Calculus Examples

$y = - x^{3} - 4 x^{2} - 3 x + 5$

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

By the Sum Rule, the derivative of $- x^{3} - 4 x^{2} - 3 x + 5$ with respect to $x$ is $\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$

Step 1.1.2

Evaluate $\frac{d}{d x} [- x^{3}]$ .

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Step 1.1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

$- \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$

Step 1.1.2.2

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

$- (3 x^{2}) + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$

Step 1.1.2.3

Multiply $3$ by $- 1$ .

$- 3 x^{2} + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

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Step 1.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$- 3 x^{2} - 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$

Step 1.1.3.2

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

$- 3 x^{2} - 4 (2 x) + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$

Step 1.1.3.3

Multiply $2$ by $- 4$ .

$- 3 x^{2} - 8 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [5]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 1.1.4.1

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

$- 3 x^{2} - 8 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 1.1.4.2

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

$- 3 x^{2} - 8 x - 3 \cdot 1 + \frac{d}{d x} [5]$

Step 1.1.4.3

Multiply $- 3$ by $1$ .

$- 3 x^{2} - 8 x - 3 + \frac{d}{d x} [5]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$- 3 x^{2} - 8 x - 3 + 0$

Step 1.1.5.2

Add $- 3 x^{2} - 8 x - 3$ and $0$ .

$f' (x) = - 3 x^{2} - 8 x - 3$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- 3 x^{2} - 8 x - 3$ .

$- 3 x^{2} - 8 x - 3$

Step 2

Set the first derivative equal to $0$ then solve the equation $- 3 x^{2} - 8 x - 3 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- 3 x^{2} - 8 x - 3 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = - 3$ , $b = - 8$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (- 3 \cdot - 3)}}{2 \cdot - 3}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 3 \cdot - 3}}{2 \cdot - 3}$

Step 2.4.1.2

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

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Step 2.4.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 12 \cdot - 3}}{2 \cdot - 3}$

Step 2.4.1.2.2

Multiply $12$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 - 36}}{2 \cdot - 3}$

Step 2.4.1.3

Subtract $36$ from $64$ .

$x = \frac{8 \pm \sqrt{28}}{2 \cdot - 3}$

Step 2.4.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 2.4.1.4.1

Factor $4$ out of $28$ .

$x = \frac{8 \pm \sqrt{4 (7)}}{2 \cdot - 3}$

Step 2.4.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot - 3}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{7}}{2 \cdot - 3}$

Step 2.4.2

Multiply $2$ by $- 3$ .

$x = \frac{8 \pm 2 \sqrt{7}}{- 6}$

Step 2.4.3

Simplify $\frac{8 \pm 2 \sqrt{7}}{- 6}$ .

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

Step 2.4.4

Move the negative in front of the fraction.

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

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 3 \cdot - 3}}{2 \cdot - 3}$

Step 2.5.1.2

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

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Step 2.5.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 12 \cdot - 3}}{2 \cdot - 3}$

Step 2.5.1.2.2

Multiply $12$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 - 36}}{2 \cdot - 3}$

Step 2.5.1.3

Subtract $36$ from $64$ .

$x = \frac{8 \pm \sqrt{28}}{2 \cdot - 3}$

Step 2.5.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 2.5.1.4.1

Factor $4$ out of $28$ .

$x = \frac{8 \pm \sqrt{4 (7)}}{2 \cdot - 3}$

Step 2.5.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot - 3}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{7}}{2 \cdot - 3}$

Step 2.5.2

Multiply $2$ by $- 3$ .

$x = \frac{8 \pm 2 \sqrt{7}}{- 6}$

Step 2.5.3

Simplify $\frac{8 \pm 2 \sqrt{7}}{- 6}$ .

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

Step 2.5.4

Move the negative in front of the fraction.

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

Step 2.5.5

Change the $\pm$ to $+$ .

$x = - \frac{4 + \sqrt{7}}{3}$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 3 \cdot - 3}}{2 \cdot - 3}$

Step 2.6.1.2

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

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Step 2.6.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 12 \cdot - 3}}{2 \cdot - 3}$

Step 2.6.1.2.2

Multiply $12$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 - 36}}{2 \cdot - 3}$

Step 2.6.1.3

Subtract $36$ from $64$ .

$x = \frac{8 \pm \sqrt{28}}{2 \cdot - 3}$

Step 2.6.1.4

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 2.6.1.4.1

Factor $4$ out of $28$ .

