Calculus Examples

$f (x) = \frac{3}{4} x^{4} + 4 x^{3} - 12 x^{2} + 24$

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 $\frac{3}{4} x^{4} + 4 x^{3} - 12 x^{2} + 24$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{4} x^{4}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$ .

$\frac{d}{d x} [\frac{3}{4} x^{4}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{3}{4} x^{4}]$ .

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Step 1.1.2.1

Since $\frac{3}{4}$ is constant with respect to $x$ , the derivative of $\frac{3}{4} x^{4}$ with respect to $x$ is $\frac{3}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{3}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

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 $n = 4$ .

$\frac{3}{4} (4 x^{3}) + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2.3

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

$\frac{4 \cdot 3}{4} x^{3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2.4

Multiply $4$ by $3$ .

$\frac{12}{4} x^{3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2.5

Combine $\frac{12}{4}$ and $x^{3}$ .

$\frac{12 x^{3}}{4} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2.6

Cancel the common factor of $12$ and $4$ .

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Step 1.1.2.6.1

Factor $4$ out of $12 x^{3}$ .

$\frac{4 (3 x^{3})}{4} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2.6.2

Cancel the common factors.

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Step 1.1.2.6.2.1

Factor $4$ out of $4$ .

$\frac{4 (3 x^{3})}{4 (1)} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2.6.2.2

Cancel the common factor.

$\frac{4 (3 x^{3})}{4 \cdot 1} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2.6.2.3

Rewrite the expression.

$\frac{3 x^{3}}{1} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.2.6.2.4

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

$3 x^{3} + \frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$3 x^{3} + 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

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 $n = 3$ .

$3 x^{3} + 4 (3 x^{2}) + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.3.3

Multiply $3$ by $4$ .

$3 x^{3} + 12 x^{2} + \frac{d}{d x} [- 12 x^{2}] + \frac{d}{d x} [24]$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 12 x^{2}]$ .

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Step 1.1.4.1

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12 x^{2}$ with respect to $x$ is $- 12 \frac{d}{d x} [x^{2}]$ .

$3 x^{3} + 12 x^{2} - 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [24]$

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 $n = 2$ .

$3 x^{3} + 12 x^{2} - 12 (2 x) + \frac{d}{d x} [24]$

Step 1.1.4.3

Multiply $2$ by $- 12$ .

$3 x^{3} + 12 x^{2} - 24 x + \frac{d}{d x} [24]$

Step 1.1.5

Differentiate using the Constant Rule.

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Step 1.1.5.1

Since $24$ is constant with respect to $x$ , the derivative of $24$ with respect to $x$ is $0$ .

$3 x^{3} + 12 x^{2} - 24 x + 0$

Step 1.1.5.2

Add $3 x^{3} + 12 x^{2} - 24 x$ and $0$ .

$f' (x) = 3 x^{3} + 12 x^{2} - 24 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{3} + 12 x^{2} - 24 x$ .

$3 x^{3} + 12 x^{2} - 24 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{3} + 12 x^{2} - 24 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{3} + 12 x^{2} - 24 x = 0$

Step 2.2

Factor $3 x$ out of $3 x^{3} + 12 x^{2} - 24 x$ .

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Step 2.2.1

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

$3 x (x^{2}) + 12 x^{2} - 24 x = 0$

Step 2.2.2

Factor $3 x$ out of $12 x^{2}$ .

$3 x (x^{2}) + 3 x (4 x) - 24 x = 0$

Step 2.2.3

Factor $3 x$ out of $- 24 x$ .

$3 x (x^{2}) + 3 x (4 x) + 3 x (- 8) = 0$

Step 2.2.4

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

$3 x (x^{2} + 4 x) + 3 x (- 8) = 0$

Step 2.2.5

Factor $3 x$ out of $3 x (x^{2} + 4 x) + 3 x (- 8)$ .

$3 x (x^{2} + 4 x - 8) = 0$

Step 2.3

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

$x = 0$

$x^{2} + 4 x - 8 = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $x^{2} + 4 x - 8$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} + 4 x - 8$ equal to $0$ .

$x^{2} + 4 x - 8 = 0$

Step 2.5.2

Solve $x^{2} + 4 x - 8 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = 1$ , $b = 4$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (1 \cdot - 8)}}{2 \cdot 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 8}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 4 \pm \sqrt{16 + 32}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Add $16$ and $32$ .

$x = \frac{- 4 \pm \sqrt{48}}{2 \cdot 1}$

Step 2.5.2.3.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.5.2.3.1.4.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.5.2.3.1.4.2

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

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

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

Step 2.5.2.3.3

Simplify $\frac{- 4 \pm 4 \sqrt{3}}{2}$ .

$x = - 2 \pm 2 \sqrt{3}$

Step 2.5.2.4

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

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Step 2.5.2.4.1

Simplify the numerator.

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Step 2.5.2.4.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 2.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 2.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 8}}{2 \cdot 1}$

Step 2.5.2.4.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 4 \pm \sqrt{16 + 32}}{2 \cdot 1}$

Step 2.5.2.4.1.3

Add $16$ and $32$ .

