqrt{7}}{3248 - 448 \sqrt{7}}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 - 448 \sqrt{7})}{(3248 + 448 \sqrt{7}) (3248 - 448 \sqrt{7})})$

Step 5.2.7

Expand the denominator using the FOIL method.

$f (\frac{1 + 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 - 448 \sqrt{7})}{10549504 - 1455104 \sqrt{7} + 1455104 \sqrt{7} - 200704 {\sqrt{7}}^{2}})$

Step 5.2.8

Simplify.

$f (\frac{1 + 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 - 448 \sqrt{7})}{9144576})$

Step 5.2.9

Cancel the common factor of $28$ .

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Step 5.2.9.1

Factor $28$ out of $9144576$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 - 448 \sqrt{7})}{28 (326592)})$

Step 5.2.9.2

Cancel the common factor.

$f (\frac{1 + 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 - 448 \sqrt{7})}{28 \cdot 326592})$

Step 5.2.9.3

Rewrite the expression.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{81 (3248 - 448 \sqrt{7})}{326592}$

Step 5.2.10

Cancel the common factors.

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Step 5.2.10.1

Factor $81$ out of $326592$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{81 (3248 - 448 \sqrt{7})}{81 \cdot 4032}$

Step 5.2.10.2

Cancel the common factor.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{81 (3248 - 448 \sqrt{7})}{81 \cdot 4032}$

Step 5.2.10.3

Rewrite the expression.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{3248 - 448 \sqrt{7}}{4032}$

Step 5.2.11

Cancel the common factor of $3248 - 448 \sqrt{7}$ and $4032$ .

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Step 5.2.11.1

Factor $112$ out of $3248$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{112 (29) - 448 \sqrt{7}}{4032}$

Step 5.2.11.2

Factor $112$ out of $- 448 \sqrt{7}$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{112 (29) + 112 (- 4 \sqrt{7})}{4032}$

Step 5.2.11.3

Factor $112$ out of $112 (29) + 112 (- 4 \sqrt{7})$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{112 (29 - 4 \sqrt{7})}{4032}$

Step 5.2.11.4

Cancel the common factors.

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Step 5.2.11.4.1

Factor $112$ out of $4032$ .

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{112 (29 - 4 \sqrt{7})}{112 \cdot 36}$

Step 5.2.11.4.2

Cancel the common factor.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{112 (29 - 4 \sqrt{7})}{112 \cdot 36}$

Step 5.2.11.4.3

Rewrite the expression.

$f (\frac{1 + 2 \sqrt{7}}{3}) = \frac{29 - 4 \sqrt{7}}{36}$

Step 5.2.12

The final answer is $\frac{29 - 4 \sqrt{7}}{36}$ .

$\frac{29 - 4 \sqrt{7}}{36}$

Step 6

Solve the original function $f (x) = \frac{{(3 x - 1)}^{2}}{{(x^{2} + 3)}^{2}}$ at $x = \frac{1 - 2 \sqrt{7}}{3}$ .

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Step 6.1

Replace the variable $x$ with $\frac{1 - 2 \sqrt{7}}{3}$ in the expression.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{{(3 (\frac{1 - 2 \sqrt{7}}{3}) - 1)}^{2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Cancel the common factor of $3$ .

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Step 6.2.1.1.1

Cancel the common factor.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{{(3 (\frac{1 - 2 \sqrt{7}}{3}) - 1)}^{2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.1.2

Rewrite the expression.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{{(1 - 2 \sqrt{7} - 1)}^{2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.2

Subtract $1$ from $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{{(0 - 2 \sqrt{7})}^{2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.3

Subtract $2 \sqrt{7}$ from $0$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{{(- 2 \sqrt{7})}^{2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.4

Apply the product rule to $- 2 \sqrt{7}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{{(- 2)}^{2} {\sqrt{7}}^{2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.5

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 {\sqrt{7}}^{2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 6.2.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 {(7^{\frac{1}{2}})}^{2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.6.2

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7^{\frac{1}{2} \cdot 2}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.6.3

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7^{\frac{2}{2}}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.6.4

Cancel the common factor of $2$ .

