Calculus Examples

$g (y) = \frac{y - 2}{{(y^{2} - 2 y + 4)}^{2}}$

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

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

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

Multiply the exponents in ${({(y^{2} - 2 y + 4)}^{2})}^{2}$ .

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

Multiply $2$ by $2$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

Since $- 2$ is constant with respect to $y$ , the derivative of $- 2$ with respect to $y$ is $0$ .

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

Step 1.1.2.5

Simplify the expression.

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Step 1.1.2.5.1

Add $1$ and $0$ .

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

Step 1.1.2.5.2

Multiply ${(y^{2} - 2 y + 4)}^{2}$ by $1$ .

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

Step 1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{2}$ and $g (y) = y^{2} - 2 y + 4$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u$ as $y^{2} - 2 y + 4$ .

$\frac{{(y^{2} - 2 y + 4)}^{2} - (y - 2) (\frac{d}{d u} [u^{2}] \frac{d}{d y} [y^{2} - 2 y + 4])}{{(y^{2} - 2 y + 4)}^{4}}$

Step 1.1.3.2

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

$\frac{{(y^{2} - 2 y + 4)}^{2} - (y - 2) (2 u \frac{d}{d y} [y^{2} - 2 y + 4])}{{(y^{2} - 2 y + 4)}^{4}}$

Step 1.1.3.3

Replace all occurrences of $u$ with $y^{2} - 2 y + 4$ .

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

Step 1.1.4

Simplify with factoring out.

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Step 1.1.4.1

Multiply $2$ by $- 1$ .

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

Step 1.1.4.2

Factor $y^{2} - 2 y + 4$ out of ${(y^{2} - 2 y + 4)}^{2} - 2 (y - 2) ((y^{2} - 2 y + 4) \frac{d}{d y} [y^{2} - 2 y + 4])$ .

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Step 1.1.4.2.1

Factor $y^{2} - 2 y + 4$ out of ${(y^{2} - 2 y + 4)}^{2}$ .

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

Step 1.1.4.2.2

Factor $y^{2} - 2 y + 4$ out of $- 2 (y - 2) ((y^{2} - 2 y + 4) \frac{d}{d y} [y^{2} - 2 y + 4])$ .

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

Step 1.1.4.2.3

Factor $y^{2} - 2 y + 4$ out of $(y^{2} - 2 y + 4) (y^{2} - 2 y + 4) + (y^{2} - 2 y + 4) (- 2 (y - 2) (\frac{d}{d y} [y^{2} - 2 y + 4]))$ .

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

Step 1.1.5

Cancel the common factors.

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Step 1.1.5.1

Factor $y^{2} - 2 y + 4$ out of ${(y^{2} - 2 y + 4)}^{4}$ .

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

Step 1.1.5.2

Cancel the common factor.

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

Step 1.1.5.3

Rewrite the expression.

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

Step 1.1.6

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

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

Step 1.1.7

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

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

Step 1.1.8

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

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

Step 1.1.9

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

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

Step 1.1.10

Multiply $- 2$ by $1$ .

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

Step 1.1.11

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

$\frac{y^{2} - 2 y + 4 - 2 (y - 2) (2 y - 2 + 0)}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.12

Add $2 y - 2$ and $0$ .

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

Step 1.1.13

Simplify.

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Step 1.1.13.1

Apply the distributive property.

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

Step 1.1.13.2

Simplify the numerator.

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Step 1.1.13.2.1

Simplify each term.

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Step 1.1.13.2.1.1

Multiply $- 2$ by $- 2$ .

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

Step 1.1.13.2.1.2

Expand $(- 2 y + 4) (2 y - 2)$ using the FOIL Method.

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Step 1.1.13.2.1.2.1

Apply the distributive property.

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

Step 1.1.13.2.1.2.2

Apply the distributive property.

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

Step 1.1.13.2.1.2.3

Apply the distributive property.

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

Step 1.1.13.2.1.3

Simplify and combine like terms.

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Step 1.1.13.2.1.3.1

Simplify each term.

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Step 1.1.13.2.1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 1.1.13.2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 1.1.13.2.1.3.1.2.1

Move $y$ .

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

Step 1.1.13.2.1.3.1.2.2

Multiply $y$ by $y$ .

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

Step 1.1.13.2.1.3.1.3

Multiply $- 2$ by $2$ .

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

Step 1.1.13.2.1.3.1.4

Multiply $- 2$ by $- 2$ .

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

Step 1.1.13.2.1.3.1.5

Multiply $2$ by $4$ .

$\frac{y^{2} - 2 y + 4 - 4 y^{2} + 4 y + 8 y + 4 \cdot - 2}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.2.1.3.1.6

Multiply $4$ by $- 2$ .

$\frac{y^{2} - 2 y + 4 - 4 y^{2} + 4 y + 8 y - 8}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.2.1.3.2

Add $4 y$ and $8 y$ .

$\frac{y^{2} - 2 y + 4 - 4 y^{2} + 12 y - 8}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.2.2

Subtract $4 y^{2}$ from $y^{2}$ .

$\frac{- 3 y^{2} - 2 y + 4 + 12 y - 8}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.2.3

Add $- 2 y$ and $12 y$ .

$\frac{- 3 y^{2} + 10 y + 4 - 8}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.2.4

Subtract $8$ from $4$ .

$\frac{- 3 y^{2} + 10 y - 4}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.3

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

$\frac{- (3 y^{2}) + 10 y - 4}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.4

Factor $- 1$ out of $10 y$ .

$\frac{- (3 y^{2}) - (- 10 y) - 4}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.5

Factor $- 1$ out of $- (3 y^{2}) - (- 10 y)$ .

$\frac{- (3 y^{2} - 10 y) - 4}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.6

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{- (3 y^{2} - 10 y) - 1 (4)}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.7

Factor $- 1$ out of $- (3 y^{2} - 10 y) - 1 (4)$ .

$\frac{- (3 y^{2} - 10 y + 4)}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.8

Rewrite $- (3 y^{2} - 10 y + 4)$ as $- 1 (3 y^{2} - 10 y + 4)$ .

$\frac{- 1 (3 y^{2} - 10 y + 4)}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.1.13.9

Move the negative in front of the fraction.

$f' (y) = - \frac{3 y^{2} - 10 y + 4}{{(y^{2} - 2 y + 4)}^{3}}$

Step 1.2

The first derivative of $g (y)$ with respect to $y$ is $- \frac{3 y^{2} - 10 y + 4}{{(y^{2} - 2 y + 4)}^{3}}$ .

$- \frac{3 y^{2} - 10 y + 4}{{(y^{2} - 2 y + 4)}^{3}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{3 y^{2} - 10 y + 4}{{(y^{2} - 2 y + 4)}^{3}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{3 y^{2} - 10 y + 4}{{(y^{2} - 2 y + 4)}^{3}} = 0$

Step 2.2

Set the numerator equal to zero.

$3 y^{2} - 10 y + 4 = 0$

Step 2.3

Solve the equation for $y$ .

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Step 2.3.1

Use the quadratic formula to find the solutions.