$x = \frac{8 \pm \sqrt{4 (7)}}{2 \cdot - 3}$

Step 2.6.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot - 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{7}}{2 \cdot - 3}$

Step 2.6.2

Multiply $2$ by $- 3$ .

$x = \frac{8 \pm 2 \sqrt{7}}{- 6}$

Step 2.6.3

Simplify $\frac{8 \pm 2 \sqrt{7}}{- 6}$ .

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

Step 2.6.4

Move the negative in front of the fraction.

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

Step 2.6.5

Change the $\pm$ to $-$ .

$x = - \frac{4 - \sqrt{7}}{3}$

Step 2.7

The final answer is the combination of both solutions.

$x = - \frac{4 + \sqrt{7}}{3}, - \frac{4 - \sqrt{7}}{3}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

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

Step 4

Evaluate $- x^{3} - 4 x^{2} - 3 x + 5$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = - \frac{4 + \sqrt{7}}{3}$ .

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Step 4.1.1

Substitute $- \frac{4 + \sqrt{7}}{3}$ for $x$ .

$- {(- \frac{4 + \sqrt{7}}{3})}^{3} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.1.1

Apply the product rule to $- \frac{4 + \sqrt{7}}{3}$ .

$- ({(- 1)}^{3} {(\frac{4 + \sqrt{7}}{3})}^{3}) - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.1.2

Apply the product rule to $\frac{4 + \sqrt{7}}{3}$ .

$- ({(- 1)}^{3} \frac{{(4 + \sqrt{7})}^{3}}{3^{3}}) - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.2

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 4.1.2.1.2.1

Move ${(- 1)}^{3}$ .

${(- 1)}^{3} \cdot - 1 \frac{{(4 + \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.2.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

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Step 4.1.2.1.2.2.1

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

${(- 1)}^{3} \cdot {(- 1)}^{1} \frac{{(4 + \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.2.2.2

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

${(- 1)}^{3 + 1} \frac{{(4 + \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.2.3

Add $3$ and $1$ .

${(- 1)}^{4} \frac{{(4 + \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.3

Raise $- 1$ to the power of $4$ .

$1 \frac{{(4 + \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.4

Multiply $\frac{{(4 + \sqrt{7})}^{3}}{3^{3}}$ by $1$ .

$\frac{{(4 + \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.5

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

$\frac{{(4 + \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.6

Use the Binomial Theorem.

$\frac{4^{3} + 3 \cdot 4^{2} \sqrt{7} + 3 \cdot 4 {\sqrt{7}}^{2} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7

Simplify each term.

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Step 4.1.2.1.7.1

Raise $4$ to the power of $3$ .

$\frac{64 + 3 \cdot 4^{2} \sqrt{7} + 3 \cdot 4 {\sqrt{7}}^{2} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.2

Raise $4$ to the power of $2$ .

$\frac{64 + 3 \cdot 16 \sqrt{7} + 3 \cdot 4 {\sqrt{7}}^{2} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.3

Multiply $3$ by $16$ .

$\frac{64 + 48 \sqrt{7} + 3 \cdot 4 {\sqrt{7}}^{2} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.4

Multiply $3$ by $4$ .

$\frac{64 + 48 \sqrt{7} + 12 {\sqrt{7}}^{2} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.5

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

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Step 4.1.2.1.7.5.1

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

$\frac{64 + 48 \sqrt{7} + 12 {(7^{\frac{1}{2}})}^{2} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.5.2

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

$\frac{64 + 48 \sqrt{7} + 12 \cdot 7^{\frac{1}{2} \cdot 2} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.5.3

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

$\frac{64 + 48 \sqrt{7} + 12 \cdot 7^{\frac{2}{2}} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.5.4

Cancel the common factor of $2$ .

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Step 4.1.2.1.7.5.4.1

Cancel the common factor.

$\frac{64 + 48 \sqrt{7} + 12 \cdot 7^{\frac{2}{2}} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.5.4.2

Rewrite the expression.

$\frac{64 + 48 \sqrt{7} + 12 \cdot 7^{1} + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.5.5

Evaluate the exponent.

$\frac{64 + 48 \sqrt{7} + 12 \cdot 7 + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.6

Multiply $12$ by $7$ .

$\frac{64 + 48 \sqrt{7} + 84 + {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.7

Rewrite ${\sqrt{7}}^{3}$ as $\sqrt{7^{3}}$ .

