Calculus Examples

$y = x^{2} + 9 x - 4$ , $y = x + 2$

Step 1

Solve by substitution to find the intersection between the curves.

Tap for more steps...

Step 1.1

Eliminate the equal sides of each equation and combine.

$x^{2} + 9 x - 4 = x + 2$

Step 1.2

Solve $x^{2} + 9 x - 4 = x + 2$ for $x$ .

Tap for more steps...

Step 1.2.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 1.2.1.1

Subtract $x$ from both sides of the equation.

$x^{2} + 9 x - 4 - x = 2$

Step 1.2.1.2

Subtract $x$ from $9 x$ .

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

Step 1.2.2

Move all terms to the left side of the equation and simplify.

Tap for more steps...

Step 1.2.2.1

Subtract $2$ from both sides of the equation.

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

Step 1.2.2.2

Subtract $2$ from $- 4$ .

$x^{2} + 8 x - 6 = 0$

Step 1.2.3

Use the quadratic formula to find the solutions.

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

Step 1.2.4

Substitute the values $a = 1$ , $b = 8$ , and $c = - 6$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (1 \cdot - 6)}}{2 \cdot 1} + y = x + 2$

Step 1.2.5

Simplify.

Tap for more steps...

Step 1.2.5.1

Simplify the numerator.

Tap for more steps...

Step 1.2.5.1.1

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

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

Tap for more steps...

Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 6}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{- 8 \pm \sqrt{64 + 24}}{2 \cdot 1}$

Step 1.2.5.1.3

Add $64$ and $24$ .

$x = \frac{- 8 \pm \sqrt{88}}{2 \cdot 1}$

Step 1.2.5.1.4

Rewrite $88$ as $2^{2} \cdot 22$ .

Tap for more steps...

Step 1.2.5.1.4.1

Factor $4$ out of $88$ .

$x = \frac{- 8 \pm \sqrt{4 (22)}}{2 \cdot 1}$

Step 1.2.5.1.4.2

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

$x = \frac{- 8 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Step 1.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 2 \sqrt{22}}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 2 \sqrt{22}}{2}$

Step 1.2.5.3

Simplify $\frac{- 8 \pm 2 \sqrt{22}}{2}$ .

$x = - 4 \pm \sqrt{22}$

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

Simplify the numerator.

Tap for more steps...

Step 1.2.6.1.1

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

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

Tap for more steps...

Step 1.2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 6}}{2 \cdot 1}$

Step 1.2.6.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{- 8 \pm \sqrt{64 + 24}}{2 \cdot 1}$

Step 1.2.6.1.3

Add $64$ and $24$ .

$x = \frac{- 8 \pm \sqrt{88}}{2 \cdot 1}$

Step 1.2.6.1.4

Rewrite $88$ as $2^{2} \cdot 22$ .

Tap for more steps...

Step 1.2.6.1.4.1

Factor $4$ out of $88$ .

$x = \frac{- 8 \pm \sqrt{4 (22)}}{2 \cdot 1}$

Step 1.2.6.1.4.2

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

$x = \frac{- 8 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 2 \sqrt{22}}{2 \cdot 1}$

Step 1.2.6.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 2 \sqrt{22}}{2}$

Step 1.2.6.3

Simplify $\frac{- 8 \pm 2 \sqrt{22}}{2}$ .

$x = - 4 \pm \sqrt{22}$

Step 1.2.6.4

Change the $\pm$ to $+$ .

$x = - 4 + \sqrt{22}$

Step 1.2.7

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

Tap for more steps...

Step 1.2.7.1

Simplify the numerator.

Tap for more steps...

Step 1.2.7.1.1

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

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 6}}{2 \cdot 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 6$ .

Tap for more steps...

Step 1.2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 6}}{2 \cdot 1}$

Step 1.2.7.1.2.2

Multiply $- 4$ by $- 6$ .

$x = \frac{- 8 \pm \sqrt{64 + 24}}{2 \cdot 1}$

Step 1.2.7.1.3

Add $64$ and $24$ .

$x = \frac{- 8 \pm \sqrt{88}}{2 \cdot 1}$

Step 1.2.7.1.4

Rewrite $88$ as $2^{2} \cdot 22$ .

Tap for more steps...

Step 1.2.7.1.4.1

Factor $4$ out of $88$ .