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

Step 2.3.2

Substitute the values $a = 3$ , $b = - 10$ , and $c = 4$ into the quadratic formula and solve for $y$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (3 \cdot 4)}}{2 \cdot 3}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the numerator.

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Step 2.3.3.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot 4}}{2 \cdot 3}$

Step 2.3.3.1.2

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

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Step 2.3.3.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{10 \pm \sqrt{100 - 12 \cdot 4}}{2 \cdot 3}$

Step 2.3.3.1.2.2

Multiply $- 12$ by $4$ .

$y = \frac{10 \pm \sqrt{100 - 48}}{2 \cdot 3}$

Step 2.3.3.1.3

Subtract $48$ from $100$ .

$y = \frac{10 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.3.3.1.4

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

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Step 2.3.3.1.4.1

Factor $4$ out of $52$ .

$y = \frac{10 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.3.3.1.4.2

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

$y = \frac{10 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.3.3.1.5

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.3.3.2

Multiply $2$ by $3$ .

$y = \frac{10 \pm 2 \sqrt{13}}{6}$

Step 2.3.3.3

Simplify $\frac{10 \pm 2 \sqrt{13}}{6}$ .

$y = \frac{5 \pm \sqrt{13}}{3}$

Step 2.3.4

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

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Step 2.3.4.1

Simplify the numerator.

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Step 2.3.4.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot 4}}{2 \cdot 3}$

Step 2.3.4.1.2

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

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Step 2.3.4.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{10 \pm \sqrt{100 - 12 \cdot 4}}{2 \cdot 3}$

Step 2.3.4.1.2.2

Multiply $- 12$ by $4$ .

$y = \frac{10 \pm \sqrt{100 - 48}}{2 \cdot 3}$

Step 2.3.4.1.3

Subtract $48$ from $100$ .

$y = \frac{10 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.3.4.1.4

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

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Step 2.3.4.1.4.1

Factor $4$ out of $52$ .

$y = \frac{10 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.3.4.1.4.2

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

$y = \frac{10 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.3.4.1.5

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.3.4.2

Multiply $2$ by $3$ .

$y = \frac{10 \pm 2 \sqrt{13}}{6}$

Step 2.3.4.3

Simplify $\frac{10 \pm 2 \sqrt{13}}{6}$ .

$y = \frac{5 \pm \sqrt{13}}{3}$

Step 2.3.4.4

Change the $\pm$ to $+$ .

$y = \frac{5 + \sqrt{13}}{3}$

Step 2.3.5

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

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Step 2.3.5.1

Simplify the numerator.

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Step 2.3.5.1.1

Raise $- 10$ to the power of $2$ .

$y = \frac{10 \pm \sqrt{100 - 4 \cdot 3 \cdot 4}}{2 \cdot 3}$

Step 2.3.5.1.2

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

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Step 2.3.5.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{10 \pm \sqrt{100 - 12 \cdot 4}}{2 \cdot 3}$

Step 2.3.5.1.2.2

Multiply $- 12$ by $4$ .

$y = \frac{10 \pm \sqrt{100 - 48}}{2 \cdot 3}$

Step 2.3.5.1.3

Subtract $48$ from $100$ .

$y = \frac{10 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.3.5.1.4

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

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Step 2.3.5.1.4.1

Factor $4$ out of $52$ .

$y = \frac{10 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.3.5.1.4.2

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

$y = \frac{10 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.3.5.1.5

Pull terms out from under the radical.

$y = \frac{10 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.3.5.2

Multiply $2$ by $3$ .

$y = \frac{10 \pm 2 \sqrt{13}}{6}$

Step 2.3.5.3

Simplify $\frac{10 \pm 2 \sqrt{13}}{6}$ .

$y = \frac{5 \pm \sqrt{13}}{3}$

Step 2.3.5.4

Change the $\pm$ to $-$ .

$y = \frac{5 - \sqrt{13}}{3}$

Step 2.3.6

The final answer is the combination of both solutions.

$y = \frac{5 + \sqrt{13}}{3}, \frac{5 - \sqrt{13}}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

Step 4

Evaluate $\frac{y - 2}{{(y^{2} - 2 y + 4)}^{2}}$ at each $y$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $y = \frac{5 + \sqrt{13}}{3}$ .

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Step 4.1.1

Substitute $\frac{5 + \sqrt{13}}{3}$ for $y$ .

$\frac{(\frac{5 + \sqrt{13}}{3}) - 2}{{({(\frac{5 + \sqrt{13}}{3})}^{2} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify the numerator.

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Step 4.1.2.1.1

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

$\frac{\frac{5 + \sqrt{13}}{3} - 2 \cdot \frac{3}{3}}{{({(\frac{5 + \sqrt{13}}{3})}^{2} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.1.2

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

$\frac{\frac{5 + \sqrt{13}}{3} + \frac{- 2 \cdot 3}{3}}{{({(\frac{5 + \sqrt{13}}{3})}^{2} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.1.3

Combine the numerators over the common denominator.

$\frac{\frac{5 + \sqrt{13} - 2 \cdot 3}{3}}{{({(\frac{5 + \sqrt{13}}{3})}^{2} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.1.4

Rewrite $\frac{5 + \sqrt{13} - 2 \cdot 3}{3}$ in a factored form.

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Step 4.1.2.1.4.1

Multiply $- 2$ by $3$ .

$\frac{\frac{5 + \sqrt{13} - 6}{3}}{{({(\frac{5 + \sqrt{13}}{3})}^{2} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.1.4.2

Subtract $6$ from $5$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{({(\frac{5 + \sqrt{13}}{3})}^{2} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2

Simplify the denominator.

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Step 4.1.2.2.1

Apply the product rule to $\frac{5 + \sqrt{13}}{3}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{{(5 + \sqrt{13})}^{2}}{3^{2}} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.2

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{{(5 + \sqrt{13})}^{2}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.3

Rewrite ${(5 + \sqrt{13})}^{2}$ as $(5 + \sqrt{13}) (5 + \sqrt{13})$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{(5 + \sqrt{13}) (5 + \sqrt{13})}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.4

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

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Step 4.1.2.2.4.1

Apply the distributive property.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{5 (5 + \sqrt{13}) + \sqrt{13} (5 + \sqrt{13})}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.4.2

Apply the distributive property.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{5 \cdot 5 + 5 \sqrt{13} + \sqrt{13} (5 + \sqrt{13})}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.4.3

Apply the distributive property.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{5 \cdot 5 + 5 \sqrt{13} + \sqrt{13} \cdot 5 + \sqrt{13} \sqrt{13}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.5

Simplify and combine like terms.

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Step 4.1.2.2.5.1

Simplify each term.