$\frac{64 + 48 \sqrt{7} + 84 + \sqrt{7^{3}}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.8

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

$\frac{64 + 48 \sqrt{7} + 84 + \sqrt{343}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.9

Rewrite $343$ as $7^{2} \cdot 7$ .

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Step 4.1.2.1.7.9.1

Factor $49$ out of $343$ .

$\frac{64 + 48 \sqrt{7} + 84 + \sqrt{49 (7)}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.9.2

Rewrite $49$ as $7^{2}$ .

$\frac{64 + 48 \sqrt{7} + 84 + \sqrt{7^{2} \cdot 7}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.7.10

Pull terms out from under the radical.

$\frac{64 + 48 \sqrt{7} + 84 + 7 \sqrt{7}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.8

Add $64$ and $84$ .

$\frac{148 + 48 \sqrt{7} + 7 \sqrt{7}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.9

Add $48 \sqrt{7}$ and $7 \sqrt{7}$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 {(- \frac{4 + \sqrt{7}}{3})}^{2} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.2.1.10.1

Apply the product rule to $- \frac{4 + \sqrt{7}}{3}$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 ({(- 1)}^{2} {(\frac{4 + \sqrt{7}}{3})}^{2}) - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.10.2

Apply the product rule to $\frac{4 + \sqrt{7}}{3}$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 ({(- 1)}^{2} \frac{{(4 + \sqrt{7})}^{2}}{3^{2}}) - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.11

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

$\frac{148 + 55 \sqrt{7}}{27} - 4 (1 \frac{{(4 + \sqrt{7})}^{2}}{3^{2}}) - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.12

Multiply $\frac{{(4 + \sqrt{7})}^{2}}{3^{2}}$ by $1$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{{(4 + \sqrt{7})}^{2}}{3^{2}} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.13

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

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{{(4 + \sqrt{7})}^{2}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.14

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

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{(4 + \sqrt{7}) (4 + \sqrt{7})}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.15

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

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Step 4.1.2.1.15.1

Apply the distributive property.

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{4 (4 + \sqrt{7}) + \sqrt{7} (4 + \sqrt{7})}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.15.2

Apply the distributive property.

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{4 \cdot 4 + 4 \sqrt{7} + \sqrt{7} (4 + \sqrt{7})}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.15.3

Apply the distributive property.

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{4 \cdot 4 + 4 \sqrt{7} + \sqrt{7} \cdot 4 + \sqrt{7} \sqrt{7}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.16

Simplify and combine like terms.

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Step 4.1.2.1.16.1

Simplify each term.

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Step 4.1.2.1.16.1.1

Multiply $4$ by $4$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{16 + 4 \sqrt{7} + \sqrt{7} \cdot 4 + \sqrt{7} \sqrt{7}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.16.1.2

Move $4$ to the left of $\sqrt{7}$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{16 + 4 \sqrt{7} + 4 \cdot \sqrt{7} + \sqrt{7} \sqrt{7}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.16.1.3

Combine using the product rule for radicals.

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{16 + 4 \sqrt{7} + 4 \sqrt{7} + \sqrt{7 \cdot 7}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.16.1.4

Multiply $7$ by $7$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{16 + 4 \sqrt{7} + 4 \sqrt{7} + \sqrt{49}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.16.1.5

Rewrite $49$ as $7^{2}$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{16 + 4 \sqrt{7} + 4 \sqrt{7} + \sqrt{7^{2}}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.16.1.6

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

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{16 + 4 \sqrt{7} + 4 \sqrt{7} + 7}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.16.2

Add $16$ and $7$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{23 + 4 \sqrt{7} + 4 \sqrt{7}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.16.3

Add $4 \sqrt{7}$ and $4 \sqrt{7}$ .

$\frac{148 + 55 \sqrt{7}}{27} - 4 \frac{23 + 8 \sqrt{7}}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.17

Combine $- 4$ and $\frac{23 + 8 \sqrt{7}}{9}$ .

$\frac{148 + 55 \sqrt{7}}{27} + \frac{- 4 (23 + 8 \sqrt{7})}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.18

Move the negative in front of the fraction.

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7})}{9} - 3 (- \frac{4 + \sqrt{7}}{3}) + 5$

Step 4.1.2.1.19

Cancel the common factor of $3$ .