$x = \frac{- 8 \pm \sqrt{4 (22)}}{2 \cdot 1}$

Step 1.2.7.1.4.2

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

$x = \frac{- 8 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 8 \pm 2 \sqrt{22}}{2 \cdot 1}$

Step 1.2.7.2

Multiply $2$ by $1$ .

$x = \frac{- 8 \pm 2 \sqrt{22}}{2}$

Step 1.2.7.3

Simplify $\frac{- 8 \pm 2 \sqrt{22}}{2}$ .

$x = - 4 \pm \sqrt{22}$

Step 1.2.7.4

Change the $\pm$ to $-$ .

$x = - 4 - \sqrt{22}$

Step 1.2.8

The final answer is the combination of both solutions.

$x = - 4 + \sqrt{22}, - 4 - \sqrt{22}$

Step 1.3

Evaluate $y$ when $x = - 4 + \sqrt{22}$ .

Tap for more steps...

Step 1.3.1

Substitute $- 4 + \sqrt{22}$ for $x$ .

$y = (- 4 + \sqrt{22}) + 2$

Step 1.3.2

Substitute $- 4 + \sqrt{22}$ for $x$ in $y = (- 4 + \sqrt{22}) + 2$ and solve for $y$ .

Tap for more steps...

Step 1.3.2.1

Remove parentheses.

$y = - 4 + \sqrt{22} + 2$

Step 1.3.2.2

Remove parentheses.

$y = (- 4 + \sqrt{22}) + 2$

Step 1.3.2.3

Add $- 4$ and $2$ .

$y = - 2 + \sqrt{22}$

Step 1.4

Evaluate $y$ when $x = - 4 - \sqrt{22}$ .

Tap for more steps...

Step 1.4.1

Substitute $- 4 - \sqrt{22}$ for $x$ .

$y = (- 4 - \sqrt{22}) + 2$

Step 1.4.2

Substitute $- 4 - \sqrt{22}$ for $x$ in $y = (- 4 - \sqrt{22}) + 2$ and solve for $y$ .

Tap for more steps...

Step 1.4.2.1

Remove parentheses.

$y = - 4 - \sqrt{22} + 2$

Step 1.4.2.2

Remove parentheses.

$y = (- 4 - \sqrt{22}) + 2$

Step 1.4.2.3

Add $- 4$ and $2$ .

$y = - 2 - \sqrt{22}$

Step 1.5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 4 + \sqrt{22}, - 2 + \sqrt{22})$

$(- 4 - \sqrt{22}, - 2 - \sqrt{22})$

$(- 4 + \sqrt{22}, - 2 + \sqrt{22})$

$(- 4 - \sqrt{22}, - 2 - \sqrt{22})$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} x + 2 d x - \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} x^{2} + 9 x - 4 d x$

Step 3

Integrate to find the area between $- 4 - \sqrt{22}$ and $- 4 + \sqrt{22}$ .

Tap for more steps...

Step 3.1

Combine the integrals into a single integral.

$\int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} x + 2 - (x^{2} + 9 x - 4) d x$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$x + 2 - x^{2} - (9 x) - - 4$

Step 3.2.2

Simplify.

Tap for more steps...

Step 3.2.2.1

Multiply $9$ by $- 1$ .

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

Step 3.2.2.2

Multiply $- 1$ by $- 4$ .

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

$\int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} x + 2 - x^{2} - 9 x + 4 d x$

Step 3.3

Simplify by adding terms.

Tap for more steps...

Step 3.3.1

Subtract $9 x$ from $x$ .

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

Step 3.3.2

Add $2$ and $4$ .

$- 8 x - x^{2} + 6$

$\int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} - 8 x - x^{2} + 6 d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} - 8 x d x + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} - x^{2} d x + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} 6 d x$

Step 3.5

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$- 8 \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} x d x + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} - x^{2} d x + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} 6 d x$

Step 3.6

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- 8 (\frac{1}{2} x^{2}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} - x^{2} d x + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} 6 d x$

Step 3.7

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

$- 8 (\frac{x^{2}}{2}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} - x^{2} d x + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} 6 d x$

Step 3.8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- 8 (\frac{x^{2}}{2}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) - \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} x^{2} d x + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} 6 d x$

Step 3.9

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$- 8 (\frac{x^{2}}{2}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) - (\frac{1}{3} x^{3}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} 6 d x$

Step 3.10

Combine $\frac{1}{3}$ and $x^{3}$ .

