Calculus Examples

$y = | x - 16 |$ , $y = \frac{x}{3}$

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.

$| x - 16 | = \frac{x}{3}$

Step 1.2

Solve $| x - 16 | = \frac{x}{3}$ for $x$ .

Tap for more steps...

Step 1.2.1

Multiply both sides by $3$ .

$| x - 16 | \cdot 3 = \frac{x}{3} \cdot 3$

Step 1.2.2

Simplify.

Tap for more steps...

Step 1.2.2.1

Simplify the left side.

Tap for more steps...

Step 1.2.2.1.1

Move $3$ to the left of $| x - 16 |$ .

$3 | x - 16 | = \frac{x}{3} \cdot 3$

Step 1.2.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$3 | x - 16 | = \frac{x}{3} \cdot 3$

Step 1.2.2.2.1.2

Rewrite the expression.

$3 | x - 16 | = x$

Step 1.2.3

Solve for $x$ .

Tap for more steps...

Step 1.2.3.1

Divide each term in $3 | x - 16 | = x$ by $3$ and simplify.

Tap for more steps...

Step 1.2.3.1.1

Divide each term in $3 | x - 16 | = x$ by $3$ .

$\frac{3 | x - 16 |}{3} = \frac{x}{3}$

Step 1.2.3.1.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.1.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.3.1.2.1.1

Cancel the common factor.

$\frac{3 | x - 16 |}{3} = \frac{x}{3}$

Step 1.2.3.1.2.1.2

Divide $| x - 16 |$ by $1$ .

$| x - 16 | = \frac{x}{3}$

Step 1.2.3.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 16 = \pm \frac{x}{3}$

Step 1.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 1.2.3.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 16 = \frac{x}{3}$

Step 1.2.3.3.2

Multiply both sides by $3$ .

$(x - 16) \cdot 3 = \frac{x}{3} \cdot 3$

Step 1.2.3.3.3

Simplify.

Tap for more steps...

Step 1.2.3.3.3.1

Simplify the left side.

Tap for more steps...

Step 1.2.3.3.3.1.1

Simplify $(x - 16) \cdot 3$ .

Tap for more steps...

Step 1.2.3.3.3.1.1.1

Apply the distributive property.

$x \cdot 3 - 16 \cdot 3 = \frac{x}{3} \cdot 3$

Step 1.2.3.3.3.1.1.2

Simplify the expression.

Tap for more steps...

Step 1.2.3.3.3.1.1.2.1

Move $3$ to the left of $x$ .

$3 \cdot x - 16 \cdot 3 = \frac{x}{3} \cdot 3$

Step 1.2.3.3.3.1.1.2.2

Multiply $- 16$ by $3$ .

$3 x - 48 = \frac{x}{3} \cdot 3$

Step 1.2.3.3.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.3.3.3.2.1.1

Cancel the common factor.

$3 x - 48 = \frac{x}{3} \cdot 3$

Step 1.2.3.3.3.2.1.2

Rewrite the expression.

$3 x - 48 = x$

Step 1.2.3.3.4

Solve for $x$ .

Tap for more steps...

Step 1.2.3.3.4.1

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

Tap for more steps...

Step 1.2.3.3.4.1.1

Subtract $x$ from both sides of the equation.

$3 x - 48 - x = 0$

Step 1.2.3.3.4.1.2

Subtract $x$ from $3 x$ .

$2 x - 48 = 0$

Step 1.2.3.3.4.2

Add $48$ to both sides of the equation.

$2 x = 48$

Step 1.2.3.3.4.3

Divide each term in $2 x = 48$ by $2$ and simplify.

Tap for more steps...

Step 1.2.3.3.4.3.1

Divide each term in $2 x = 48$ by $2$ .

$\frac{2 x}{2} = \frac{48}{2}$

Step 1.2.3.3.4.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.3.4.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.3.4.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{48}{2}$

Step 1.2.3.3.4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{48}{2}$

Step 1.2.3.3.4.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.4.3.3.1

Divide $48$ by $2$ .

$x = 24$

Step 1.2.3.3.5

Next, use the negative value of the $\pm$ to find the second solution.

$x - 16 = - \frac{x}{3}$

Step 1.2.3.3.6

Multiply both sides by $3$ .

$(x - 16) \cdot 3 = - \frac{x}{3} \cdot 3$

Step 1.2.3.3.7

Simplify.

Tap for more steps...

Step 1.2.3.3.7.1

Simplify the left side.