$x = \frac{- 4 \pm \sqrt{48}}{2 \cdot 1}$

Step 2.5.2.4.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.5.2.4.1.4.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.5.2.4.1.4.2

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

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.4.2

Multiply $2$ by $1$ .

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

Step 2.5.2.4.3

Simplify $\frac{- 4 \pm 4 \sqrt{3}}{2}$ .

$x = - 2 \pm 2 \sqrt{3}$

Step 2.5.2.4.4

Change the $\pm$ to $+$ .

$x = - 2 + 2 \sqrt{3}$

Step 2.5.2.5

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

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Step 2.5.2.5.1

Simplify the numerator.

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Step 2.5.2.5.1.1

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 2.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 2.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 8}}{2 \cdot 1}$

Step 2.5.2.5.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{- 4 \pm \sqrt{16 + 32}}{2 \cdot 1}$

Step 2.5.2.5.1.3

Add $16$ and $32$ .

$x = \frac{- 4 \pm \sqrt{48}}{2 \cdot 1}$

Step 2.5.2.5.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.5.2.5.1.4.1

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.5.2.5.1.4.2

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

$x = \frac{- 4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 4 \pm 4 \sqrt{3}}{2 \cdot 1}$

Step 2.5.2.5.2

Multiply $2$ by $1$ .

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

Step 2.5.2.5.3

Simplify $\frac{- 4 \pm 4 \sqrt{3}}{2}$ .

$x = - 2 \pm 2 \sqrt{3}$

Step 2.5.2.5.4

Change the $\pm$ to $-$ .

$x = - 2 - 2 \sqrt{3}$

Step 2.5.2.6

The final answer is the combination of both solutions.

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

Step 2.6

The final solution is all the values that make $3 x (x^{2} + 4 x - 8) = 0$ true.

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

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 $\frac{3}{4} x^{4} + 4 x^{3} - 12 x^{2} + 24$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

$\frac{3}{4} \cdot {(0)}^{4} + 4 {(0)}^{3} - 12 {(0)}^{2} + 24$

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

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

$\frac{3}{4} \cdot 0 + 4 {(0)}^{3} - 12 {(0)}^{2} + 24$

Step 4.1.2.1.2

Multiply $\frac{3}{4}$ by $0$ .

$0 + 4 {(0)}^{3} - 12 {(0)}^{2} + 24$

Step 4.1.2.1.3

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

$0 + 4 \cdot 0 - 12 {(0)}^{2} + 24$

Step 4.1.2.1.4

Multiply $4$ by $0$ .

$0 + 0 - 12 {(0)}^{2} + 24$

Step 4.1.2.1.5

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

$0 + 0 - 12 \cdot 0 + 24$

Step 4.1.2.1.6

Multiply $- 12$ by $0$ .

$0 + 0 + 0 + 24$

Step 4.1.2.2

Simplify by adding numbers.

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Step 4.1.2.2.1

Add $0$ and $0$ .

$0 + 0 + 24$

Step 4.1.2.2.2

Add $0$ and $0$ .

$0 + 24$

Step 4.1.2.2.3

Add $0$ and $24$ .

$24$

Step 4.2

Evaluate at $x = - 2 + 2 \sqrt{3}$ .

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Step 4.2.1

Substitute $- 2 + 2 \sqrt{3}$ for $x$ .

$\frac{3}{4} \cdot {(- 2 + 2 \sqrt{3})}^{4} + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

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 Binomial Theorem.

$\frac{3}{4} \cdot ({(- 2)}^{4} + 4 {(- 2)}^{3} (2 \sqrt{3}) + 6 {(- 2)}^{2} {(2 \sqrt{3})}^{2} + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2

Simplify each term.

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Step 4.2.2.1.2.1

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

$\frac{3}{4} \cdot (16 + 4 {(- 2)}^{3} (2 \sqrt{3}) + 6 {(- 2)}^{2} {(2 \sqrt{3})}^{2} + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.2

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

$\frac{3}{4} \cdot (16 + 4 \cdot - 8 (2 \sqrt{3}) + 6 {(- 2)}^{2} {(2 \sqrt{3})}^{2} + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.3

Multiply $4$ by $- 8$ .

$\frac{3}{4} \cdot (16 - 32 (2 \sqrt{3}) + 6 {(- 2)}^{2} {(2 \sqrt{3})}^{2} + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.4

Multiply $2$ by $- 32$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 6 {(- 2)}^{2} {(2 \sqrt{3})}^{2} + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.5

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 6 \cdot 4 {(2 \sqrt{3})}^{2} + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.6

Multiply $6$ by $4$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 {(2 \sqrt{3})}^{2} + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.7

Apply the product rule to $2 \sqrt{3}$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 (2^{2} {\sqrt{3}}^{2}) + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.8

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 (4 {\sqrt{3}}^{2}) + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.9

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.2.1.2.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 (4 {(3^{\frac{1}{2}})}^{2}) + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.9.2

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 (4 \cdot 3^{\frac{1}{2} \cdot 2}) + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.9.3

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 (4 \cdot 3^{\frac{2}{2}}) + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.9.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.2.9.4.1

Cancel the common factor.