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Step 4.1.2.2.5.1.1

Multiply $5$ by $5$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{25 + 5 \sqrt{13} + \sqrt{13} \cdot 5 + \sqrt{13} \sqrt{13}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.5.1.2

Move $5$ to the left of $\sqrt{13}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{25 + 5 \sqrt{13} + 5 \cdot \sqrt{13} + \sqrt{13} \sqrt{13}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.5.1.3

Combine using the product rule for radicals.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{25 + 5 \sqrt{13} + 5 \sqrt{13} + \sqrt{13 \cdot 13}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.5.1.4

Multiply $13$ by $13$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{25 + 5 \sqrt{13} + 5 \sqrt{13} + \sqrt{169}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.5.1.5

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{25 + 5 \sqrt{13} + 5 \sqrt{13} + \sqrt{13^{2}}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.5.1.6

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{25 + 5 \sqrt{13} + 5 \sqrt{13} + 13}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.5.2

Add $25$ and $13$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 5 \sqrt{13} + 5 \sqrt{13}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.5.3

Add $5 \sqrt{13}$ and $5 \sqrt{13}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13}}{9} - 2 \frac{5 + \sqrt{13}}{3} + 4)}^{2}}$

Step 4.1.2.2.6

Combine $- 2$ and $\frac{5 + \sqrt{13}}{3}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13}}{9} + \frac{- 2 (5 + \sqrt{13})}{3} + 4)}^{2}}$

Step 4.1.2.2.7

Move the negative in front of the fraction.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13}}{9} - \frac{2 (5 + \sqrt{13})}{3} + 4)}^{2}}$

Step 4.1.2.2.8

To write $- \frac{2 (5 + \sqrt{13})}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13}}{9} - \frac{2 (5 + \sqrt{13})}{3} \cdot \frac{3}{3} + 4)}^{2}}$

Step 4.1.2.2.9

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.2.9.1

Multiply $\frac{2 (5 + \sqrt{13})}{3}$ by $\frac{3}{3}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13}}{9} - \frac{2 (5 + \sqrt{13}) \cdot 3}{3 \cdot 3} + 4)}^{2}}$

Step 4.1.2.2.9.2

Multiply $3$ by $3$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13}}{9} - \frac{2 (5 + \sqrt{13}) \cdot 3}{9} + 4)}^{2}}$

Step 4.1.2.2.10

Combine the numerators over the common denominator.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} - 2 (5 + \sqrt{13}) \cdot 3}{9} + 4)}^{2}}$

Step 4.1.2.2.11

To write $4$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} - 2 (5 + \sqrt{13}) \cdot 3}{9} + 4 \cdot \frac{9}{9})}^{2}}$

Step 4.1.2.2.12

Combine $4$ and $\frac{9}{9}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} - 2 (5 + \sqrt{13}) \cdot 3}{9} + \frac{4 \cdot 9}{9})}^{2}}$

Step 4.1.2.2.13

Combine the numerators over the common denominator.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} - 2 (5 + \sqrt{13}) \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.1.2.2.14

Rewrite $\frac{38 + 10 \sqrt{13} - 2 (5 + \sqrt{13}) \cdot 3 + 4 \cdot 9}{9}$ in a factored form.

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Step 4.1.2.2.14.1

Apply the distributive property.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} + (- 2 \cdot 5 - 2 \sqrt{13}) \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.1.2.2.14.2

Multiply $- 2$ by $5$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} + (- 10 - 2 \sqrt{13}) \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.1.2.2.14.3

Apply the distributive property.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} - 10 \cdot 3 - 2 \sqrt{13} \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.1.2.2.14.4

Multiply $- 10$ by $3$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} - 30 - 2 \sqrt{13} \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.1.2.2.14.5

Multiply $3$ by $- 2$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} - 30 - 6 \sqrt{13} + 4 \cdot 9}{9})}^{2}}$

Step 4.1.2.2.14.6

Multiply $4$ by $9$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{38 + 10 \sqrt{13} - 30 - 6 \sqrt{13} + 36}{9})}^{2}}$

Step 4.1.2.2.14.7

Subtract $30$ from $38$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{8 + 10 \sqrt{13} - 6 \sqrt{13} + 36}{9})}^{2}}$

Step 4.1.2.2.14.8

Add $8$ and $36$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{44 + 10 \sqrt{13} - 6 \sqrt{13}}{9})}^{2}}$

Step 4.1.2.2.14.9

Subtract $6 \sqrt{13}$ from $10 \sqrt{13}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{{(\frac{44 + 4 \sqrt{13}}{9})}^{2}}$

Step 4.1.2.2.15

Apply the product rule to $\frac{44 + 4 \sqrt{13}}{9}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{{(44 + 4 \sqrt{13})}^{2}}{9^{2}}}$

Step 4.1.2.2.16

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{{(44 + 4 \sqrt{13})}^{2}}{81}}$

Step 4.1.2.2.17

Rewrite ${(44 + 4 \sqrt{13})}^{2}$ as $(44 + 4 \sqrt{13}) (44 + 4 \sqrt{13})$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{(44 + 4 \sqrt{13}) (44 + 4 \sqrt{13})}{81}}$

Step 4.1.2.2.18

Expand $(44 + 4 \sqrt{13}) (44 + 4 \sqrt{13})$ using the FOIL Method.

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Step 4.1.2.2.18.1

Apply the distributive property.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{44 (44 + 4 \sqrt{13}) + 4 \sqrt{13} (44 + 4 \sqrt{13})}{81}}$

Step 4.1.2.2.18.2

Apply the distributive property.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{44 \cdot 44 + 44 (4 \sqrt{13}) + 4 \sqrt{13} (44 + 4 \sqrt{13})}{81}}$

Step 4.1.2.2.18.3

Apply the distributive property.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{44 \cdot 44 + 44 (4 \sqrt{13}) + 4 \sqrt{13} \cdot 44 + 4 \sqrt{13} (4 \sqrt{13})}{81}}$

Step 4.1.2.2.19

Simplify and combine like terms.

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Step 4.1.2.2.19.1

Simplify each term.

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Step 4.1.2.2.19.1.1

Multiply $44$ by $44$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 44 (4 \sqrt{13}) + 4 \sqrt{13} \cdot 44 + 4 \sqrt{13} (4 \sqrt{13})}{81}}$

Step 4.1.2.2.19.1.2

Multiply $4$ by $44$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 4 \sqrt{13} \cdot 44 + 4 \sqrt{13} (4 \sqrt{13})}{81}}$

Step 4.1.2.2.19.1.3

Multiply $44$ by $4$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 4 \sqrt{13} (4 \sqrt{13})}{81}}$

Step 4.1.2.2.19.1.4

Multiply $4 \sqrt{13} (4 \sqrt{13})$ .

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Step 4.1.2.2.19.1.4.1

Multiply $4$ by $4$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 \sqrt{13} \sqrt{13}}{81}}$

Step 4.1.2.2.19.1.4.2

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 ({\sqrt{13}}^{1} \sqrt{13})}{81}}$

Step 4.1.2.2.19.1.4.3

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{81}}$

Step 4.1.2.2.19.1.4.4

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 {\sqrt{13}}^{1 + 1}}{81}}$

Step 4.1.2.2.19.1.4.5

Add $1$ and $1$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 {\sqrt{13}}^{2}}{81}}$

Step 4.1.2.2.19.1.5

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

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Step 4.1.2.2.19.1.5.1

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 {(13^{\frac{1}{2}})}^{2}}{81}}$

Step 4.1.2.2.19.1.5.2

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 \cdot 13^{\frac{1}{2} \cdot 2}}{81}}$

Step 4.1.2.2.19.1.5.3

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

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 \cdot 13^{\frac{2}{2}}}{81}}$

Step 4.1.2.2.19.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.2.2.19.1.5.4.1

Cancel the common factor.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 \cdot 13^{\frac{2}{2}}}{81}}$

Step 4.1.2.2.19.1.5.4.2

Rewrite the expression.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 \cdot 13^{1}}{81}}$

Step 4.1.2.2.19.1.5.5

Evaluate the exponent.