$- 8 (\frac{x^{2}}{2}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) - (\frac{x^{3}}{3}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) + \int_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}} 6 d x$

Step 3.11

Apply the constant rule.

$- 8 (\frac{x^{2}}{2}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) - (\frac{x^{3}}{3}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) + 6 x]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}$

Step 3.12

Simplify the answer.

Tap for more steps...

Step 3.12.1

Substitute and simplify.

Tap for more steps...

Step 3.12.1.1

Evaluate $\frac{x^{2}}{2}$ at $- 4 + \sqrt{22}$ and at $- 4 - \sqrt{22}$ .

$- 8 ((\frac{{(- 4 + \sqrt{22})}^{2}}{2}) - \frac{{(- 4 - \sqrt{22})}^{2}}{2}) - (\frac{x^{3}}{3}]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}) + 6 x]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}$

Step 3.12.1.2

Evaluate $\frac{x^{3}}{3}$ at $- 4 + \sqrt{22}$ and at $- 4 - \sqrt{22}$ .

$- 8 (\frac{{(- 4 + \sqrt{22})}^{2}}{2} - \frac{{(- 4 - \sqrt{22})}^{2}}{2}) - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + 6 x]_{- 4 - \sqrt{22}}^{- 4 + \sqrt{22}}$

Step 3.12.1.3

Evaluate $6 x$ at $- 4 + \sqrt{22}$ and at $- 4 - \sqrt{22}$ .

$- 8 (\frac{{(- 4 + \sqrt{22})}^{2}}{2} - \frac{{(- 4 - \sqrt{22})}^{2}}{2}) - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4

Simplify.

Tap for more steps...

Step 3.12.1.4.1

Combine the numerators over the common denominator.

$- 8 \frac{{(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}}{2} - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.2

Combine $- 8$ and $\frac{{(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}}{2}$ .

$\frac{- 8 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})}{2} - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.3

Cancel the common factor of $- 8$ and $2$ .

Tap for more steps...

Step 3.12.1.4.3.1

Factor $2$ out of $- 8 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})$ .

$\frac{2 (- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}))}{2} - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.3.2

Cancel the common factors.

Tap for more steps...

Step 3.12.1.4.3.2.1

Factor $2$ out of $2$ .

$\frac{2 (- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}))}{2 (1)} - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.3.2.2

Cancel the common factor.

$\frac{2 (- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}))}{2 \cdot 1} - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.3.2.3

Rewrite the expression.

$\frac{- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})}{1} - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.3.2.4

Divide $- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})$ by $1$ .

$- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - (\frac{{(- 4 + \sqrt{22})}^{3}}{3} - \frac{{(- 4 - \sqrt{22})}^{3}}{3}) + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.4

Combine the numerators over the common denominator.

$- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - \frac{{(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}}{3} + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.5

To write $- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) \cdot \frac{3}{3} - \frac{{(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}}{3} + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.6

Combine $- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})$ and $\frac{3}{3}$ .

$\frac{- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) \cdot 3}{3} - \frac{{(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}}{3} + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.7

Combine the numerators over the common denominator.

$\frac{- 4 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) \cdot 3 - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3})}{3} + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.8

Multiply $3$ by $- 4$ .

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3})}{3} + (6 (- 4 + \sqrt{22})) - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.9

To write $6 (- 4 + \sqrt{22})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3})}{3} + 6 (- 4 + \sqrt{22}) \cdot \frac{3}{3} - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.10

Combine $6 (- 4 + \sqrt{22})$ and $\frac{3}{3}$ .

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3})}{3} + \frac{6 (- 4 + \sqrt{22}) \cdot 3}{3} - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.11

Combine the numerators over the common denominator.

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) + 6 (- 4 + \sqrt{22}) \cdot 3}{3} - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.12

Multiply $3$ by $6$ .

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) + 18 (- 4 + \sqrt{22})}{3} - 6 (- 4 - \sqrt{22})$

Step 3.12.1.4.13

To write $- 6 (- 4 - \sqrt{22})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) + 18 (- 4 + \sqrt{22})}{3} - 6 (- 4 - \sqrt{22}) \cdot \frac{3}{3}$

Step 3.12.1.4.14

Combine $- 6 (- 4 - \sqrt{22})$ and $\frac{3}{3}$ .