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 (4 \cdot 3^{\frac{2}{2}}) + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.9.4.2

Rewrite the expression.

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 (4 \cdot 3^{1}) + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.9.5

Evaluate the exponent.

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 (4 \cdot 3) + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.10

Multiply $24 (4 \cdot 3)$ .

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Step 4.2.2.1.2.10.1

Multiply $4$ by $3$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 24 \cdot 12 + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.10.2

Multiply $24$ by $12$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 + 4 \cdot - 2 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.11

Multiply $4$ by $- 2$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 {(2 \sqrt{3})}^{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.12

Apply the product rule to $2 \sqrt{3}$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 (2^{3} {\sqrt{3}}^{3}) + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.13

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 (8 {\sqrt{3}}^{3}) + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.14

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 (8 \sqrt{3^{3}}) + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.15

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 (8 \sqrt{27}) + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.16

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.2.2.1.2.16.1

Factor $9$ out of $27$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 (8 \sqrt{9 (3)}) + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.16.2

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 (8 \sqrt{3^{2} \cdot 3}) + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.17

Pull terms out from under the radical.

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 (8 (3 \sqrt{3})) + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.18

Multiply $3$ by $8$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 8 (24 \sqrt{3}) + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.19

Multiply $24$ by $- 8$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + {(2 \sqrt{3})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.20

Apply the product rule to $2 \sqrt{3}$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 2^{4} {\sqrt{3}}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.21

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 {\sqrt{3}}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.22

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 4.2.2.1.2.22.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 {(3^{\frac{1}{2}})}^{4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.22.2

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 \cdot 3^{\frac{1}{2} \cdot 4}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.22.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 \cdot 3^{\frac{4}{2}}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.22.4

Cancel the common factor of $4$ and $2$ .

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Step 4.2.2.1.2.22.4.1

Factor $2$ out of $4$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 \cdot 3^{\frac{2 \cdot 2}{2}}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.22.4.2

Cancel the common factors.

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Step 4.2.2.1.2.22.4.2.1

Factor $2$ out of $2$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 \cdot 3^{\frac{2 \cdot 2}{2 (1)}}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.22.4.2.2

Cancel the common factor.

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.22.4.2.3

Rewrite the expression.

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 \cdot 3^{\frac{2}{1}}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.22.4.2.4

Divide $2$ by $1$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 \cdot 3^{2}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.23

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

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 16 \cdot 9) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.2.24

Multiply $16$ by $9$ .

$\frac{3}{4} \cdot (16 - 64 \sqrt{3} + 288 - 192 \sqrt{3} + 144) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.3

Add $16$ and $288$ .

$\frac{3}{4} \cdot (304 - 64 \sqrt{3} - 192 \sqrt{3} + 144) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.4

Add $304$ and $144$ .

$\frac{3}{4} \cdot (448 - 64 \sqrt{3} - 192 \sqrt{3}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.5

Subtract $192 \sqrt{3}$ from $- 64 \sqrt{3}$ .

$\frac{3}{4} \cdot (448 - 256 \sqrt{3}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.6

Apply the distributive property.

$\frac{3}{4} \cdot 448 + \frac{3}{4} (- 256 \sqrt{3}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.7

Cancel the common factor of $4$ .

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Step 4.2.2.1.7.1

Factor $4$ out of $448$ .

$\frac{3}{4} \cdot (4 (112)) + \frac{3}{4} (- 256 \sqrt{3}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.7.2

Cancel the common factor.

$\frac{3}{4} \cdot (4 \cdot 112) + \frac{3}{4} (- 256 \sqrt{3}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.7.3

Rewrite the expression.

$3 \cdot 112 + \frac{3}{4} (- 256 \sqrt{3}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.8

Multiply $3$ by $112$ .

$336 + \frac{3}{4} (- 256 \sqrt{3}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.9

Cancel the common factor of $4$ .

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Step 4.2.2.1.9.1

Factor $4$ out of $- 256 \sqrt{3}$ .

$336 + \frac{3}{4} (4 (- 64 \sqrt{3})) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.9.2

Cancel the common factor.

$336 + \frac{3}{4} (4 (- 64 \sqrt{3})) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.9.3

Rewrite the expression.

$336 + 3 (- 64 \sqrt{3}) + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.10

Multiply $- 64$ by $3$ .

$336 - 192 \sqrt{3} + 4 {(- 2 + 2 \sqrt{3})}^{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.11

Use the Binomial Theorem.

$336 - 192 \sqrt{3} + 4 ({(- 2)}^{3} + 3 {(- 2)}^{2} (2 \sqrt{3}) + 3 \cdot - 2 {(2 \sqrt{3})}^{2} + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12

Simplify each term.