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 16 \cdot 13}{81}}$

Step 4.1.2.2.19.1.6

Multiply $16$ by $13$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{1936 + 176 \sqrt{13} + 176 \sqrt{13} + 208}{81}}$

Step 4.1.2.2.19.2

Add $1936$ and $208$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{2144 + 176 \sqrt{13} + 176 \sqrt{13}}{81}}$

Step 4.1.2.2.19.3

Add $176 \sqrt{13}$ and $176 \sqrt{13}$ .

$\frac{\frac{- 1 + \sqrt{13}}{3}}{\frac{2144 + 352 \sqrt{13}}{81}}$

Step 4.1.2.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{- 1 + \sqrt{13}}{3} \cdot \frac{81}{2144 + 352 \sqrt{13}}$

Step 4.1.2.4

Cancel the common factor of $3$ .

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Step 4.1.2.4.1

Factor $3$ out of $81$ .

$\frac{- 1 + \sqrt{13}}{3} \cdot \frac{3 (27)}{2144 + 352 \sqrt{13}}$

Step 4.1.2.4.2

Cancel the common factor.

$\frac{- 1 + \sqrt{13}}{3} \cdot \frac{3 \cdot 27}{2144 + 352 \sqrt{13}}$

Step 4.1.2.4.3

Rewrite the expression.

$(- 1 + \sqrt{13}) \frac{27}{2144 + 352 \sqrt{13}}$

Step 4.1.2.5

Multiply $\frac{27}{2144 + 352 \sqrt{13}}$ by $\frac{2144 - 352 \sqrt{13}}{2144 - 352 \sqrt{13}}$ .

$(- 1 + \sqrt{13}) (\frac{27}{2144 + 352 \sqrt{13}} \cdot \frac{2144 - 352 \sqrt{13}}{2144 - 352 \sqrt{13}})$

Step 4.1.2.6

Multiply $\frac{27}{2144 + 352 \sqrt{13}}$ by $\frac{2144 - 352 \sqrt{13}}{2144 - 352 \sqrt{13}}$ .

$(- 1 + \sqrt{13}) \frac{27 (2144 - 352 \sqrt{13})}{(2144 + 352 \sqrt{13}) (2144 - 352 \sqrt{13})}$

Step 4.1.2.7

Expand the denominator using the FOIL method.

$(- 1 + \sqrt{13}) \frac{27 (2144 - 352 \sqrt{13})}{4596736 - 754688 \sqrt{13} + 754688 \sqrt{13} - 123904 {\sqrt{13}}^{2}}$

Step 4.1.2.8

Simplify.

$(- 1 + \sqrt{13}) \frac{27 (2144 - 352 \sqrt{13})}{2985984}$

Step 4.1.2.9

Cancel the common factors.

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Step 4.1.2.9.1

Factor $27$ out of $2985984$ .

$(- 1 + \sqrt{13}) \frac{27 (2144 - 352 \sqrt{13})}{27 \cdot 110592}$

Step 4.1.2.9.2

Cancel the common factor.

$(- 1 + \sqrt{13}) \frac{27 (2144 - 352 \sqrt{13})}{27 \cdot 110592}$

Step 4.1.2.9.3

Rewrite the expression.

$(- 1 + \sqrt{13}) \frac{2144 - 352 \sqrt{13}}{110592}$

Step 4.1.2.10

Simplify terms.

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Step 4.1.2.10.1

Cancel the common factor of $2144 - 352 \sqrt{13}$ and $110592$ .

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Step 4.1.2.10.1.1

Factor $32$ out of $2144$ .

$(- 1 + \sqrt{13}) \frac{32 (67) - 352 \sqrt{13}}{110592}$

Step 4.1.2.10.1.2

Factor $32$ out of $- 352 \sqrt{13}$ .

$(- 1 + \sqrt{13}) \frac{32 (67) + 32 (- 11 \sqrt{13})}{110592}$

Step 4.1.2.10.1.3

Factor $32$ out of $32 (67) + 32 (- 11 \sqrt{13})$ .

$(- 1 + \sqrt{13}) \frac{32 (67 - 11 \sqrt{13})}{110592}$

Step 4.1.2.10.1.4

Cancel the common factors.

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Step 4.1.2.10.1.4.1

Factor $32$ out of $110592$ .

$(- 1 + \sqrt{13}) \frac{32 (67 - 11 \sqrt{13})}{32 \cdot 3456}$

Step 4.1.2.10.1.4.2

Cancel the common factor.

$(- 1 + \sqrt{13}) \frac{32 (67 - 11 \sqrt{13})}{32 \cdot 3456}$

Step 4.1.2.10.1.4.3

Rewrite the expression.

$(- 1 + \sqrt{13}) \frac{67 - 11 \sqrt{13}}{3456}$

Step 4.1.2.10.2

Apply the distributive property.

$- 1 \frac{67 - 11 \sqrt{13}}{3456} + \sqrt{13} \frac{67 - 11 \sqrt{13}}{3456}$

Step 4.1.2.10.3

Rewrite $- 1 \frac{67 - 11 \sqrt{13}}{3456}$ as $- \frac{67 - 11 \sqrt{13}}{3456}$ .

$- \frac{67 - 11 \sqrt{13}}{3456} + \sqrt{13} \frac{67 - 11 \sqrt{13}}{3456}$

Step 4.1.2.10.4

Combine $\sqrt{13}$ and $\frac{67 - 11 \sqrt{13}}{3456}$ .

$- \frac{67 - 11 \sqrt{13}}{3456} + \frac{\sqrt{13} (67 - 11 \sqrt{13})}{3456}$

Step 4.1.2.10.5

Combine the numerators over the common denominator.

$\frac{- (67 - 11 \sqrt{13}) + \sqrt{13} (67 - 11 \sqrt{13})}{3456}$

Step 4.1.2.11

Simplify each term.

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Step 4.1.2.11.1

Apply the distributive property.

$\frac{- 1 \cdot 67 - (- 11 \sqrt{13}) + \sqrt{13} (67 - 11 \sqrt{13})}{3456}$

Step 4.1.2.11.2

Multiply $- 1$ by $67$ .

$\frac{- 67 - (- 11 \sqrt{13}) + \sqrt{13} (67 - 11 \sqrt{13})}{3456}$

Step 4.1.2.11.3

Multiply $- 11$ by $- 1$ .