Tap for more steps...

Step 1.2.3.3.7.1.1

Simplify $(x - 16) \cdot 3$ .

Tap for more steps...

Step 1.2.3.3.7.1.1.1

Apply the distributive property.

$x \cdot 3 - 16 \cdot 3 = - \frac{x}{3} \cdot 3$

Step 1.2.3.3.7.1.1.2

Simplify the expression.

Tap for more steps...

Step 1.2.3.3.7.1.1.2.1

Move $3$ to the left of $x$ .

$3 \cdot x - 16 \cdot 3 = - \frac{x}{3} \cdot 3$

Step 1.2.3.3.7.1.1.2.2

Multiply $- 16$ by $3$ .

$3 x - 48 = - \frac{x}{3} \cdot 3$

Step 1.2.3.3.7.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.7.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.3.3.7.2.1.1

Move the leading negative in $- \frac{x}{3}$ into the numerator.

$3 x - 48 = \frac{- x}{3} \cdot 3$

Step 1.2.3.3.7.2.1.2

Cancel the common factor.

$3 x - 48 = \frac{- x}{3} \cdot 3$

Step 1.2.3.3.7.2.1.3

Rewrite the expression.

$3 x - 48 = - x$

Step 1.2.3.3.8

Solve for $x$ .

Tap for more steps...

Step 1.2.3.3.8.1

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

Tap for more steps...

Step 1.2.3.3.8.1.1

Add $x$ to both sides of the equation.

$3 x - 48 + x = 0$

Step 1.2.3.3.8.1.2

Add $3 x$ and $x$ .

$4 x - 48 = 0$

Step 1.2.3.3.8.2

Add $48$ to both sides of the equation.

$4 x = 48$

Step 1.2.3.3.8.3

Divide each term in $4 x = 48$ by $4$ and simplify.

Tap for more steps...

Step 1.2.3.3.8.3.1

Divide each term in $4 x = 48$ by $4$ .

$\frac{4 x}{4} = \frac{48}{4}$

Step 1.2.3.3.8.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.3.8.3.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.3.3.8.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{48}{4}$

Step 1.2.3.3.8.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{48}{4}$

Step 1.2.3.3.8.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.8.3.3.1

Divide $48$ by $4$ .

$x = 12$

Step 1.2.3.3.9

The complete solution is the result of both the positive and negative portions of the solution.

$x = 24, 12$

Step 1.3

Evaluate $y$ when $x = 24$ .

Tap for more steps...

Step 1.3.1

Substitute $24$ for $x$ .

$y = \frac{24}{3}$

Step 1.3.2

Substitute $24$ for $x$ in $y = \frac{24}{3}$ and solve for $y$ .

Tap for more steps...

Step 1.3.2.1

Remove parentheses.

$y = \frac{24}{3}$

Step 1.3.2.2

Divide $24$ by $3$ .

$y = 8$

Step 1.4

Evaluate $y$ when $x = 12$ .

Tap for more steps...

Step 1.4.1

Substitute $12$ for $x$ .

$y = \frac{12}{3}$

Step 1.4.2

Substitute $12$ for $x$ in $y = \frac{12}{3}$ and solve for $y$ .

Tap for more steps...

Step 1.4.2.1

Remove parentheses.

$y = \frac{12}{3}$

Step 1.4.2.2

Divide $12$ by $3$ .

$y = 4$

Step 1.5

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

$(24, 8)$

$(12, 4)$

$(24, 8)$

$(12, 4)$

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.

$A r e a = \int_{12}^{24} \frac{x}{3} d x - \int_{12}^{24} | x - 16 | d x$

Step 3

Integrate to find the area between $12$ and $24$ .

Tap for more steps...

Step 3.1

Combine the integrals into a single integral.

$\int_{12}^{24} \frac{x}{3} - (| x - 16 |) d x$

Step 3.2

To write $- | x - 16 |$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\int_{12}^{24} \frac{x}{3} - | x - 16 | \cdot \frac{3}{3} d x$

Step 3.3

Combine $- | x - 16 |$ and $\frac{3}{3}$ .

$\int_{12}^{24} \frac{x}{3} + \frac{- | x - 16 | \cdot 3}{3} d x$

Step 3.4

Combine the numerators over the common denominator.

$\int_{12}^{24} \frac{x - | x - 16 | \cdot 3}{3} d x$

Step 3.5

Multiply $3$ by $- 1$ .