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Step 4.2.2.1.12.1

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

$336 - 192 \sqrt{3} + 4 (- 8 + 3 {(- 2)}^{2} (2 \sqrt{3}) + 3 \cdot - 2 {(2 \sqrt{3})}^{2} + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.2

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

$336 - 192 \sqrt{3} + 4 (- 8 + 3 \cdot 4 (2 \sqrt{3}) + 3 \cdot - 2 {(2 \sqrt{3})}^{2} + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.3

Multiply $3$ by $4$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 12 (2 \sqrt{3}) + 3 \cdot - 2 {(2 \sqrt{3})}^{2} + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.4

Multiply $2$ by $12$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} + 3 \cdot - 2 {(2 \sqrt{3})}^{2} + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.5

Multiply $3$ by $- 2$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 {(2 \sqrt{3})}^{2} + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.6

Apply the product rule to $2 \sqrt{3}$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 (2^{2} {\sqrt{3}}^{2}) + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.7

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

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 (4 {\sqrt{3}}^{2}) + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.8

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.2.1.12.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 (4 {(3^{\frac{1}{2}})}^{2}) + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.8.2

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

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 (4 \cdot 3^{\frac{1}{2} \cdot 2}) + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.8.3

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

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 (4 \cdot 3^{\frac{2}{2}}) + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.8.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.12.8.4.1

Cancel the common factor.

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 (4 \cdot 3^{\frac{2}{2}}) + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.8.4.2

Rewrite the expression.

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 (4 \cdot 3^{1}) + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.8.5

Evaluate the exponent.

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 (4 \cdot 3) + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.9

Multiply $- 6 (4 \cdot 3)$ .

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Step 4.2.2.1.12.9.1

Multiply $4$ by $3$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 6 \cdot 12 + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.9.2

Multiply $- 6$ by $12$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + {(2 \sqrt{3})}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.10

Apply the product rule to $2 \sqrt{3}$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + 2^{3} {\sqrt{3}}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.11

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

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + 8 {\sqrt{3}}^{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.12

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

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + 8 \sqrt{3^{3}}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.13

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

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + 8 \sqrt{27}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.14

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.2.2.1.12.14.1

Factor $9$ out of $27$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + 8 \sqrt{9 (3)}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.14.2

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

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + 8 \sqrt{3^{2} \cdot 3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.15

Pull terms out from under the radical.

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + 8 (3 \sqrt{3})) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.12.16

Multiply $3$ by $8$ .

$336 - 192 \sqrt{3} + 4 (- 8 + 24 \sqrt{3} - 72 + 24 \sqrt{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.13

Subtract $72$ from $- 8$ .

$336 - 192 \sqrt{3} + 4 (- 80 + 24 \sqrt{3} + 24 \sqrt{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.14

Add $24 \sqrt{3}$ and $24 \sqrt{3}$ .

$336 - 192 \sqrt{3} + 4 (- 80 + 48 \sqrt{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.15

Apply the distributive property.

$336 - 192 \sqrt{3} + 4 \cdot - 80 + 4 (48 \sqrt{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.16

Multiply $4$ by $- 80$ .

$336 - 192 \sqrt{3} - 320 + 4 (48 \sqrt{3}) - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.17

Multiply $48$ by $4$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 {(- 2 + 2 \sqrt{3})}^{2} + 24$

Step 4.2.2.1.18

Rewrite ${(- 2 + 2 \sqrt{3})}^{2}$ as $(- 2 + 2 \sqrt{3}) (- 2 + 2 \sqrt{3})$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 ((- 2 + 2 \sqrt{3}) (- 2 + 2 \sqrt{3})) + 24$

Step 4.2.2.1.19

Expand $(- 2 + 2 \sqrt{3}) (- 2 + 2 \sqrt{3})$ using the FOIL Method.

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Step 4.2.2.1.19.1

Apply the distributive property.

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (- 2 (- 2 + 2 \sqrt{3}) + 2 \sqrt{3} (- 2 + 2 \sqrt{3})) + 24$

Step 4.2.2.1.19.2

Apply the distributive property.

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (- 2 \cdot - 2 - 2 (2 \sqrt{3}) + 2 \sqrt{3} (- 2 + 2 \sqrt{3})) + 24$

Step 4.2.2.1.19.3

Apply the distributive property.

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (- 2 \cdot - 2 - 2 (2 \sqrt{3}) + 2 \sqrt{3} \cdot - 2 + 2 \sqrt{3} (2 \sqrt{3})) + 24$

Step 4.2.2.1.20

Simplify and combine like terms.

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Step 4.2.2.1.20.1

Simplify each term.

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Step 4.2.2.1.20.1.1

Multiply $- 2$ by $- 2$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 2 (2 \sqrt{3}) + 2 \sqrt{3} \cdot - 2 + 2 \sqrt{3} (2 \sqrt{3})) + 24$

Step 4.2.2.1.20.1.2

Multiply $2$ by $- 2$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} + 2 \sqrt{3} \cdot - 2 + 2 \sqrt{3} (2 \sqrt{3})) + 24$

Step 4.2.2.1.20.1.3

Multiply $- 2$ by $2$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 2 \sqrt{3} (2 \sqrt{3})) + 24$

Step 4.2.2.1.20.1.4

Multiply $2 \sqrt{3} (2 \sqrt{3})$ .