$\frac{- 67 + 11 \sqrt{13} + \sqrt{13} (67 - 11 \sqrt{13})}{3456}$

Step 4.1.2.11.4

Apply the distributive property.

$\frac{- 67 + 11 \sqrt{13} + \sqrt{13} \cdot 67 + \sqrt{13} (- 11 \sqrt{13})}{3456}$

Step 4.1.2.11.5

Move $67$ to the left of $\sqrt{13}$ .

$\frac{- 67 + 11 \sqrt{13} + 67 \cdot \sqrt{13} + \sqrt{13} (- 11 \sqrt{13})}{3456}$

Step 4.1.2.11.6

Multiply $\sqrt{13} (- 11 \sqrt{13})$ .

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Step 4.1.2.11.6.1

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

$\frac{- 67 + 11 \sqrt{13} + 67 \cdot \sqrt{13} - 11 ({\sqrt{13}}^{1} \sqrt{13})}{3456}$

Step 4.1.2.11.6.2

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

$\frac{- 67 + 11 \sqrt{13} + 67 \cdot \sqrt{13} - 11 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{3456}$

Step 4.1.2.11.6.3

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

$\frac{- 67 + 11 \sqrt{13} + 67 \cdot \sqrt{13} - 11 {\sqrt{13}}^{1 + 1}}{3456}$

Step 4.1.2.11.6.4

Add $1$ and $1$ .

$\frac{- 67 + 11 \sqrt{13} + 67 \cdot \sqrt{13} - 11 {\sqrt{13}}^{2}}{3456}$

Step 4.1.2.11.7

Simplify each term.

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Step 4.1.2.11.7.1

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

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Step 4.1.2.11.7.1.1

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

$\frac{- 67 + 11 \sqrt{13} + 67 \sqrt{13} - 11 {(13^{\frac{1}{2}})}^{2}}{3456}$

Step 4.1.2.11.7.1.2

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

$\frac{- 67 + 11 \sqrt{13} + 67 \sqrt{13} - 11 \cdot 13^{\frac{1}{2} \cdot 2}}{3456}$

Step 4.1.2.11.7.1.3

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

$\frac{- 67 + 11 \sqrt{13} + 67 \sqrt{13} - 11 \cdot 13^{\frac{2}{2}}}{3456}$

Step 4.1.2.11.7.1.4

Cancel the common factor of $2$ .

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Step 4.1.2.11.7.1.4.1

Cancel the common factor.

$\frac{- 67 + 11 \sqrt{13} + 67 \sqrt{13} - 11 \cdot 13^{\frac{2}{2}}}{3456}$

Step 4.1.2.11.7.1.4.2

Rewrite the expression.

$\frac{- 67 + 11 \sqrt{13} + 67 \sqrt{13} - 11 \cdot 13^{1}}{3456}$

Step 4.1.2.11.7.1.5

Evaluate the exponent.

$\frac{- 67 + 11 \sqrt{13} + 67 \sqrt{13} - 11 \cdot 13}{3456}$

Step 4.1.2.11.7.2

Multiply $- 11$ by $13$ .

$\frac{- 67 + 11 \sqrt{13} + 67 \sqrt{13} - 143}{3456}$

Step 4.1.2.12

Simplify terms.

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Step 4.1.2.12.1

Subtract $143$ from $- 67$ .

$\frac{- 210 + 11 \sqrt{13} + 67 \sqrt{13}}{3456}$

Step 4.1.2.12.2

Add $11 \sqrt{13}$ and $67 \sqrt{13}$ .

$\frac{- 210 + 78 \sqrt{13}}{3456}$

Step 4.1.2.12.3

Cancel the common factor of $- 210 + 78 \sqrt{13}$ and $3456$ .

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Step 4.1.2.12.3.1

Factor $6$ out of $- 210$ .

$\frac{6 (- 35) + 78 \sqrt{13}}{3456}$

Step 4.1.2.12.3.2

Factor $6$ out of $78 \sqrt{13}$ .

$\frac{6 (- 35) + 6 (13 \sqrt{13})}{3456}$

Step 4.1.2.12.3.3

Factor $6$ out of $6 (- 35) + 6 (13 \sqrt{13})$ .

$\frac{6 (- 35 + 13 \sqrt{13})}{3456}$

Step 4.1.2.12.3.4

Cancel the common factors.

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Step 4.1.2.12.3.4.1

Factor $6$ out of $3456$ .

$\frac{6 (- 35 + 13 \sqrt{13})}{6 \cdot 576}$

Step 4.1.2.12.3.4.2

Cancel the common factor.

$\frac{6 (- 35 + 13 \sqrt{13})}{6 \cdot 576}$

Step 4.1.2.12.3.4.3

Rewrite the expression.

$\frac{- 35 + 13 \sqrt{13}}{576}$

Step 4.1.2.12.4

Rewrite $- 35$ as $- 1 (35)$ .

$\frac{- 1 (35) + 13 \sqrt{13}}{576}$

Step 4.1.2.12.5

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

$\frac{- 1 (35) - (- 13 \sqrt{13})}{576}$

Step 4.1.2.12.6

Factor $- 1$ out of $- 1 (35) - (- 13 \sqrt{13})$ .

$\frac{- 1 (35 - 13 \sqrt{13})}{576}$

Step 4.1.2.12.7

Move the negative in front of the fraction.

$- \frac{35 - 13 \sqrt{13}}{576}$

Step 4.2

Evaluate at $y = \frac{5 - \sqrt{13}}{3}$ .

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Step 4.2.1

Substitute $\frac{5 - \sqrt{13}}{3}$ for $y$ .

$\frac{(\frac{5 - \sqrt{13}}{3}) - 2}{{({(\frac{5 - \sqrt{13}}{3})}^{2} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the numerator.

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Step 4.2.2.1.1

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

$\frac{\frac{5 - \sqrt{13}}{3} - 2 \cdot \frac{3}{3}}{{({(\frac{5 - \sqrt{13}}{3})}^{2} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.1.2

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

$\frac{\frac{5 - \sqrt{13}}{3} + \frac{- 2 \cdot 3}{3}}{{({(\frac{5 - \sqrt{13}}{3})}^{2} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.1.3

Combine the numerators over the common denominator.

$\frac{\frac{5 - \sqrt{13} - 2 \cdot 3}{3}}{{({(\frac{5 - \sqrt{13}}{3})}^{2} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.1.4

Rewrite $\frac{5 - \sqrt{13} - 2 \cdot 3}{3}$ in a factored form.

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Step 4.2.2.1.4.1

Multiply $- 2$ by $3$ .

$\frac{\frac{5 - \sqrt{13} - 6}{3}}{{({(\frac{5 - \sqrt{13}}{3})}^{2} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.1.4.2

Subtract $6$ from $5$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{({(\frac{5 - \sqrt{13}}{3})}^{2} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2

Simplify the denominator.