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Step 4.1.2.1.19.1

Move the leading negative in $- \frac{4 + \sqrt{7}}{3}$ into the numerator.

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7})}{9} - 3 \frac{- (4 + \sqrt{7})}{3} + 5$

Step 4.1.2.1.19.2

Factor $3$ out of $- 3$ .

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7})}{9} + 3 (- 1) \frac{- (4 + \sqrt{7})}{3} + 5$

Step 4.1.2.1.19.3

Cancel the common factor.

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7})}{9} + 3 \cdot - 1 \frac{- (4 + \sqrt{7})}{3} + 5$

Step 4.1.2.1.19.4

Rewrite the expression.

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7})}{9} - 1 (- (4 + \sqrt{7})) + 5$

Step 4.1.2.1.20

Multiply $- 1$ by $- 1$ .

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7})}{9} + 1 (4 + \sqrt{7}) + 5$

Step 4.1.2.1.21

Multiply $4 + \sqrt{7}$ by $1$ .

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7})}{9} + 4 + \sqrt{7} + 5$

Step 4.1.2.2

To write $- \frac{4 (23 + 8 \sqrt{7})}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7})}{9} \cdot \frac{3}{3} + 4 + \sqrt{7} + 5$

Step 4.1.2.3

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.3.1

Multiply $\frac{4 (23 + 8 \sqrt{7})}{9}$ by $\frac{3}{3}$ .

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7}) \cdot 3}{9 \cdot 3} + 4 + \sqrt{7} + 5$

Step 4.1.2.3.2

Multiply $9$ by $3$ .

$\frac{148 + 55 \sqrt{7}}{27} - \frac{4 (23 + 8 \sqrt{7}) \cdot 3}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.4

Combine the numerators over the common denominator.

$\frac{148 + 55 \sqrt{7} - 4 (23 + 8 \sqrt{7}) \cdot 3}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Apply the distributive property.

$\frac{148 + 55 \sqrt{7} + (- 4 \cdot 23 - 4 (8 \sqrt{7})) \cdot 3}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.5.2

Multiply $- 4$ by $23$ .

$\frac{148 + 55 \sqrt{7} + (- 92 - 4 (8 \sqrt{7})) \cdot 3}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.5.3

Multiply $8$ by $- 4$ .

$\frac{148 + 55 \sqrt{7} + (- 92 - 32 \sqrt{7}) \cdot 3}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.5.4

Apply the distributive property.

$\frac{148 + 55 \sqrt{7} - 92 \cdot 3 - 32 \sqrt{7} \cdot 3}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.5.5

Multiply $- 92$ by $3$ .

$\frac{148 + 55 \sqrt{7} - 276 - 32 \sqrt{7} \cdot 3}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.5.6

Multiply $3$ by $- 32$ .

$\frac{148 + 55 \sqrt{7} - 276 - 96 \sqrt{7}}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.5.7

Subtract $276$ from $148$ .

$\frac{- 128 + 55 \sqrt{7} - 96 \sqrt{7}}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.5.8

Subtract $96 \sqrt{7}$ from $55 \sqrt{7}$ .

$\frac{- 128 - 41 \sqrt{7}}{27} + 4 + \sqrt{7} + 5$

Step 4.1.2.6

To write $4$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 128 - 41 \sqrt{7}}{27} + 4 \cdot \frac{27}{27} + \sqrt{7} + 5$

Step 4.1.2.7

Combine $4$ and $\frac{27}{27}$ .

$\frac{- 128 - 41 \sqrt{7}}{27} + \frac{4 \cdot 27}{27} + \sqrt{7} + 5$

Step 4.1.2.8

Simplify the expression.

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Step 4.1.2.8.1

Combine the numerators over the common denominator.

$\frac{- 128 - 41 \sqrt{7} + 4 \cdot 27}{27} + \sqrt{7} + 5$

Step 4.1.2.8.2

Multiply $4$ by $27$ .

$\frac{- 128 - 41 \sqrt{7} + 108}{27} + \sqrt{7} + 5$

Step 4.1.2.8.3

Add $- 128$ and $108$ .

$\frac{- 20 - 41 \sqrt{7}}{27} + \sqrt{7} + 5$

Step 4.1.2.9

To write $\sqrt{7}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 20 - 41 \sqrt{7}}{27} + \sqrt{7} \cdot \frac{27}{27} + 5$

Step 4.1.2.10

Combine $\sqrt{7}$ and $\frac{27}{27}$ .