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Step 4.2.2.1.20.1.4.1

Multiply $2$ by $2$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 \sqrt{3} \sqrt{3}) + 24$

Step 4.2.2.1.20.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 ({\sqrt{3}}^{1} \sqrt{3})) + 24$

Step 4.2.2.1.20.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})) + 24$

Step 4.2.2.1.20.1.4.4

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

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 {\sqrt{3}}^{1 + 1}) + 24$

Step 4.2.2.1.20.1.4.5

Add $1$ and $1$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 {\sqrt{3}}^{2}) + 24$

Step 4.2.2.1.20.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.2.1.20.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 {(3^{\frac{1}{2}})}^{2}) + 24$

Step 4.2.2.1.20.1.5.2

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

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 \cdot 3^{\frac{1}{2} \cdot 2}) + 24$

Step 4.2.2.1.20.1.5.3

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

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}) + 24$

Step 4.2.2.1.20.1.5.4

Cancel the common factor of $2$ .

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Step 4.2.2.1.20.1.5.4.1

Cancel the common factor.

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}) + 24$

Step 4.2.2.1.20.1.5.4.2

Rewrite the expression.

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 \cdot 3^{1}) + 24$

Step 4.2.2.1.20.1.5.5

Evaluate the exponent.

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 4 \cdot 3) + 24$

Step 4.2.2.1.20.1.6

Multiply $4$ by $3$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (4 - 4 \sqrt{3} - 4 \sqrt{3} + 12) + 24$

Step 4.2.2.1.20.2

Add $4$ and $12$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (16 - 4 \sqrt{3} - 4 \sqrt{3}) + 24$

Step 4.2.2.1.20.3

Subtract $4 \sqrt{3}$ from $- 4 \sqrt{3}$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 (16 - 8 \sqrt{3}) + 24$

Step 4.2.2.1.21

Apply the distributive property.

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 12 \cdot 16 - 12 (- 8 \sqrt{3}) + 24$

Step 4.2.2.1.22

Multiply $- 12$ by $16$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 192 - 12 (- 8 \sqrt{3}) + 24$

Step 4.2.2.1.23

Multiply $- 8$ by $- 12$ .

$336 - 192 \sqrt{3} - 320 + 192 \sqrt{3} - 192 + 96 \sqrt{3} + 24$

Step 4.2.2.2

Simplify by adding terms.

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Step 4.2.2.2.1

Subtract $320$ from $336$ .

$16 - 192 \sqrt{3} + 192 \sqrt{3} - 192 + 96 \sqrt{3} + 24$

Step 4.2.2.2.2

Simplify by adding and subtracting.

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Step 4.2.2.2.2.1

Subtract $192$ from $16$ .

$- 176 - 192 \sqrt{3} + 192 \sqrt{3} + 96 \sqrt{3} + 24$

Step 4.2.2.2.2.2

Add $- 176$ and $24$ .

$- 152 - 192 \sqrt{3} + 192 \sqrt{3} + 96 \sqrt{3}$

Step 4.2.2.2.3

Add $- 192 \sqrt{3}$ and $192 \sqrt{3}$ .

$- 152 + 0 + 96 \sqrt{3}$

Step 4.2.2.2.4

Add $- 152$ and $0$ .

$- 152 + 96 \sqrt{3}$

Step 4.3

Evaluate at $x = - 2 - 2 \sqrt{3}$ .

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Step 4.3.1

Substitute $- 2 - 2 \sqrt{3}$ for $x$ .

$\frac{3}{4} \cdot {(- 2 - 2 \sqrt{3})}^{4} + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Use the Binomial Theorem.

$\frac{3}{4} \cdot ({(- 2)}^{4} + 4 {(- 2)}^{3} (- 2 \sqrt{3}) + 6 {(- 2)}^{2} {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2

Simplify each term.

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Step 4.3.2.1.2.1

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

$\frac{3}{4} \cdot (16 + 4 {(- 2)}^{3} (- 2 \sqrt{3}) + 6 {(- 2)}^{2} {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.2

Multiply ${(- 2)}^{3}$ by $- 2$ by adding the exponents.

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Step 4.3.2.1.2.2.1

Move $- 2$ .

$\frac{3}{4} \cdot (16 + 4 (- 2 {(- 2)}^{3}) \sqrt{3} + 6 {(- 2)}^{2} {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.2.2

Multiply $- 2$ by ${(- 2)}^{3}$ .

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Step 4.3.2.1.2.2.2.1

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

$\frac{3}{4} \cdot (16 + 4 ({(- 2)}^{1} {(- 2)}^{3}) \sqrt{3} + 6 {(- 2)}^{2} {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.2.2.2

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

$\frac{3}{4} \cdot (16 + 4 {(- 2)}^{1 + 3} \sqrt{3} + 6 {(- 2)}^{2} {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.2.3

Add $1$ and $3$ .