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Step 4.2.2.2.1

Apply the product rule to $\frac{5 - \sqrt{13}}{3}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{{(5 - \sqrt{13})}^{2}}{3^{2}} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.2

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{{(5 - \sqrt{13})}^{2}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.3

Rewrite ${(5 - \sqrt{13})}^{2}$ as $(5 - \sqrt{13}) (5 - \sqrt{13})$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{(5 - \sqrt{13}) (5 - \sqrt{13})}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.4

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

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Step 4.2.2.2.4.1

Apply the distributive property.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{5 (5 - \sqrt{13}) - \sqrt{13} (5 - \sqrt{13})}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.4.2

Apply the distributive property.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{5 \cdot 5 + 5 (- \sqrt{13}) - \sqrt{13} (5 - \sqrt{13})}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.4.3

Apply the distributive property.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{5 \cdot 5 + 5 (- \sqrt{13}) - \sqrt{13} \cdot 5 - \sqrt{13} (- \sqrt{13})}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5

Simplify and combine like terms.

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Step 4.2.2.2.5.1

Simplify each term.

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Step 4.2.2.2.5.1.1

Multiply $5$ by $5$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 + 5 (- \sqrt{13}) - \sqrt{13} \cdot 5 - \sqrt{13} (- \sqrt{13})}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.2

Multiply $- 1$ by $5$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - \sqrt{13} \cdot 5 - \sqrt{13} (- \sqrt{13})}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.3

Multiply $5$ by $- 1$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} - \sqrt{13} (- \sqrt{13})}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.4

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

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Step 4.2.2.2.5.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + 1 \sqrt{13} \sqrt{13}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.4.2

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + \sqrt{13} \sqrt{13}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.4.3

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + {\sqrt{13}}^{1} \sqrt{13}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.4.4

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + {\sqrt{13}}^{1} {\sqrt{13}}^{1}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.4.5

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + {\sqrt{13}}^{1 + 1}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.4.6

Add $1$ and $1$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + {\sqrt{13}}^{2}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.5

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

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Step 4.2.2.2.5.1.5.1

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + {(13^{\frac{1}{2}})}^{2}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.5.2

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + 13^{\frac{1}{2} \cdot 2}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.5.3

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + 13^{\frac{2}{2}}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.5.1.5.4.1

Cancel the common factor.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + 13^{\frac{2}{2}}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.5.4.2

Rewrite the expression.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + 13^{1}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.1.5.5

Evaluate the exponent.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{25 - 5 \sqrt{13} - 5 \sqrt{13} + 13}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.2

Add $25$ and $13$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 5 \sqrt{13} - 5 \sqrt{13}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.5.3

Subtract $5 \sqrt{13}$ from $- 5 \sqrt{13}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13}}{9} - 2 \frac{5 - \sqrt{13}}{3} + 4)}^{2}}$

Step 4.2.2.2.6

Combine $- 2$ and $\frac{5 - \sqrt{13}}{3}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13}}{9} + \frac{- 2 (5 - \sqrt{13})}{3} + 4)}^{2}}$

Step 4.2.2.2.7

Move the negative in front of the fraction.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13}}{9} - \frac{2 (5 - \sqrt{13})}{3} + 4)}^{2}}$

Step 4.2.2.2.8

To write $- \frac{2 (5 - \sqrt{13})}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13}}{9} - \frac{2 (5 - \sqrt{13})}{3} \cdot \frac{3}{3} + 4)}^{2}}$

Step 4.2.2.2.9

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.2.9.1

Multiply $\frac{2 (5 - \sqrt{13})}{3}$ by $\frac{3}{3}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13}}{9} - \frac{2 (5 - \sqrt{13}) \cdot 3}{3 \cdot 3} + 4)}^{2}}$

Step 4.2.2.2.9.2

Multiply $3$ by $3$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13}}{9} - \frac{2 (5 - \sqrt{13}) \cdot 3}{9} + 4)}^{2}}$

Step 4.2.2.2.10

Combine the numerators over the common denominator.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} - 2 (5 - \sqrt{13}) \cdot 3}{9} + 4)}^{2}}$

Step 4.2.2.2.11

To write $4$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} - 2 (5 - \sqrt{13}) \cdot 3}{9} + 4 \cdot \frac{9}{9})}^{2}}$

Step 4.2.2.2.12

Combine $4$ and $\frac{9}{9}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} - 2 (5 - \sqrt{13}) \cdot 3}{9} + \frac{4 \cdot 9}{9})}^{2}}$

Step 4.2.2.2.13

Combine the numerators over the common denominator.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} - 2 (5 - \sqrt{13}) \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.2.2.2.14

Rewrite $\frac{38 - 10 \sqrt{13} - 2 (5 - \sqrt{13}) \cdot 3 + 4 \cdot 9}{9}$ in a factored form.

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Step 4.2.2.2.14.1

Apply the distributive property.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} + (- 2 \cdot 5 - 2 (- \sqrt{13})) \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.2.2.2.14.2

Multiply $- 2$ by $5$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} + (- 10 - 2 (- \sqrt{13})) \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.2.2.2.14.3

Multiply $- 1$ by $- 2$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} + (- 10 + 2 \sqrt{13}) \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.2.2.2.14.4

Apply the distributive property.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} - 10 \cdot 3 + 2 \sqrt{13} \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.2.2.2.14.5

Multiply $- 10$ by $3$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} - 30 + 2 \sqrt{13} \cdot 3 + 4 \cdot 9}{9})}^{2}}$

Step 4.2.2.2.14.6

Multiply $3$ by $2$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} - 30 + 6 \sqrt{13} + 4 \cdot 9}{9})}^{2}}$

Step 4.2.2.2.14.7

Multiply $4$ by $9$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{38 - 10 \sqrt{13} - 30 + 6 \sqrt{13} + 36}{9})}^{2}}$

Step 4.2.2.2.14.8

Subtract $30$ from $38$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{8 - 10 \sqrt{13} + 6 \sqrt{13} + 36}{9})}^{2}}$

Step 4.2.2.2.14.9

Add $8$ and $36$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{44 - 10 \sqrt{13} + 6 \sqrt{13}}{9})}^{2}}$

Step 4.2.2.2.14.10

Add $- 10 \sqrt{13}$ and $6 \sqrt{13}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{{(\frac{44 - 4 \sqrt{13}}{9})}^{2}}$

Step 4.2.2.2.15

Apply the product rule to $\frac{44 - 4 \sqrt{13}}{9}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{{(44 - 4 \sqrt{13})}^{2}}{9^{2}}}$

Step 4.2.2.2.16

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{{(44 - 4 \sqrt{13})}^{2}}{81}}$

Step 4.2.2.2.17

Rewrite ${(44 - 4 \sqrt{13})}^{2}$ as $(44 - 4 \sqrt{13}) (44 - 4 \sqrt{13})$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{(44 - 4 \sqrt{13}) (44 - 4 \sqrt{13})}{81}}$

Step 4.2.2.2.18

Expand $(44 - 4 \sqrt{13}) (44 - 4 \sqrt{13})$ using the FOIL Method.