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) + 18 (- 4 + \sqrt{22})}{3} + \frac{- 6 (- 4 - \sqrt{22}) \cdot 3}{3}$

Step 3.12.1.4.15

Combine the numerators over the common denominator.

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) + 18 (- 4 + \sqrt{22}) - 6 (- 4 - \sqrt{22}) \cdot 3}{3}$

Step 3.12.1.4.16

Multiply $3$ by $- 6$ .

$\frac{- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) + 18 (- 4 + \sqrt{22}) - 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.2

Simplify.

Tap for more steps...

Step 3.12.2.1

Factor $- 1$ out of $- 12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})$ .

$\frac{- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) + 18 (- 4 + \sqrt{22}) - 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.2.2

Factor $- 1$ out of $- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2})) - ({(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3})$ .

$\frac{- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) + 18 (- 4 + \sqrt{22}) - 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.2.3

Factor $- 1$ out of $18 (- 4 + \sqrt{22})$ .

$\frac{- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) - (- 18 (- 4 + \sqrt{22})) - 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.2.4

Factor $- 1$ out of $- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3}) - (- 18 (- 4 + \sqrt{22}))$ .

$\frac{- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22})) - 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.2.5

Factor $- 1$ out of $- 18 (- 4 - \sqrt{22})$ .

$\frac{- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22})) - (18 (- 4 - \sqrt{22}))}{3}$

Step 3.12.2.6

Factor $- 1$ out of $- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22})) - (18 (- 4 - \sqrt{22}))$ .

$\frac{- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22}))}{3}$

Step 3.12.2.7

Rewrite $- (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22}))$ as $- 1 (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22}))$ .

$\frac{- 1 (12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22}))}{3}$

Step 3.12.2.8

Move the negative in front of the fraction.

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - {(- 4 - \sqrt{22})}^{2}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3

Simplify.

Tap for more steps...

Step 3.12.3.1

Rewrite ${(- 4 - \sqrt{22})}^{2}$ as $(- 4 - \sqrt{22}) (- 4 - \sqrt{22})$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - ((- 4 - \sqrt{22}) (- 4 - \sqrt{22}))) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.2

Expand $(- 4 - \sqrt{22}) (- 4 - \sqrt{22})$ using the FOIL Method.

Tap for more steps...

Step 3.12.3.2.1

Apply the distributive property.

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (- 4 (- 4 - \sqrt{22}) - \sqrt{22} (- 4 - \sqrt{22}))) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.2.2

Apply the distributive property.

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (- 4 \cdot - 4 - 4 (- \sqrt{22}) - \sqrt{22} (- 4 - \sqrt{22}))) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.2.3

Apply the distributive property.

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (- 4 \cdot - 4 - 4 (- \sqrt{22}) - \sqrt{22} \cdot - 4 - \sqrt{22} (- \sqrt{22}))) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3

Simplify and combine like terms.

Tap for more steps...

Step 3.12.3.3.1

Simplify each term.

Tap for more steps...

Step 3.12.3.3.1.1

Multiply $- 4$ by $- 4$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 - 4 (- \sqrt{22}) - \sqrt{22} \cdot - 4 - \sqrt{22} (- \sqrt{22}))) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.2

Multiply $- 1$ by $- 4$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} - \sqrt{22} \cdot - 4 - \sqrt{22} (- \sqrt{22}))) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.3

Multiply $- 4$ by $- 1$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} - \sqrt{22} (- \sqrt{22}))) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.4

Multiply $- \sqrt{22} (- \sqrt{22})$ .

Tap for more steps...

Step 3.12.3.3.1.4.1

Multiply $- 1$ by $- 1$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + 1 \sqrt{22} \sqrt{22})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.4.2

Multiply $\sqrt{22}$ by $1$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + \sqrt{22} \sqrt{22})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.4.3

Raise $\sqrt{22}$ to the power of $1$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + {\sqrt{22}}^{1} \sqrt{22})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.4.4

Raise $\sqrt{22}$ to the power of $1$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + {\sqrt{22}}^{1} {\sqrt{22}}^{1})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.4.5

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

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + {\sqrt{22}}^{1 + 1})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.4.6

Add $1$ and $1$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + {\sqrt{22}}^{2})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.5

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

Tap for more steps...