$\int_{12}^{24} \frac{x - 3 | x - 16 |}{3} d x$

Step 3.6

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{12}^{24} x - 3 | x - 16 | d x$

Step 3.7

Split the single integral into multiple integrals.

$\frac{1}{3} (\int_{12}^{24} x d x + \int_{12}^{24} - 3 | x - 16 | d x)$

Step 3.8

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

$\frac{1}{3} (\frac{1}{2} x^{2}]_{12}^{24} + \int_{12}^{24} - 3 | x - 16 | d x)$

Step 3.9

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{3} (\frac{1}{2} x^{2}]_{12}^{24} - 3 \int_{12}^{24} | x - 16 | d x)$

Step 3.10

Split up the integral depending on where $x - 16$ is positive and negative.

$\frac{1}{3} (\frac{1}{2} x^{2}]_{12}^{24} - 3 (\int_{12}^{16} - x + 16 d x + \int_{16}^{24} x - 16 d x))$

Step 3.11

Split the single integral into multiple integrals.

$\frac{1}{3} (\frac{1}{2} x^{2}]_{12}^{24} - 3 (\int_{12}^{16} - x d x + \int_{12}^{16} 16 d x + \int_{16}^{24} x - 16 d x))$

Step 3.12

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

$\frac{1}{3} (\frac{1}{2} x^{2}]_{12}^{24} - 3 (- \int_{12}^{16} x d x + \int_{12}^{16} 16 d x + \int_{16}^{24} x - 16 d x))$

Step 3.13

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

$\frac{1}{3} (\frac{1}{2} x^{2}]_{12}^{24} - 3 (- (\frac{1}{2} x^{2}]_{12}^{16}) + \int_{12}^{16} 16 d x + \int_{16}^{24} x - 16 d x))$

Step 3.14

Simplify.

Tap for more steps...

Step 3.14.1

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

$\frac{1}{3} (\frac{1}{2} x^{2}]_{12}^{24} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + \int_{12}^{16} 16 d x + \int_{16}^{24} x - 16 d x))$

Step 3.14.2

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

$\frac{1}{3} (\frac{x^{2}}{2}]_{12}^{24} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + \int_{12}^{16} 16 d x + \int_{16}^{24} x - 16 d x))$

Step 3.15

Apply the constant rule.

$\frac{1}{3} (\frac{x^{2}}{2}]_{12}^{24} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + 16 x]_{12}^{16} + \int_{16}^{24} x - 16 d x))$

Step 3.16

Split the single integral into multiple integrals.

$\frac{1}{3} (\frac{x^{2}}{2}]_{12}^{24} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + 16 x]_{12}^{16} + \int_{16}^{24} x d x + \int_{16}^{24} - 16 d x))$

Step 3.17

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

$\frac{1}{3} (\frac{x^{2}}{2}]_{12}^{24} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + 16 x]_{12}^{16} + \frac{1}{2} x^{2}]_{16}^{24} + \int_{16}^{24} - 16 d x))$

Step 3.18

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

$\frac{1}{3} (\frac{x^{2}}{2}]_{12}^{24} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + 16 x]_{12}^{16} + \frac{x^{2}}{2}]_{16}^{24} + \int_{16}^{24} - 16 d x))$

Step 3.19

Apply the constant rule.

$\frac{1}{3} (\frac{x^{2}}{2}]_{12}^{24} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + 16 x]_{12}^{16} + \frac{x^{2}}{2}]_{16}^{24} + - 16 x]_{16}^{24}))$

Step 3.20

Simplify the answer.

Tap for more steps...

Step 3.20.1

Combine $\frac{x^{2}}{2}]_{16}^{24}$ and $- 16 x]_{16}^{24}$ .

$\frac{1}{3} (\frac{x^{2}}{2}]_{12}^{24} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + 16 x]_{12}^{16} + \frac{x^{2}}{2} - 16 x]_{16}^{24}))$

Step 3.20.2

Substitute and simplify.

Tap for more steps...

Step 3.20.2.1

Evaluate $\frac{x^{2}}{2}$ at $24$ and at $12$ .

$\frac{1}{3} ((\frac{24^{2}}{2}) - \frac{12^{2}}{2} - 3 (- (\frac{x^{2}}{2}]_{12}^{16}) + 16 x]_{12}^{16} + \frac{x^{2}}{2} - 16 x]_{16}^{24}))$

Step 3.20.2.2

Evaluate $\frac{x^{2}}{2}$ at $16$ and at $12$ .