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Step 4.2.2.2.18.1

Apply the distributive property.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{44 (44 - 4 \sqrt{13}) - 4 \sqrt{13} (44 - 4 \sqrt{13})}{81}}$

Step 4.2.2.2.18.2

Apply the distributive property.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{44 \cdot 44 + 44 (- 4 \sqrt{13}) - 4 \sqrt{13} (44 - 4 \sqrt{13})}{81}}$

Step 4.2.2.2.18.3

Apply the distributive property.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{44 \cdot 44 + 44 (- 4 \sqrt{13}) - 4 \sqrt{13} \cdot 44 - 4 \sqrt{13} (- 4 \sqrt{13})}{81}}$

Step 4.2.2.2.19

Simplify and combine like terms.

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Step 4.2.2.2.19.1

Simplify each term.

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Step 4.2.2.2.19.1.1

Multiply $44$ by $44$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 + 44 (- 4 \sqrt{13}) - 4 \sqrt{13} \cdot 44 - 4 \sqrt{13} (- 4 \sqrt{13})}{81}}$

Step 4.2.2.2.19.1.2

Multiply $- 4$ by $44$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 4 \sqrt{13} \cdot 44 - 4 \sqrt{13} (- 4 \sqrt{13})}{81}}$

Step 4.2.2.2.19.1.3

Multiply $44$ by $- 4$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} - 4 \sqrt{13} (- 4 \sqrt{13})}{81}}$

Step 4.2.2.2.19.1.4

Multiply $- 4 \sqrt{13} (- 4 \sqrt{13})$ .

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Step 4.2.2.2.19.1.4.1

Multiply $- 4$ by $- 4$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 \sqrt{13} \sqrt{13}}{81}}$

Step 4.2.2.2.19.1.4.2

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 ({\sqrt{13}}^{1} \sqrt{13})}{81}}$

Step 4.2.2.2.19.1.4.3

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{81}}$

Step 4.2.2.2.19.1.4.4

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 {\sqrt{13}}^{1 + 1}}{81}}$

Step 4.2.2.2.19.1.4.5

Add $1$ and $1$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 {\sqrt{13}}^{2}}{81}}$

Step 4.2.2.2.19.1.5

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

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Step 4.2.2.2.19.1.5.1

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 {(13^{\frac{1}{2}})}^{2}}{81}}$

Step 4.2.2.2.19.1.5.2

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 \cdot 13^{\frac{1}{2} \cdot 2}}{81}}$

Step 4.2.2.2.19.1.5.3

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

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 \cdot 13^{\frac{2}{2}}}{81}}$

Step 4.2.2.2.19.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.2.19.1.5.4.1

Cancel the common factor.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 \cdot 13^{\frac{2}{2}}}{81}}$

Step 4.2.2.2.19.1.5.4.2

Rewrite the expression.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 \cdot 13^{1}}{81}}$

Step 4.2.2.2.19.1.5.5

Evaluate the exponent.

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 16 \cdot 13}{81}}$

Step 4.2.2.2.19.1.6

Multiply $16$ by $13$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{1936 - 176 \sqrt{13} - 176 \sqrt{13} + 208}{81}}$

Step 4.2.2.2.19.2

Add $1936$ and $208$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{2144 - 176 \sqrt{13} - 176 \sqrt{13}}{81}}$

Step 4.2.2.2.19.3

Subtract $176 \sqrt{13}$ from $- 176 \sqrt{13}$ .

$\frac{\frac{- 1 - \sqrt{13}}{3}}{\frac{2144 - 352 \sqrt{13}}{81}}$

Step 4.2.2.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{- 1 - \sqrt{13}}{3} \cdot \frac{81}{2144 - 352 \sqrt{13}}$

Step 4.2.2.4

Cancel the common factor of $3$ .

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Step 4.2.2.4.1

Factor $3$ out of $81$ .

$\frac{- 1 - \sqrt{13}}{3} \cdot \frac{3 (27)}{2144 - 352 \sqrt{13}}$

Step 4.2.2.4.2

Cancel the common factor.

$\frac{- 1 - \sqrt{13}}{3} \cdot \frac{3 \cdot 27}{2144 - 352 \sqrt{13}}$

Step 4.2.2.4.3

Rewrite the expression.

$(- 1 - \sqrt{13}) \frac{27}{2144 - 352 \sqrt{13}}$

Step 4.2.2.5

Multiply $\frac{27}{2144 - 352 \sqrt{13}}$ by $\frac{2144 + 352 \sqrt{13}}{2144 + 352 \sqrt{13}}$ .

$(- 1 - \sqrt{13}) (\frac{27}{2144 - 352 \sqrt{13}} \cdot \frac{2144 + 352 \sqrt{13}}{2144 + 352 \sqrt{13}})$

Step 4.2.2.6

Multiply $\frac{27}{2144 - 352 \sqrt{13}}$ by $\frac{2144 + 352 \sqrt{13}}{2144 + 352 \sqrt{13}}$ .

$(- 1 - \sqrt{13}) \frac{27 (2144 + 352 \sqrt{13})}{(2144 - 352 \sqrt{13}) (2144 + 352 \sqrt{13})}$

Step 4.2.2.7

Expand the denominator using the FOIL method.

$(- 1 - \sqrt{13}) \frac{27 (2144 + 352 \sqrt{13})}{4596736 + 754688 \sqrt{13} - 754688 \sqrt{13} - 123904 {\sqrt{13}}^{2}}$

Step 4.2.2.8

Simplify.

$(- 1 - \sqrt{13}) \frac{27 (2144 + 352 \sqrt{13})}{2985984}$

Step 4.2.2.9

Cancel the common factors.

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Step 4.2.2.9.1

Factor $27$ out of $2985984$ .

$(- 1 - \sqrt{13}) \frac{27 (2144 + 352 \sqrt{13})}{27 \cdot 110592}$

Step 4.2.2.9.2

Cancel the common factor.

$(- 1 - \sqrt{13}) \frac{27 (2144 + 352 \sqrt{13})}{27 \cdot 110592}$

Step 4.2.2.9.3

Rewrite the expression.

$(- 1 - \sqrt{13}) \frac{2144 + 352 \sqrt{13}}{110592}$

Step 4.2.2.10

Simplify terms.

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Step 4.2.2.10.1

Cancel the common factor of $2144 + 352 \sqrt{13}$ and $110592$ .

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Step 4.2.2.10.1.1

Factor $32$ out of $2144$ .

$(- 1 - \sqrt{13}) \frac{32 (67) + 352 \sqrt{13}}{110592}$

Step 4.2.2.10.1.2

Factor $32$ out of $352 \sqrt{13}$ .

$(- 1 - \sqrt{13}) \frac{32 (67) + 32 (11 \sqrt{13})}{110592}$

Step 4.2.2.10.1.3

Factor $32$ out of $32 (67) + 32 (11 \sqrt{13})$ .

$(- 1 - \sqrt{13}) \frac{32 (67 + 11 \sqrt{13})}{110592}$

Step 4.2.2.10.1.4

Cancel the common factors.

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Step 4.2.2.10.1.4.1

Factor $32$ out of $110592$ .

$(- 1 - \sqrt{13}) \frac{32 (67 + 11 \sqrt{13})}{32 \cdot 3456}$

Step 4.2.2.10.1.4.2

Cancel the common factor.