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Step 6.2.1.6.4.1

Cancel the common factor.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7^{\frac{2}{2}}}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.6.4.2

Rewrite the expression.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.1.6.5

Evaluate the exponent.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{({(\frac{1 - 2 \sqrt{7}}{3})}^{2} + 3)}^{2}}$

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

Apply the product rule to $\frac{1 - 2 \sqrt{7}}{3}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{{(1 - 2 \sqrt{7})}^{2}}{3^{2}} + 3)}^{2}}$

Step 6.2.2.2

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{{(1 - 2 \sqrt{7})}^{2}}{9} + 3)}^{2}}$

Step 6.2.2.3

Rewrite ${(1 - 2 \sqrt{7})}^{2}$ as $(1 - 2 \sqrt{7}) (1 - 2 \sqrt{7})$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{(1 - 2 \sqrt{7}) (1 - 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.4

Expand $(1 - 2 \sqrt{7}) (1 - 2 \sqrt{7})$ using the FOIL Method.

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Step 6.2.2.4.1

Apply the distributive property.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 (1 - 2 \sqrt{7}) - 2 \sqrt{7} (1 - 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.4.2

Apply the distributive property.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 \cdot 1 + 1 (- 2 \sqrt{7}) - 2 \sqrt{7} (1 - 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.4.3

Apply the distributive property.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 \cdot 1 + 1 (- 2 \sqrt{7}) - 2 \sqrt{7} \cdot 1 - 2 \sqrt{7} (- 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.5

Simplify and combine like terms.

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Step 6.2.2.5.1

Simplify each term.

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Step 6.2.2.5.1.1

Multiply $1$ by $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 + 1 (- 2 \sqrt{7}) - 2 \sqrt{7} \cdot 1 - 2 \sqrt{7} (- 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.5.1.2

Multiply $- 2 \sqrt{7}$ by $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} \cdot 1 - 2 \sqrt{7} (- 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.5.1.3

Multiply $- 2$ by $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} - 2 \sqrt{7} (- 2 \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.5.1.4

Multiply $- 2 \sqrt{7} (- 2 \sqrt{7})$ .

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Step 6.2.2.5.1.4.1

Multiply $- 2$ by $- 2$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 \sqrt{7} \sqrt{7}}{9} + 3)}^{2}}$

Step 6.2.2.5.1.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 (\sqrt{7} \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.5.1.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 (\sqrt{7} \sqrt{7})}{9} + 3)}^{2}}$

Step 6.2.2.5.1.4.4

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 {\sqrt{7}}^{1 + 1}}{9} + 3)}^{2}}$

Step 6.2.2.5.1.4.5

Add $1$ and $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 {\sqrt{7}}^{2}}{9} + 3)}^{2}}$

Step 6.2.2.5.1.5

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 6.2.2.5.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 {(7^{\frac{1}{2}})}^{2}}{9} + 3)}^{2}}$

Step 6.2.2.5.1.5.2

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 \cdot 7^{\frac{1}{2} \cdot 2}}{9} + 3)}^{2}}$

Step 6.2.2.5.1.5.3

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 \cdot 7^{\frac{2}{2}}}{9} + 3)}^{2}}$

Step 6.2.2.5.1.5.4

Cancel the common factor of $2$ .

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Step 6.2.2.5.1.5.4.1

Cancel the common factor.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 \cdot 7^{\frac{2}{2}}}{9} + 3)}^{2}}$

Step 6.2.2.5.1.5.4.2

Rewrite the expression.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 \cdot 7}{9} + 3)}^{2}}$

Step 6.2.2.5.1.5.5

Evaluate the exponent.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 4 \cdot 7}{9} + 3)}^{2}}$

Step 6.2.2.5.1.6

Multiply $4$ by $7$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{1 - 2 \sqrt{7} - 2 \sqrt{7} + 28}{9} + 3)}^{2}}$

Step 6.2.2.5.2

Add $1$ and $28$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 - 2 \sqrt{7} - 2 \sqrt{7}}{9} + 3)}^{2}}$

Step 6.2.2.5.3

Subtract $2 \sqrt{7}$ from $- 2 \sqrt{7}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 - 4 \sqrt{7}}{9} + 3)}^{2}}$

Step 6.2.2.6

To write $3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 - 4 \sqrt{7}}{9} + 3 \cdot \frac{9}{9})}^{2}}$

Step 6.2.2.7

Combine $3$ and $\frac{9}{9}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 - 4 \sqrt{7}}{9} + \frac{3 \cdot 9}{9})}^{2}}$

Step 6.2.2.8

Combine the numerators over the common denominator.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 - 4 \sqrt{7} + 3 \cdot 9}{9})}^{2}}$

Step 6.2.2.9

Rewrite $\frac{29 - 4 \sqrt{7} + 3 \cdot 9}{9}$ in a factored form.