$\frac{- 20 - 41 \sqrt{7}}{27} + \frac{\sqrt{7} \cdot 27}{27} + 5$

Step 4.1.2.11

Simplify the expression.

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Step 4.1.2.11.1

Combine the numerators over the common denominator.

$\frac{- 20 - 41 \sqrt{7} + \sqrt{7} \cdot 27}{27} + 5$

Step 4.1.2.11.2

Reorder the factors of $\sqrt{7} \cdot 27$ .

$\frac{- 20 - 41 \sqrt{7} + 27 \sqrt{7}}{27} + 5$

Step 4.1.2.12

Add $- 41 \sqrt{7}$ and $27 \sqrt{7}$ .

$\frac{- 20 - 14 \sqrt{7}}{27} + 5$

Step 4.1.2.13

To write $5$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 20 - 14 \sqrt{7}}{27} + 5 \cdot \frac{27}{27}$

Step 4.1.2.14

Combine fractions.

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Step 4.1.2.14.1

Combine $5$ and $\frac{27}{27}$ .

$\frac{- 20 - 14 \sqrt{7}}{27} + \frac{5 \cdot 27}{27}$

Step 4.1.2.14.2

Combine the numerators over the common denominator.

$\frac{- 20 - 14 \sqrt{7} + 5 \cdot 27}{27}$

Step 4.1.2.15

Simplify the numerator.

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Step 4.1.2.15.1

Multiply $5$ by $27$ .

$\frac{- 20 - 14 \sqrt{7} + 135}{27}$

Step 4.1.2.15.2

Add $- 20$ and $135$ .

$\frac{115 - 14 \sqrt{7}}{27}$

Step 4.2

Evaluate at $x = - \frac{4 - \sqrt{7}}{3}$ .

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Step 4.2.1

Substitute $- \frac{4 - \sqrt{7}}{3}$ for $x$ .

$- {(- \frac{4 - \sqrt{7}}{3})}^{3} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.1.1.1

Apply the product rule to $- \frac{4 - \sqrt{7}}{3}$ .

$- ({(- 1)}^{3} {(\frac{4 - \sqrt{7}}{3})}^{3}) - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.1.2

Apply the product rule to $\frac{4 - \sqrt{7}}{3}$ .

$- ({(- 1)}^{3} \frac{{(4 - \sqrt{7})}^{3}}{3^{3}}) - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.2

Multiply $- 1$ by ${(- 1)}^{3}$ by adding the exponents.

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Step 4.2.2.1.2.1

Move ${(- 1)}^{3}$ .

${(- 1)}^{3} \cdot - 1 \frac{{(4 - \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.2.2

Multiply ${(- 1)}^{3}$ by $- 1$ .

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Step 4.2.2.1.2.2.1

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

${(- 1)}^{3} \cdot {(- 1)}^{1} \frac{{(4 - \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.2.2.2

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

${(- 1)}^{3 + 1} \frac{{(4 - \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.2.3

Add $3$ and $1$ .

${(- 1)}^{4} \frac{{(4 - \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.3

Raise $- 1$ to the power of $4$ .

$1 \frac{{(4 - \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.4

Multiply $\frac{{(4 - \sqrt{7})}^{3}}{3^{3}}$ by $1$ .

$\frac{{(4 - \sqrt{7})}^{3}}{3^{3}} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.5

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

$\frac{{(4 - \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.6

Use the Binomial Theorem.

$\frac{4^{3} + 3 \cdot 4^{2} (- \sqrt{7}) + 3 \cdot 4 {(- \sqrt{7})}^{2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7

Simplify each term.

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Step 4.2.2.1.7.1

Raise $4$ to the power of $3$ .

$\frac{64 + 3 \cdot 4^{2} (- \sqrt{7}) + 3 \cdot 4 {(- \sqrt{7})}^{2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.2

Raise $4$ to the power of $2$ .

$\frac{64 + 3 \cdot 16 (- \sqrt{7}) + 3 \cdot 4 {(- \sqrt{7})}^{2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.3

Multiply $3$ by $16$ .

$\frac{64 + 48 (- \sqrt{7}) + 3 \cdot 4 {(- \sqrt{7})}^{2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.4

Multiply $- 1$ by $48$ .

$\frac{64 - 48 \sqrt{7} + 3 \cdot 4 {(- \sqrt{7})}^{2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.5

Multiply $3$ by $4$ .