Step 3.12.3.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + {(22^{\frac{1}{2}})}^{2})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.5.2

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

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + 22^{\frac{1}{2} \cdot 2})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.5.3

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

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + 22^{\frac{2}{2}})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.12.3.3.1.5.4.1

Cancel the common factor.

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + 22^{\frac{2}{2}})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.5.4.2

Rewrite the expression.

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + 22^{1})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.1.5.5

Evaluate the exponent.

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (16 + 4 \sqrt{22} + 4 \sqrt{22} + 22)) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.2

Add $16$ and $22$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (38 + 4 \sqrt{22} + 4 \sqrt{22})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.3.3

Add $4 \sqrt{22}$ and $4 \sqrt{22}$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - (38 + 8 \sqrt{22})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.4

Apply the distributive property.

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - 1 \cdot 38 - (8 \sqrt{22})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.5

Multiply $- 1$ by $38$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - 38 - (8 \sqrt{22})) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.6

Multiply $8$ by $- 1$ .

$- \frac{12 ({(- 4 + \sqrt{22})}^{2} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7

Simplify each term.

Tap for more steps...

Step 3.12.3.7.1

Rewrite ${(- 4 + \sqrt{22})}^{2}$ as $(- 4 + \sqrt{22}) (- 4 + \sqrt{22})$ .

$- \frac{12 ((- 4 + \sqrt{22}) (- 4 + \sqrt{22}) - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.2

Expand $(- 4 + \sqrt{22}) (- 4 + \sqrt{22})$ using the FOIL Method.

Tap for more steps...

Step 3.12.3.7.2.1

Apply the distributive property.

$- \frac{12 (- 4 (- 4 + \sqrt{22}) + \sqrt{22} (- 4 + \sqrt{22}) - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.2.2

Apply the distributive property.

$- \frac{12 (- 4 \cdot - 4 - 4 \sqrt{22} + \sqrt{22} (- 4 + \sqrt{22}) - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.2.3

Apply the distributive property.

$- \frac{12 (- 4 \cdot - 4 - 4 \sqrt{22} + \sqrt{22} \cdot - 4 + \sqrt{22} \sqrt{22} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.3

Simplify and combine like terms.

Tap for more steps...

Step 3.12.3.7.3.1

Simplify each term.

Tap for more steps...

Step 3.12.3.7.3.1.1

Multiply $- 4$ by $- 4$ .

$- \frac{12 (16 - 4 \sqrt{22} + \sqrt{22} \cdot - 4 + \sqrt{22} \sqrt{22} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.3.1.2

Move $- 4$ to the left of $\sqrt{22}$ .

$- \frac{12 (16 - 4 \sqrt{22} - 4 \cdot \sqrt{22} + \sqrt{22} \sqrt{22} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.3.1.3

Combine using the product rule for radicals.

$- \frac{12 (16 - 4 \sqrt{22} - 4 \sqrt{22} + \sqrt{22 \cdot 22} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.3.1.4

Multiply $22$ by $22$ .

$- \frac{12 (16 - 4 \sqrt{22} - 4 \sqrt{22} + \sqrt{484} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.3.1.5

Rewrite $484$ as $22^{2}$ .

$- \frac{12 (16 - 4 \sqrt{22} - 4 \sqrt{22} + \sqrt{22^{2}} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.3.1.6

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

$- \frac{12 (16 - 4 \sqrt{22} - 4 \sqrt{22} + 22 - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.3.2

Add $16$ and $22$ .

$- \frac{12 (38 - 4 \sqrt{22} - 4 \sqrt{22} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.7.3.3

Subtract $4 \sqrt{22}$ from $- 4 \sqrt{22}$ .

$- \frac{12 (38 - 8 \sqrt{22} - 38 - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.8

Subtract $38$ from $38$ .

$- \frac{12 (0 - 8 \sqrt{22} - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.9

Subtract $8 \sqrt{22}$ from $0$ .

$- \frac{12 (- 8 \sqrt{22} - 8 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.10

Subtract $8 \sqrt{22}$ from $- 8 \sqrt{22}$ .

$- \frac{12 (- 16 \sqrt{22}) + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.11

Multiply $- 16$ by $12$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - {(- 4 - \sqrt{22})}^{3} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.12

Use the Binomial Theorem.