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Step 6.2.2.9.1

Multiply $3$ by $9$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{29 - 4 \sqrt{7} + 27}{9})}^{2}}$

Step 6.2.2.9.2

Add $29$ and $27$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{{(\frac{56 - 4 \sqrt{7}}{9})}^{2}}$

Step 6.2.2.10

Apply the product rule to $\frac{56 - 4 \sqrt{7}}{9}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{{(56 - 4 \sqrt{7})}^{2}}{9^{2}}}$

Step 6.2.2.11

Raise $9$ to the power of $2$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{{(56 - 4 \sqrt{7})}^{2}}{81}}$

Step 6.2.2.12

Rewrite ${(56 - 4 \sqrt{7})}^{2}$ as $(56 - 4 \sqrt{7}) (56 - 4 \sqrt{7})$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{(56 - 4 \sqrt{7}) (56 - 4 \sqrt{7})}{81}}$

Step 6.2.2.13

Expand $(56 - 4 \sqrt{7}) (56 - 4 \sqrt{7})$ using the FOIL Method.

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Step 6.2.2.13.1

Apply the distributive property.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{56 (56 - 4 \sqrt{7}) - 4 \sqrt{7} (56 - 4 \sqrt{7})}{81}}$

Step 6.2.2.13.2

Apply the distributive property.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{56 \cdot 56 + 56 (- 4 \sqrt{7}) - 4 \sqrt{7} (56 - 4 \sqrt{7})}{81}}$

Step 6.2.2.13.3

Apply the distributive property.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{56 \cdot 56 + 56 (- 4 \sqrt{7}) - 4 \sqrt{7} \cdot 56 - 4 \sqrt{7} (- 4 \sqrt{7})}{81}}$

Step 6.2.2.14

Simplify and combine like terms.

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Step 6.2.2.14.1

Simplify each term.

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Step 6.2.2.14.1.1

Multiply $56$ by $56$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 + 56 (- 4 \sqrt{7}) - 4 \sqrt{7} \cdot 56 - 4 \sqrt{7} (- 4 \sqrt{7})}{81}}$

Step 6.2.2.14.1.2

Multiply $- 4$ by $56$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 4 \sqrt{7} \cdot 56 - 4 \sqrt{7} (- 4 \sqrt{7})}{81}}$

Step 6.2.2.14.1.3

Multiply $56$ by $- 4$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} - 4 \sqrt{7} (- 4 \sqrt{7})}{81}}$

Step 6.2.2.14.1.4

Multiply $- 4 \sqrt{7} (- 4 \sqrt{7})$ .

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Step 6.2.2.14.1.4.1

Multiply $- 4$ by $- 4$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 \sqrt{7} \sqrt{7}}{81}}$

Step 6.2.2.14.1.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 (\sqrt{7} \sqrt{7})}{81}}$

Step 6.2.2.14.1.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 (\sqrt{7} \sqrt{7})}{81}}$

Step 6.2.2.14.1.4.4

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 {\sqrt{7}}^{1 + 1}}{81}}$

Step 6.2.2.14.1.4.5

Add $1$ and $1$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 {\sqrt{7}}^{2}}{81}}$

Step 6.2.2.14.1.5

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 6.2.2.14.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 {(7^{\frac{1}{2}})}^{2}}{81}}$

Step 6.2.2.14.1.5.2

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 \cdot 7^{\frac{1}{2} \cdot 2}}{81}}$

Step 6.2.2.14.1.5.3

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

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 \cdot 7^{\frac{2}{2}}}{81}}$

Step 6.2.2.14.1.5.4

Cancel the common factor of $2$ .