$\frac{3}{4} \cdot (16 + 4 {(- 2)}^{4} \sqrt{3} + 6 {(- 2)}^{2} {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.3

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

$\frac{3}{4} \cdot (16 + 4 \cdot 16 \sqrt{3} + 6 {(- 2)}^{2} {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.4

Multiply $4$ by $16$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 6 {(- 2)}^{2} {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.5

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 6 \cdot 4 {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.6

Multiply $6$ by $4$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 {(- 2 \sqrt{3})}^{2} + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.7

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 ({(- 2)}^{2} {\sqrt{3}}^{2}) + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.8

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 (4 {\sqrt{3}}^{2}) + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.9

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.3.2.1.2.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 (4 {(3^{\frac{1}{2}})}^{2}) + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.9.2

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 (4 \cdot 3^{\frac{1}{2} \cdot 2}) + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.9.3

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 (4 \cdot 3^{\frac{2}{2}}) + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.9.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.2.9.4.1

Cancel the common factor.

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 (4 \cdot 3^{\frac{2}{2}}) + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.9.4.2

Rewrite the expression.

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 (4 \cdot 3^{1}) + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.9.5

Evaluate the exponent.

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 (4 \cdot 3) + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.10

Multiply $24 (4 \cdot 3)$ .

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Step 4.3.2.1.2.10.1

Multiply $4$ by $3$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 24 \cdot 12 + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.10.2

Multiply $24$ by $12$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 4 \cdot - 2 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.11

Multiply $4$ by $- 2$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 {(- 2 \sqrt{3})}^{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.12

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 ({(- 2)}^{3} {\sqrt{3}}^{3}) + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.13

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 (- 8 {\sqrt{3}}^{3}) + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.14

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 (- 8 \sqrt{3^{3}}) + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.15

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 (- 8 \sqrt{27}) + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.16

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.3.2.1.2.16.1

Factor $9$ out of $27$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 (- 8 \sqrt{9 (3)}) + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.16.2

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 (- 8 \sqrt{3^{2} \cdot 3}) + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.17

Pull terms out from under the radical.

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 (- 8 (3 \sqrt{3})) + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.18

Multiply $3$ by $- 8$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 - 8 (- 24 \sqrt{3}) + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.19

Multiply $- 24$ by $- 8$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + {(- 2 \sqrt{3})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.20

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + {(- 2)}^{4} {\sqrt{3}}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.21

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 {\sqrt{3}}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.22

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 4.3.2.1.2.22.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 {(3^{\frac{1}{2}})}^{4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.22.2

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 \cdot 3^{\frac{1}{2} \cdot 4}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.22.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 \cdot 3^{\frac{4}{2}}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.22.4

Cancel the common factor of $4$ and $2$ .

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Step 4.3.2.1.2.22.4.1

Factor $2$ out of $4$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 \cdot 3^{\frac{2 \cdot 2}{2}}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.22.4.2

Cancel the common factors.

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Step 4.3.2.1.2.22.4.2.1

Factor $2$ out of $2$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 \cdot 3^{\frac{2 \cdot 2}{2 (1)}}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.22.4.2.2

Cancel the common factor.

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.22.4.2.3

Rewrite the expression.

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 \cdot 3^{\frac{2}{1}}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.22.4.2.4

Divide $2$ by $1$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 \cdot 3^{2}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.23

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

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 16 \cdot 9) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.2.24

Multiply $16$ by $9$ .

$\frac{3}{4} \cdot (16 + 64 \sqrt{3} + 288 + 192 \sqrt{3} + 144) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.3

Add $16$ and $288$ .

$\frac{3}{4} \cdot (304 + 64 \sqrt{3} + 192 \sqrt{3} + 144) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.4

Add $304$ and $144$ .

$\frac{3}{4} \cdot (448 + 64 \sqrt{3} + 192 \sqrt{3}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.5

Add $64 \sqrt{3}$ and $192 \sqrt{3}$ .

$\frac{3}{4} \cdot (448 + 256 \sqrt{3}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.6

Apply the distributive property.

$\frac{3}{4} \cdot 448 + \frac{3}{4} (256 \sqrt{3}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.7

Cancel the common factor of $4$ .

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Step 4.3.2.1.7.1

Factor $4$ out of $448$ .

$\frac{3}{4} \cdot (4 (112)) + \frac{3}{4} (256 \sqrt{3}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.7.2

Cancel the common factor.

$\frac{3}{4} \cdot (4 \cdot 112) + \frac{3}{4} (256 \sqrt{3}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.7.3

Rewrite the expression.

$3 \cdot 112 + \frac{3}{4} (256 \sqrt{3}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.8

Multiply $3$ by $112$ .

$336 + \frac{3}{4} (256 \sqrt{3}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.9

Cancel the common factor of $4$ .

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Step 4.3.2.1.9.1

Factor $4$ out of $256 \sqrt{3}$ .

$336 + \frac{3}{4} (4 (64 \sqrt{3})) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.9.2

Cancel the common factor.