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} + 3 {(- 4)}^{2} (- \sqrt{22}) + 3 \cdot - 4 {(- \sqrt{22})}^{2} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.13

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

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} + 3 \cdot 16 (- \sqrt{22}) + 3 \cdot - 4 {(- \sqrt{22})}^{2} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.14

Multiply $3$ by $16$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} + 48 (- \sqrt{22}) + 3 \cdot - 4 {(- \sqrt{22})}^{2} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.15

Multiply $- 1$ by $48$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} + 3 \cdot - 4 {(- \sqrt{22})}^{2} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.16

Multiply $3$ by $- 4$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 {(- \sqrt{22})}^{2} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.17

Apply the product rule to $- \sqrt{22}$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 ({(- 1)}^{2} {\sqrt{22}}^{2}) + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.18

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

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 (1 {\sqrt{22}}^{2}) + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.19

Multiply ${\sqrt{22}}^{2}$ by $1$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 {\sqrt{22}}^{2} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.20

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

Tap for more steps...

Step 3.12.3.20.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 {(22^{\frac{1}{2}})}^{2} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.20.2

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

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 \cdot 22^{\frac{1}{2} \cdot 2} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.20.3

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

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 \cdot 22^{\frac{2}{2}} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.20.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.12.3.20.4.1

Cancel the common factor.

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 \cdot 22^{\frac{2}{2}} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.20.4.2

Rewrite the expression.

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 \cdot 22^{1} + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.20.5

Evaluate the exponent.

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 12 \cdot 22 + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.21

Multiply $- 12$ by $22$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - ({(- 4)}^{3} - 48 \sqrt{22} - 264 + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22

Simplify each term.

Tap for more steps...

Step 3.12.3.22.1

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

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 + {(- \sqrt{22})}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22.2

Apply the product rule to $- \sqrt{22}$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 + {(- 1)}^{3} {\sqrt{22}}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22.3

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

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 - {\sqrt{22}}^{3}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22.4

Rewrite ${\sqrt{22}}^{3}$ as $\sqrt{22^{3}}$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 - \sqrt{22^{3}}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22.5

Raise $22$ to the power of $3$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 - \sqrt{10648}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22.6

Rewrite $10648$ as $22^{2} \cdot 22$ .

Tap for more steps...

Step 3.12.3.22.6.1

Factor $484$ out of $10648$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 - \sqrt{484 (22)}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22.6.2

Rewrite $484$ as $22^{2}$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 - \sqrt{22^{2} \cdot 22}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22.7

Pull terms out from under the radical.

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 - (22 \sqrt{22})) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.22.8

Multiply $22$ by $- 1$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 64 - 48 \sqrt{22} - 264 - 22 \sqrt{22}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.23

Subtract $264$ from $- 64$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 328 - 48 \sqrt{22} - 22 \sqrt{22}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.24

Subtract $22 \sqrt{22}$ from $- 48 \sqrt{22}$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - (- 328 - 70 \sqrt{22}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.25

Apply the distributive property.

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} - - 328 - (- 70 \sqrt{22}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.26

Multiply $- 1$ by $- 328$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} + 328 - (- 70 \sqrt{22}) - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.27

Multiply $- 70$ by $- 1$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} + 328 + 70 \sqrt{22} - 18 (- 4 + \sqrt{22}) + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.28

Apply the distributive property.

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} + 328 + 70 \sqrt{22} - 18 \cdot - 4 - 18 \sqrt{22} + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.29

Multiply $- 18$ by $- 4$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} + 18 (- 4 - \sqrt{22})}{3}$

Step 3.12.3.30

Apply the distributive property.

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} + 18 \cdot - 4 + 18 (- \sqrt{22})}{3}$

Step 3.12.3.31

Multiply $18$ by $- 4$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 + 18 (- \sqrt{22})}{3}$

Step 3.12.3.32

Multiply $- 1$ by $18$ .

$- \frac{- 192 \sqrt{22} + {(- 4 + \sqrt{22})}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33

Simplify the numerator.

Tap for more steps...

Step 3.12.3.33.1

Use the Binomial Theorem.

$- \frac{- 192 \sqrt{22} + {(- 4)}^{3} + 3 {(- 4)}^{2} \sqrt{22} + 3 \cdot - 4 {\sqrt{22}}^{2} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2

Simplify each term.

Tap for more steps...