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Step 6.2.2.14.1.5.4.1

Cancel the common factor.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 \cdot 7^{\frac{2}{2}}}{81}}$

Step 6.2.2.14.1.5.4.2

Rewrite the expression.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 \cdot 7}{81}}$

Step 6.2.2.14.1.5.5

Evaluate the exponent.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 16 \cdot 7}{81}}$

Step 6.2.2.14.1.6

Multiply $16$ by $7$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3136 - 224 \sqrt{7} - 224 \sqrt{7} + 112}{81}}$

Step 6.2.2.14.2

Add $3136$ and $112$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3248 - 224 \sqrt{7} - 224 \sqrt{7}}{81}}$

Step 6.2.2.14.3

Subtract $224 \sqrt{7}$ from $- 224 \sqrt{7}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{4 \cdot 7}{\frac{3248 - 448 \sqrt{7}}{81}}$

Step 6.2.3

Multiply $4$ by $7$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{28}{\frac{3248 - 448 \sqrt{7}}{81}}$

Step 6.2.4

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{1 - 2 \sqrt{7}}{3}) = 28 (\frac{81}{3248 - 448 \sqrt{7}})$

Step 6.2.5

Multiply $\frac{81}{3248 - 448 \sqrt{7}}$ by $\frac{3248 + 448 \sqrt{7}}{3248 + 448 \sqrt{7}}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = 28 (\frac{81}{3248 - 448 \sqrt{7}} \cdot \frac{3248 + 448 \sqrt{7}}{3248 + 448 \sqrt{7}})$

Step 6.2.6

Multiply $\frac{81}{3248 - 448 \sqrt{7}}$ by $\frac{3248 + 448 \sqrt{7}}{3248 + 448 \sqrt{7}}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 + 448 \sqrt{7})}{(3248 - 448 \sqrt{7}) (3248 + 448 \sqrt{7})})$

Step 6.2.7

Expand the denominator using the FOIL method.

$f (\frac{1 - 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 + 448 \sqrt{7})}{10549504 + 1455104 \sqrt{7} - 1455104 \sqrt{7} - 200704 {\sqrt{7}}^{2}})$

Step 6.2.8

Simplify.

$f (\frac{1 - 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 + 448 \sqrt{7})}{9144576})$

Step 6.2.9

Cancel the common factor of $28$ .

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Step 6.2.9.1

Factor $28$ out of $9144576$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 + 448 \sqrt{7})}{28 (326592)})$

Step 6.2.9.2

Cancel the common factor.

$f (\frac{1 - 2 \sqrt{7}}{3}) = 28 (\frac{81 (3248 + 448 \sqrt{7})}{28 \cdot 326592})$

Step 6.2.9.3

Rewrite the expression.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{81 (3248 + 448 \sqrt{7})}{326592}$

Step 6.2.10

Cancel the common factors.

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Step 6.2.10.1

Factor $81$ out of $326592$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{81 (3248 + 448 \sqrt{7})}{81 \cdot 4032}$

Step 6.2.10.2

Cancel the common factor.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{81 (3248 + 448 \sqrt{7})}{81 \cdot 4032}$

Step 6.2.10.3

Rewrite the expression.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{3248 + 448 \sqrt{7}}{4032}$

Step 6.2.11

Cancel the common factor of $3248 + 448 \sqrt{7}$ and $4032$ .

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Step 6.2.11.1

Factor $112$ out of $3248$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{112 (29) + 448 \sqrt{7}}{4032}$

Step 6.2.11.2

Factor $112$ out of $448 \sqrt{7}$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{112 (29) + 112 (4 \sqrt{7})}{4032}$

Step 6.2.11.3

Factor $112$ out of $112 (29) + 112 (4 \sqrt{7})$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{112 (29 + 4 \sqrt{7})}{4032}$

Step 6.2.11.4

Cancel the common factors.

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Step 6.2.11.4.1

Factor $112$ out of $4032$ .

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{112 (29 + 4 \sqrt{7})}{112 \cdot 36}$

Step 6.2.11.4.2

Cancel the common factor.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{112 (29 + 4 \sqrt{7})}{112 \cdot 36}$

Step 6.2.11.4.3

Rewrite the expression.

$f (\frac{1 - 2 \sqrt{7}}{3}) = \frac{29 + 4 \sqrt{7}}{36}$

Step 6.2.12

The final answer is $\frac{29 + 4 \sqrt{7}}{36}$ .