$\frac{64 - 48 \sqrt{7} + 12 {(- \sqrt{7})}^{2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.6

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

$\frac{64 - 48 \sqrt{7} + 12 ({(- 1)}^{2} {\sqrt{7}}^{2}) + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.7

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

$\frac{64 - 48 \sqrt{7} + 12 (1 {\sqrt{7}}^{2}) + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.8

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

$\frac{64 - 48 \sqrt{7} + 12 {\sqrt{7}}^{2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.9

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

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Step 4.2.2.1.7.9.1

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

$\frac{64 - 48 \sqrt{7} + 12 {(7^{\frac{1}{2}})}^{2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.9.2

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

$\frac{64 - 48 \sqrt{7} + 12 \cdot 7^{\frac{1}{2} \cdot 2} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.9.3

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

$\frac{64 - 48 \sqrt{7} + 12 \cdot 7^{\frac{2}{2}} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.9.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.7.9.4.1

Cancel the common factor.

$\frac{64 - 48 \sqrt{7} + 12 \cdot 7^{\frac{2}{2}} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.9.4.2

Rewrite the expression.

$\frac{64 - 48 \sqrt{7} + 12 \cdot 7^{1} + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.9.5

Evaluate the exponent.

$\frac{64 - 48 \sqrt{7} + 12 \cdot 7 + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.10

Multiply $12$ by $7$ .

$\frac{64 - 48 \sqrt{7} + 84 + {(- \sqrt{7})}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.11

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

$\frac{64 - 48 \sqrt{7} + 84 + {(- 1)}^{3} {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.12

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

$\frac{64 - 48 \sqrt{7} + 84 - {\sqrt{7}}^{3}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.13

Rewrite ${\sqrt{7}}^{3}$ as $\sqrt{7^{3}}$ .

$\frac{64 - 48 \sqrt{7} + 84 - \sqrt{7^{3}}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.14

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

$\frac{64 - 48 \sqrt{7} + 84 - \sqrt{343}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.15

Rewrite $343$ as $7^{2} \cdot 7$ .

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Step 4.2.2.1.7.15.1

Factor $49$ out of $343$ .

$\frac{64 - 48 \sqrt{7} + 84 - \sqrt{49 (7)}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.15.2

Rewrite $49$ as $7^{2}$ .

$\frac{64 - 48 \sqrt{7} + 84 - \sqrt{7^{2} \cdot 7}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.16

Pull terms out from under the radical.

$\frac{64 - 48 \sqrt{7} + 84 - (7 \sqrt{7})}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.7.17

Multiply $7$ by $- 1$ .

$\frac{64 - 48 \sqrt{7} + 84 - 7 \sqrt{7}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.8

Add $64$ and $84$ .

$\frac{148 - 48 \sqrt{7} - 7 \sqrt{7}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.9

Subtract $7 \sqrt{7}$ from $- 48 \sqrt{7}$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 {(- \frac{4 - \sqrt{7}}{3})}^{2} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.2.1.10.1

Apply the product rule to $- \frac{4 - \sqrt{7}}{3}$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 ({(- 1)}^{2} {(\frac{4 - \sqrt{7}}{3})}^{2}) - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.10.2

Apply the product rule to $\frac{4 - \sqrt{7}}{3}$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 ({(- 1)}^{2} \frac{{(4 - \sqrt{7})}^{2}}{3^{2}}) - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.11

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 (1 \frac{{(4 - \sqrt{7})}^{2}}{3^{2}}) - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.12

Multiply $\frac{{(4 - \sqrt{7})}^{2}}{3^{2}}$ by $1$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{{(4 - \sqrt{7})}^{2}}{3^{2}} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.13

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{{(4 - \sqrt{7})}^{2}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.14

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{(4 - \sqrt{7}) (4 - \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.15

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

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Step 4.2.2.1.15.1

Apply the distributive property.

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{4 (4 - \sqrt{7}) - \sqrt{7} (4 - \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.15.2

Apply the distributive property.

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{4 \cdot 4 + 4 (- \sqrt{7}) - \sqrt{7} (4 - \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.15.3

Apply the distributive property.

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{4 \cdot 4 + 4 (- \sqrt{7}) - \sqrt{7} \cdot 4 - \sqrt{7} (- \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16

Simplify and combine like terms.

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Step 4.2.2.1.16.1

Simplify each term.