$336 + \frac{3}{4} (4 (64 \sqrt{3})) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.9.3

Rewrite the expression.

$336 + 3 (64 \sqrt{3}) + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.10

Multiply $64$ by $3$ .

$336 + 192 \sqrt{3} + 4 {(- 2 - 2 \sqrt{3})}^{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.11

Use the Binomial Theorem.

$336 + 192 \sqrt{3} + 4 ({(- 2)}^{3} + 3 {(- 2)}^{2} (- 2 \sqrt{3}) + 3 \cdot - 2 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12

Simplify each term.

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Step 4.3.2.1.12.1

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

$336 + 192 \sqrt{3} + 4 (- 8 + 3 {(- 2)}^{2} (- 2 \sqrt{3}) + 3 \cdot - 2 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.2

Multiply ${(- 2)}^{2}$ by $- 2$ by adding the exponents.

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Step 4.3.2.1.12.2.1

Move $- 2$ .

$336 + 192 \sqrt{3} + 4 (- 8 + 3 (- 2 {(- 2)}^{2}) \sqrt{3} + 3 \cdot - 2 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.2.2

Multiply $- 2$ by ${(- 2)}^{2}$ .

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Step 4.3.2.1.12.2.2.1

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

$336 + 192 \sqrt{3} + 4 (- 8 + 3 ({(- 2)}^{1} {(- 2)}^{2}) \sqrt{3} + 3 \cdot - 2 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.2.2.2

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

$336 + 192 \sqrt{3} + 4 (- 8 + 3 {(- 2)}^{1 + 2} \sqrt{3} + 3 \cdot - 2 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.2.3

Add $1$ and $2$ .

$336 + 192 \sqrt{3} + 4 (- 8 + 3 {(- 2)}^{3} \sqrt{3} + 3 \cdot - 2 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.3

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

$336 + 192 \sqrt{3} + 4 (- 8 + 3 \cdot - 8 \sqrt{3} + 3 \cdot - 2 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.4

Multiply $3$ by $- 8$ .

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} + 3 \cdot - 2 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.5

Multiply $3$ by $- 2$ .

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.6

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 ({(- 2)}^{2} {\sqrt{3}}^{2}) + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.7

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 (4 {\sqrt{3}}^{2}) + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.8

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.3.2.1.12.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 (4 {(3^{\frac{1}{2}})}^{2}) + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.8.2

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 (4 \cdot 3^{\frac{1}{2} \cdot 2}) + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.8.3

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 (4 \cdot 3^{\frac{2}{2}}) + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.8.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.12.8.4.1

Cancel the common factor.

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 (4 \cdot 3^{\frac{2}{2}}) + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.8.4.2

Rewrite the expression.

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 (4 \cdot 3^{1}) + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.8.5

Evaluate the exponent.

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 (4 \cdot 3) + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.9

Multiply $- 6 (4 \cdot 3)$ .

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Step 4.3.2.1.12.9.1

Multiply $4$ by $3$ .

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 6 \cdot 12 + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.9.2

Multiply $- 6$ by $12$ .

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 + {(- 2 \sqrt{3})}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.10

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 + {(- 2)}^{3} {\sqrt{3}}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.11

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 - 8 {\sqrt{3}}^{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.12

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 - 8 \sqrt{3^{3}}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.13

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 - 8 \sqrt{27}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.14

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.3.2.1.12.14.1

Factor $9$ out of $27$ .

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 - 8 \sqrt{9 (3)}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.14.2

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

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 - 8 \sqrt{3^{2} \cdot 3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.15

Pull terms out from under the radical.

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 - 8 (3 \sqrt{3})) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.12.16

Multiply $3$ by $- 8$ .

$336 + 192 \sqrt{3} + 4 (- 8 - 24 \sqrt{3} - 72 - 24 \sqrt{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.13

Subtract $72$ from $- 8$ .

$336 + 192 \sqrt{3} + 4 (- 80 - 24 \sqrt{3} - 24 \sqrt{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.14

Subtract $24 \sqrt{3}$ from $- 24 \sqrt{3}$ .

$336 + 192 \sqrt{3} + 4 (- 80 - 48 \sqrt{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.15

Apply the distributive property.

$336 + 192 \sqrt{3} + 4 \cdot - 80 + 4 (- 48 \sqrt{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.16

Multiply $4$ by $- 80$ .

$336 + 192 \sqrt{3} - 320 + 4 (- 48 \sqrt{3}) - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.17

Multiply $- 48$ by $4$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 {(- 2 - 2 \sqrt{3})}^{2} + 24$

Step 4.3.2.1.18

Rewrite ${(- 2 - 2 \sqrt{3})}^{2}$ as $(- 2 - 2 \sqrt{3}) (- 2 - 2 \sqrt{3})$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 ((- 2 - 2 \sqrt{3}) (- 2 - 2 \sqrt{3})) + 24$

Step 4.3.2.1.19

Expand $(- 2 - 2 \sqrt{3}) (- 2 - 2 \sqrt{3})$ using the FOIL Method.