$\frac{29 + 4 \sqrt{7}}{36}$

Step 7

Solve the original function $f (x) = \frac{{(3 x - 1)}^{2}}{{(x^{2} + 3)}^{2}}$ at $x = \frac{1}{3}$ .

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Step 7.1

Replace the variable $x$ with $\frac{1}{3}$ in the expression.

$f (\frac{1}{3}) = \frac{{(3 (\frac{1}{3}) - 1)}^{2}}{{({(\frac{1}{3})}^{2} + 3)}^{2}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify the numerator.

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Step 7.2.1.1

Cancel the common factor of $3$ .

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Step 7.2.1.1.1

Cancel the common factor.

$f (\frac{1}{3}) = \frac{{(3 (\frac{1}{3}) - 1)}^{2}}{{({(\frac{1}{3})}^{2} + 3)}^{2}}$

Step 7.2.1.1.2

Rewrite the expression.

$f (\frac{1}{3}) = \frac{{(1 - 1)}^{2}}{{({(\frac{1}{3})}^{2} + 3)}^{2}}$

Step 7.2.1.2

Subtract $1$ from $1$ .

$f (\frac{1}{3}) = \frac{0^{2}}{{({(\frac{1}{3})}^{2} + 3)}^{2}}$

Step 7.2.1.3

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

$f (\frac{1}{3}) = \frac{0}{{({(\frac{1}{3})}^{2} + 3)}^{2}}$

Step 7.2.2

Simplify the denominator.

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Step 7.2.2.1

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

$f (\frac{1}{3}) = \frac{0}{{(\frac{1^{2}}{3^{2}} + 3)}^{2}}$

Step 7.2.2.2

One to any power is one.

$f (\frac{1}{3}) = \frac{0}{{(\frac{1}{3^{2}} + 3)}^{2}}$

Step 7.2.2.3

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

$f (\frac{1}{3}) = \frac{0}{{(\frac{1}{9} + 3)}^{2}}$

Step 7.2.2.4

To write $3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{1}{3}) = \frac{0}{{(\frac{1}{9} + 3 \cdot \frac{9}{9})}^{2}}$

Step 7.2.2.5

Combine $3$ and $\frac{9}{9}$ .

$f (\frac{1}{3}) = \frac{0}{{(\frac{1}{9} + \frac{3 \cdot 9}{9})}^{2}}$

Step 7.2.2.6

Combine the numerators over the common denominator.

$f (\frac{1}{3}) = \frac{0}{{(\frac{1 + 3 \cdot 9}{9})}^{2}}$

Step 7.2.2.7

Simplify the numerator.

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Step 7.2.2.7.1

Multiply $3$ by $9$ .

$f (\frac{1}{3}) = \frac{0}{{(\frac{1 + 27}{9})}^{2}}$

Step 7.2.2.7.2

Add $1$ and $27$ .

$f (\frac{1}{3}) = \frac{0}{{(\frac{28}{9})}^{2}}$

Step 7.2.2.8

Apply the product rule to $\frac{28}{9}$ .

$f (\frac{1}{3}) = \frac{0}{\frac{28^{2}}{9^{2}}}$

Step 7.2.2.9

Raise $28$ to the power of $2$ .

$f (\frac{1}{3}) = \frac{0}{\frac{784}{9^{2}}}$

Step 7.2.2.10

Raise $9$ to the power of $2$ .

$f (\frac{1}{3}) = \frac{0}{\frac{784}{81}}$

Step 7.2.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{1}{3}) = 0 (\frac{81}{784})$

Step 7.2.4

Multiply $0$ by $\frac{81}{784}$ .

$f (\frac{1}{3}) = 0$

Step 7.2.5

The final answer is $0$ .

$0$

Step 8

The horizontal tangent lines on function $f (x) = \frac{{(3 x - 1)}^{2}}{{(x^{2} + 3)}^{2}}$ are $y = \frac{29 - 4 \sqrt{7}}{36}, y = \frac{29 + 4 \sqrt{7}}{36}, y = 0$ .

$y = \frac{29 - 4 \sqrt{7}}{36}, y = \frac{29 + 4 \sqrt{7}}{36}, y = 0$

Step 9