Step 3.12.3.33.2.1

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

$- \frac{- 192 \sqrt{22} - 64 + 3 {(- 4)}^{2} \sqrt{22} + 3 \cdot - 4 {\sqrt{22}}^{2} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.2

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

$- \frac{- 192 \sqrt{22} - 64 + 3 \cdot 16 \sqrt{22} + 3 \cdot - 4 {\sqrt{22}}^{2} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.3

Multiply $3$ by $16$ .

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} + 3 \cdot - 4 {\sqrt{22}}^{2} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.4

Multiply $3$ by $- 4$ .

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 12 {\sqrt{22}}^{2} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.5

Rewrite ${\sqrt{22}}^{2}$ as $22$ .

Tap for more steps...

Step 3.12.3.33.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{22}$ as $22^{\frac{1}{2}}$ .

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 12 {(22^{\frac{1}{2}})}^{2} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.5.2

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

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 12 \cdot 22^{\frac{1}{2} \cdot 2} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.5.3

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

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 12 \cdot 22^{\frac{2}{2}} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.12.3.33.2.5.4.1

Cancel the common factor.

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 12 \cdot 22^{\frac{2}{2}} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.5.4.2

Rewrite the expression.

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 12 \cdot 22^{1} + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.5.5

Evaluate the exponent.

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 12 \cdot 22 + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.6

Multiply $- 12$ by $22$ .

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 264 + {\sqrt{22}}^{3} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.7

Rewrite ${\sqrt{22}}^{3}$ as $\sqrt{22^{3}}$ .

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 264 + \sqrt{22^{3}} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.8

Raise $22$ to the power of $3$ .

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 264 + \sqrt{10648} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.9

Rewrite $10648$ as $22^{2} \cdot 22$ .

Tap for more steps...

Step 3.12.3.33.2.9.1

Factor $484$ out of $10648$ .

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 264 + \sqrt{484 (22)} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.9.2

Rewrite $484$ as $22^{2}$ .

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 264 + \sqrt{22^{2} \cdot 22} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.2.10

Pull terms out from under the radical.

$- \frac{- 192 \sqrt{22} - 64 + 48 \sqrt{22} - 264 + 22 \sqrt{22} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.3

Subtract $264$ from $- 64$ .

$- \frac{- 192 \sqrt{22} - 328 + 48 \sqrt{22} + 22 \sqrt{22} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.4

Add $48 \sqrt{22}$ and $22 \sqrt{22}$ .

$- \frac{- 192 \sqrt{22} - 328 + 70 \sqrt{22} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.5

Add $- 192 \sqrt{22}$ and $70 \sqrt{22}$ .

$- \frac{- 328 - 122 \sqrt{22} + 328 + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.6

Add $- 328$ and $328$ .

$- \frac{0 - 122 \sqrt{22} + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.7

Subtract $122 \sqrt{22}$ from $0$ .

$- \frac{- 122 \sqrt{22} + 70 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.8

Add $- 122 \sqrt{22}$ and $70 \sqrt{22}$ .

$- \frac{- 52 \sqrt{22} + 72 - 18 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.9

Subtract $18 \sqrt{22}$ from $- 52 \sqrt{22}$ .

$- \frac{72 - 70 \sqrt{22} - 72 - 18 \sqrt{22}}{3}$

Step 3.12.3.33.10

Subtract $72$ from $72$ .

$- \frac{0 - 70 \sqrt{22} - 18 \sqrt{22}}{3}$

Step 3.12.3.33.11

Subtract $70 \sqrt{22}$ from $0$ .

$- \frac{- 70 \sqrt{22} - 18 \sqrt{22}}{3}$

Step 3.12.3.33.12

Subtract $18 \sqrt{22}$ from $- 70 \sqrt{22}$ .

$- \frac{- 88 \sqrt{22}}{3}$

Step 3.12.3.34

Move the negative in front of the fraction.

$- - \frac{88 \sqrt{22}}{3}$

Step 3.12.3.35

Multiply $- - \frac{88 \sqrt{22}}{3}$ .

Tap for more steps...

Step 3.12.3.35.1

Multiply $- 1$ by $- 1$ .

$1 \frac{88 \sqrt{22}}{3}$

Step 3.12.3.35.2

Multiply $\frac{88 \sqrt{22}}{3}$ by $1$ .

$\frac{88 \sqrt{22}}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{88 \sqrt{22}}{3}$

Decimal Form:

$137.58552895 \dots$

Step 5