$(- 1 - \sqrt{13}) \frac{32 (67 + 11 \sqrt{13})}{32 \cdot 3456}$

Step 4.2.2.10.1.4.3

Rewrite the expression.

$(- 1 - \sqrt{13}) \frac{67 + 11 \sqrt{13}}{3456}$

Step 4.2.2.10.2

Apply the distributive property.

$- 1 \frac{67 + 11 \sqrt{13}}{3456} - \sqrt{13} \frac{67 + 11 \sqrt{13}}{3456}$

Step 4.2.2.10.3

Rewrite $- 1 \frac{67 + 11 \sqrt{13}}{3456}$ as $- \frac{67 + 11 \sqrt{13}}{3456}$ .

$- \frac{67 + 11 \sqrt{13}}{3456} - \sqrt{13} \frac{67 + 11 \sqrt{13}}{3456}$

Step 4.2.2.10.4

Combine $\frac{67 + 11 \sqrt{13}}{3456}$ and $\sqrt{13}$ .

$- \frac{67 + 11 \sqrt{13}}{3456} - \frac{(67 + 11 \sqrt{13}) \sqrt{13}}{3456}$

Step 4.2.2.10.5

Combine the numerators over the common denominator.

$\frac{- (67 + 11 \sqrt{13}) - (67 + 11 \sqrt{13}) \sqrt{13}}{3456}$

Step 4.2.2.11

Simplify each term.

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Step 4.2.2.11.1

Apply the distributive property.

$\frac{- 1 \cdot 67 - (11 \sqrt{13}) - (67 + 11 \sqrt{13}) \sqrt{13}}{3456}$

Step 4.2.2.11.2

Multiply $- 1$ by $67$ .

$\frac{- 67 - (11 \sqrt{13}) - (67 + 11 \sqrt{13}) \sqrt{13}}{3456}$

Step 4.2.2.11.3

Multiply $11$ by $- 1$ .

$\frac{- 67 - 11 \sqrt{13} - (67 + 11 \sqrt{13}) \sqrt{13}}{3456}$

Step 4.2.2.11.4

Apply the distributive property.

$\frac{- 67 - 11 \sqrt{13} + (- 1 \cdot 67 - (11 \sqrt{13})) \sqrt{13}}{3456}$

Step 4.2.2.11.5

Multiply $- 1$ by $67$ .

$\frac{- 67 - 11 \sqrt{13} + (- 67 - (11 \sqrt{13})) \sqrt{13}}{3456}$

Step 4.2.2.11.6

Multiply $11$ by $- 1$ .

$\frac{- 67 - 11 \sqrt{13} + (- 67 - 11 \sqrt{13}) \sqrt{13}}{3456}$

Step 4.2.2.11.7

Apply the distributive property.

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 \sqrt{13} \sqrt{13}}{3456}$

Step 4.2.2.11.8

Multiply $- 11 \sqrt{13} \sqrt{13}$ .

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Step 4.2.2.11.8.1

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

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 ({\sqrt{13}}^{1} \sqrt{13})}{3456}$

Step 4.2.2.11.8.2

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

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 ({\sqrt{13}}^{1} {\sqrt{13}}^{1})}{3456}$

Step 4.2.2.11.8.3

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

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 {\sqrt{13}}^{1 + 1}}{3456}$

Step 4.2.2.11.8.4

Add $1$ and $1$ .

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 {\sqrt{13}}^{2}}{3456}$

Step 4.2.2.11.9

Simplify each term.

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Step 4.2.2.11.9.1

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

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Step 4.2.2.11.9.1.1

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

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 {(13^{\frac{1}{2}})}^{2}}{3456}$

Step 4.2.2.11.9.1.2

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

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 \cdot 13^{\frac{1}{2} \cdot 2}}{3456}$

Step 4.2.2.11.9.1.3

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

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 \cdot 13^{\frac{2}{2}}}{3456}$

Step 4.2.2.11.9.1.4

Cancel the common factor of $2$ .

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Step 4.2.2.11.9.1.4.1

Cancel the common factor.

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 \cdot 13^{\frac{2}{2}}}{3456}$

Step 4.2.2.11.9.1.4.2

Rewrite the expression.

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 \cdot 13^{1}}{3456}$

Step 4.2.2.11.9.1.5

Evaluate the exponent.

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 11 \cdot 13}{3456}$

Step 4.2.2.11.9.2

Multiply $- 11$ by $13$ .

$\frac{- 67 - 11 \sqrt{13} - 67 \sqrt{13} - 143}{3456}$

Step 4.2.2.12

Simplify terms.

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Step 4.2.2.12.1

Subtract $143$ from $- 67$ .

$\frac{- 210 - 11 \sqrt{13} - 67 \sqrt{13}}{3456}$

Step 4.2.2.12.2

Subtract $67 \sqrt{13}$ from $- 11 \sqrt{13}$ .

$\frac{- 210 - 78 \sqrt{13}}{3456}$

Step 4.2.2.12.3

Cancel the common factor of $- 210 - 78 \sqrt{13}$ and $3456$ .

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Step 4.2.2.12.3.1

Factor $6$ out of $- 210$ .

$\frac{6 (- 35) - 78 \sqrt{13}}{3456}$

Step 4.2.2.12.3.2

Factor $6$ out of $- 78 \sqrt{13}$ .

$\frac{6 (- 35) + 6 (- 13 \sqrt{13})}{3456}$

Step 4.2.2.12.3.3

Factor $6$ out of $6 (- 35) + 6 (- 13 \sqrt{13})$ .

$\frac{6 (- 35 - 13 \sqrt{13})}{3456}$

Step 4.2.2.12.3.4

Cancel the common factors.

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Step 4.2.2.12.3.4.1

Factor $6$ out of $3456$ .

$\frac{6 (- 35 - 13 \sqrt{13})}{6 \cdot 576}$

Step 4.2.2.12.3.4.2

Cancel the common factor.

$\frac{6 (- 35 - 13 \sqrt{13})}{6 \cdot 576}$

Step 4.2.2.12.3.4.3

Rewrite the expression.

$\frac{- 35 - 13 \sqrt{13}}{576}$

Step 4.2.2.12.4

Rewrite $- 35$ as $- 1 (35)$ .

$\frac{- 1 (35) - 13 \sqrt{13}}{576}$

Step 4.2.2.12.5

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

$\frac{- 1 (35) - (13 \sqrt{13})}{576}$

Step 4.2.2.12.6

Factor $- 1$ out of $- 1 (35) - (13 \sqrt{13})$ .

$\frac{- 1 (35 + 13 \sqrt{13})}{576}$

Step 4.2.2.12.7

Move the negative in front of the fraction.

$- \frac{35 + 13 \sqrt{13}}{576}$

Step 4.3

List all of the points.

$(\frac{5 + \sqrt{13}}{3}, - \frac{35 - 13 \sqrt{13}}{576}), (\frac{5 - \sqrt{13}}{3}, - \frac{35 + 13 \sqrt{13}}{576})$

Step 5