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Step 4.2.2.1.16.1.1

Multiply $4$ by $4$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 + 4 (- \sqrt{7}) - \sqrt{7} \cdot 4 - \sqrt{7} (- \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.2

Multiply $- 1$ by $4$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - \sqrt{7} \cdot 4 - \sqrt{7} (- \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.3

Multiply $4$ by $- 1$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} - \sqrt{7} (- \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.4

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

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Step 4.2.2.1.16.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + 1 \sqrt{7} \sqrt{7}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.4.2

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + \sqrt{7} \sqrt{7}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.4.3

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + {\sqrt{7}}^{1} \sqrt{7}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.4.4

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + {\sqrt{7}}^{1} {\sqrt{7}}^{1}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.4.5

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + {\sqrt{7}}^{1 + 1}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.4.6

Add $1$ and $1$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + {\sqrt{7}}^{2}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.5

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

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Step 4.2.2.1.16.1.5.1

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + {(7^{\frac{1}{2}})}^{2}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.5.2

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + 7^{\frac{1}{2} \cdot 2}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.5.3

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

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + 7^{\frac{2}{2}}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.16.1.5.4.1

Cancel the common factor.

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + 7^{\frac{2}{2}}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.5.4.2

Rewrite the expression.

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + 7^{1}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.1.5.5

Evaluate the exponent.

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{16 - 4 \sqrt{7} - 4 \sqrt{7} + 7}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.2

Add $16$ and $7$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{23 - 4 \sqrt{7} - 4 \sqrt{7}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.16.3

Subtract $4 \sqrt{7}$ from $- 4 \sqrt{7}$ .

$\frac{148 - 55 \sqrt{7}}{27} - 4 \frac{23 - 8 \sqrt{7}}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.17

Combine $- 4$ and $\frac{23 - 8 \sqrt{7}}{9}$ .

$\frac{148 - 55 \sqrt{7}}{27} + \frac{- 4 (23 - 8 \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.18

Move the negative in front of the fraction.

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7})}{9} - 3 (- \frac{4 - \sqrt{7}}{3}) + 5$

Step 4.2.2.1.19

Cancel the common factor of $3$ .

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Step 4.2.2.1.19.1

Move the leading negative in $- \frac{4 - \sqrt{7}}{3}$ into the numerator.

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7})}{9} - 3 \frac{- (4 - \sqrt{7})}{3} + 5$

Step 4.2.2.1.19.2

Factor $3$ out of $- 3$ .

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7})}{9} + 3 (- 1) \frac{- (4 - \sqrt{7})}{3} + 5$

Step 4.2.2.1.19.3

Cancel the common factor.

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7})}{9} + 3 \cdot - 1 \frac{- (4 - \sqrt{7})}{3} + 5$

Step 4.2.2.1.19.4

Rewrite the expression.

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7})}{9} - 1 (- (4 - \sqrt{7})) + 5$

Step 4.2.2.1.20

Multiply $- 1$ by $- 1$ .

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7})}{9} + 1 (4 - \sqrt{7}) + 5$

Step 4.2.2.1.21

Multiply $4 - \sqrt{7}$ by $1$ .

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7})}{9} + 4 - \sqrt{7} + 5$

Step 4.2.2.2

To write $- \frac{4 (23 - 8 \sqrt{7})}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7})}{9} \cdot \frac{3}{3} + 4 - \sqrt{7} + 5$

Step 4.2.2.3

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.3.1

Multiply $\frac{4 (23 - 8 \sqrt{7})}{9}$ by $\frac{3}{3}$ .

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7}) \cdot 3}{9 \cdot 3} + 4 - \sqrt{7} + 5$

Step 4.2.2.3.2

Multiply $9$ by $3$ .

$\frac{148 - 55 \sqrt{7}}{27} - \frac{4 (23 - 8 \sqrt{7}) \cdot 3}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.4

Combine the numerators over the common denominator.

$\frac{148 - 55 \sqrt{7} - 4 (23 - 8 \sqrt{7}) \cdot 3}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.5

Simplify the numerator.

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Step 4.2.2.5.1

Apply the distributive property.

$\frac{148 - 55 \sqrt{7} + (- 4 \cdot 23 - 4 (- 8 \sqrt{7})) \cdot 3}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.5.2

Multiply $- 4$ by $23$ .