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Step 4.3.2.1.19.1

Apply the distributive property.

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (- 2 (- 2 - 2 \sqrt{3}) - 2 \sqrt{3} (- 2 - 2 \sqrt{3})) + 24$

Step 4.3.2.1.19.2

Apply the distributive property.

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (- 2 \cdot - 2 - 2 (- 2 \sqrt{3}) - 2 \sqrt{3} (- 2 - 2 \sqrt{3})) + 24$

Step 4.3.2.1.19.3

Apply the distributive property.

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (- 2 \cdot - 2 - 2 (- 2 \sqrt{3}) - 2 \sqrt{3} \cdot - 2 - 2 \sqrt{3} (- 2 \sqrt{3})) + 24$

Step 4.3.2.1.20

Simplify and combine like terms.

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Step 4.3.2.1.20.1

Simplify each term.

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Step 4.3.2.1.20.1.1

Multiply $- 2$ by $- 2$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 - 2 (- 2 \sqrt{3}) - 2 \sqrt{3} \cdot - 2 - 2 \sqrt{3} (- 2 \sqrt{3})) + 24$

Step 4.3.2.1.20.1.2

Multiply $- 2$ by $- 2$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} - 2 \sqrt{3} \cdot - 2 - 2 \sqrt{3} (- 2 \sqrt{3})) + 24$

Step 4.3.2.1.20.1.3

Multiply $- 2$ by $- 2$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} - 2 \sqrt{3} (- 2 \sqrt{3})) + 24$

Step 4.3.2.1.20.1.4

Multiply $- 2 \sqrt{3} (- 2 \sqrt{3})$ .

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Step 4.3.2.1.20.1.4.1

Multiply $- 2$ by $- 2$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 \sqrt{3} \sqrt{3}) + 24$

Step 4.3.2.1.20.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 ({\sqrt{3}}^{1} \sqrt{3})) + 24$

Step 4.3.2.1.20.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})) + 24$

Step 4.3.2.1.20.1.4.4

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

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 {\sqrt{3}}^{1 + 1}) + 24$

Step 4.3.2.1.20.1.4.5

Add $1$ and $1$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 {\sqrt{3}}^{2}) + 24$

Step 4.3.2.1.20.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.3.2.1.20.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 {(3^{\frac{1}{2}})}^{2}) + 24$

Step 4.3.2.1.20.1.5.2

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

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 \cdot 3^{\frac{1}{2} \cdot 2}) + 24$

Step 4.3.2.1.20.1.5.3

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

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}) + 24$

Step 4.3.2.1.20.1.5.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.20.1.5.4.1

Cancel the common factor.

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}) + 24$

Step 4.3.2.1.20.1.5.4.2

Rewrite the expression.

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 \cdot 3^{1}) + 24$

Step 4.3.2.1.20.1.5.5

Evaluate the exponent.

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 4 \cdot 3) + 24$

Step 4.3.2.1.20.1.6

Multiply $4$ by $3$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (4 + 4 \sqrt{3} + 4 \sqrt{3} + 12) + 24$

Step 4.3.2.1.20.2

Add $4$ and $12$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (16 + 4 \sqrt{3} + 4 \sqrt{3}) + 24$

Step 4.3.2.1.20.3

Add $4 \sqrt{3}$ and $4 \sqrt{3}$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 (16 + 8 \sqrt{3}) + 24$

Step 4.3.2.1.21

Apply the distributive property.

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 12 \cdot 16 - 12 (8 \sqrt{3}) + 24$

Step 4.3.2.1.22

Multiply $- 12$ by $16$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 192 - 12 (8 \sqrt{3}) + 24$

Step 4.3.2.1.23

Multiply $8$ by $- 12$ .

$336 + 192 \sqrt{3} - 320 - 192 \sqrt{3} - 192 - 96 \sqrt{3} + 24$

Step 4.3.2.2

Simplify by adding terms.

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Step 4.3.2.2.1

Subtract $320$ from $336$ .

$16 + 192 \sqrt{3} - 192 \sqrt{3} - 192 - 96 \sqrt{3} + 24$

Step 4.3.2.2.2

Simplify by adding and subtracting.

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Step 4.3.2.2.2.1

Subtract $192$ from $16$ .

$- 176 + 192 \sqrt{3} - 192 \sqrt{3} - 96 \sqrt{3} + 24$

Step 4.3.2.2.2.2

Add $- 176$ and $24$ .

$- 152 + 192 \sqrt{3} - 192 \sqrt{3} - 96 \sqrt{3}$

Step 4.3.2.2.3

Subtract $192 \sqrt{3}$ from $192 \sqrt{3}$ .

$- 152 + 0 - 96 \sqrt{3}$

Step 4.3.2.2.4

Add $- 152$ and $0$ .

$- 152 - 96 \sqrt{3}$

Step 4.4

List all of the points.

$(0, 24), (- 2 + 2 \sqrt{3}, - 152 + 96 \sqrt{3}), (- 2 - 2 \sqrt{3}, - 152 - 96 \sqrt{3})$

Step 5