$\frac{148 - 55 \sqrt{7} + (- 92 - 4 (- 8 \sqrt{7})) \cdot 3}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.5.3

Multiply $- 8$ by $- 4$ .

$\frac{148 - 55 \sqrt{7} + (- 92 + 32 \sqrt{7}) \cdot 3}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.5.4

Apply the distributive property.

$\frac{148 - 55 \sqrt{7} - 92 \cdot 3 + 32 \sqrt{7} \cdot 3}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.5.5

Multiply $- 92$ by $3$ .

$\frac{148 - 55 \sqrt{7} - 276 + 32 \sqrt{7} \cdot 3}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.5.6

Multiply $3$ by $32$ .

$\frac{148 - 55 \sqrt{7} - 276 + 96 \sqrt{7}}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.5.7

Subtract $276$ from $148$ .

$\frac{- 128 - 55 \sqrt{7} + 96 \sqrt{7}}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.5.8

Add $- 55 \sqrt{7}$ and $96 \sqrt{7}$ .

$\frac{- 128 + 41 \sqrt{7}}{27} + 4 - \sqrt{7} + 5$

Step 4.2.2.6

To write $4$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 128 + 41 \sqrt{7}}{27} + 4 \cdot \frac{27}{27} - \sqrt{7} + 5$

Step 4.2.2.7

Combine $4$ and $\frac{27}{27}$ .

$\frac{- 128 + 41 \sqrt{7}}{27} + \frac{4 \cdot 27}{27} - \sqrt{7} + 5$

Step 4.2.2.8

Simplify the expression.

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Step 4.2.2.8.1

Combine the numerators over the common denominator.

$\frac{- 128 + 41 \sqrt{7} + 4 \cdot 27}{27} - \sqrt{7} + 5$

Step 4.2.2.8.2

Multiply $4$ by $27$ .

$\frac{- 128 + 41 \sqrt{7} + 108}{27} - \sqrt{7} + 5$

Step 4.2.2.8.3

Add $- 128$ and $108$ .

$\frac{- 20 + 41 \sqrt{7}}{27} - \sqrt{7} + 5$

Step 4.2.2.9

To write $- \sqrt{7}$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 20 + 41 \sqrt{7}}{27} - \sqrt{7} \cdot \frac{27}{27} + 5$

Step 4.2.2.10

Combine fractions.

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Step 4.2.2.10.1

Combine $- \sqrt{7}$ and $\frac{27}{27}$ .

$\frac{- 20 + 41 \sqrt{7}}{27} + \frac{- \sqrt{7} \cdot 27}{27} + 5$

Step 4.2.2.10.2

Combine the numerators over the common denominator.

$\frac{- 20 + 41 \sqrt{7} - \sqrt{7} \cdot 27}{27} + 5$

Step 4.2.2.11

Simplify the numerator.

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Step 4.2.2.11.1

Multiply $27$ by $- 1$ .

$\frac{- 20 + 41 \sqrt{7} - 27 \sqrt{7}}{27} + 5$

Step 4.2.2.11.2

Subtract $27 \sqrt{7}$ from $41 \sqrt{7}$ .

$\frac{- 20 + 14 \sqrt{7}}{27} + 5$

Step 4.2.2.12

To write $5$ as a fraction with a common denominator, multiply by $\frac{27}{27}$ .

$\frac{- 20 + 14 \sqrt{7}}{27} + 5 \cdot \frac{27}{27}$

Step 4.2.2.13

Combine fractions.

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Step 4.2.2.13.1

Combine $5$ and $\frac{27}{27}$ .

$\frac{- 20 + 14 \sqrt{7}}{27} + \frac{5 \cdot 27}{27}$

Step 4.2.2.13.2

Combine the numerators over the common denominator.

$\frac{- 20 + 14 \sqrt{7} + 5 \cdot 27}{27}$

Step 4.2.2.14

Simplify the numerator.

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Step 4.2.2.14.1

Multiply $5$ by $27$ .

$\frac{- 20 + 14 \sqrt{7} + 135}{27}$

Step 4.2.2.14.2

Add $- 20$ and $135$ .

$\frac{115 + 14 \sqrt{7}}{27}$

Step 4.3

List all of the points.

$(- \frac{4 + \sqrt{7}}{3}, \frac{115 - 14 \sqrt{7}}{27}), (- \frac{4 - \sqrt{7}}{3}, \frac{115 + 14 \sqrt{7}}{27})$

Step 5