$\frac{1}{3} ((\frac{24^{2}}{2}) - \frac{12^{2}}{2} - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + 16 x]_{12}^{16} + \frac{x^{2}}{2} - 16 x]_{16}^{24}))$

Step 3.20.2.3

Evaluate $16 x$ at $16$ and at $12$ .

$\frac{1}{3} ((\frac{24^{2}}{2}) - \frac{12^{2}}{2} - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + \frac{x^{2}}{2} - 16 x]_{16}^{24}))$

Step 3.20.2.4

Evaluate $\frac{x^{2}}{2} - 16 x$ at $24$ and at $16$ .

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

Step 3.20.2.5

Simplify.

Tap for more steps...

Step 3.20.2.5.1

Raise $24$ to the power of $2$ .

$\frac{1}{3} (\frac{576}{2} - \frac{12^{2}}{2} - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.2

Cancel the common factor of $576$ and $2$ .

Tap for more steps...

Step 3.20.2.5.2.1

Factor $2$ out of $576$ .

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

Step 3.20.2.5.2.2

Cancel the common factors.

Tap for more steps...

Step 3.20.2.5.2.2.1

Factor $2$ out of $2$ .

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

Step 3.20.2.5.2.2.2

Cancel the common factor.

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

Step 3.20.2.5.2.2.3

Rewrite the expression.

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

Step 3.20.2.5.2.2.4

Divide $288$ by $1$ .

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

Step 3.20.2.5.3

Raise $12$ to the power of $2$ .

$\frac{1}{3} (288 - \frac{144}{2} - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.4

Cancel the common factor of $144$ and $2$ .

Tap for more steps...

Step 3.20.2.5.4.1

Factor $2$ out of $144$ .

$\frac{1}{3} (288 - \frac{2 \cdot 72}{2} - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.4.2

Cancel the common factors.

Tap for more steps...

Step 3.20.2.5.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{3} (288 - \frac{2 \cdot 72}{2 (1)} - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.4.2.2

Cancel the common factor.

$\frac{1}{3} (288 - \frac{2 \cdot 72}{2 \cdot 1} - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.4.2.3

Rewrite the expression.

$\frac{1}{3} (288 - \frac{72}{1} - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.4.2.4

Divide $72$ by $1$ .

$\frac{1}{3} (288 - 1 \cdot 72 - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.5

Multiply $- 1$ by $72$ .

$\frac{1}{3} (288 - 72 - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.6

Subtract $72$ from $288$ .

$\frac{1}{3} (216 - 3 (- ((\frac{16^{2}}{2}) - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.7

Raise $16$ to the power of $2$ .

$\frac{1}{3} (216 - 3 (- (\frac{256}{2} - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.8

Cancel the common factor of $256$ and $2$ .

Tap for more steps...

Step 3.20.2.5.8.1

Factor $2$ out of $256$ .

$\frac{1}{3} (216 - 3 (- (\frac{2 \cdot 128}{2} - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.8.2

Cancel the common factors.

Tap for more steps...

Step 3.20.2.5.8.2.1

Factor $2$ out of $2$ .

$\frac{1}{3} (216 - 3 (- (\frac{2 \cdot 128}{2 (1)} - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.8.2.2

Cancel the common factor.

$\frac{1}{3} (216 - 3 (- (\frac{2 \cdot 128}{2 \cdot 1} - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.8.2.3

Rewrite the expression.

$\frac{1}{3} (216 - 3 (- (\frac{128}{1} - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.8.2.4

Divide $128$ by $1$ .

$\frac{1}{3} (216 - 3 (- (128 - \frac{12^{2}}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.9

Raise $12$ to the power of $2$ .

$\frac{1}{3} (216 - 3 (- (128 - \frac{144}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.10

Cancel the common factor of $144$ and $2$ .

Tap for more steps...

Step 3.20.2.5.10.1

Factor $2$ out of $144$ .

$\frac{1}{3} (216 - 3 (- (128 - \frac{2 \cdot 72}{2}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.10.2

Cancel the common factors.

Tap for more steps...

Step 3.20.2.5.10.2.1

Factor $2$ out of $2$ .

$\frac{1}{3} (216 - 3 (- (128 - \frac{2 \cdot 72}{2 (1)}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.10.2.2

Cancel the common factor.

$\frac{1}{3} (216 - 3 (- (128 - \frac{2 \cdot 72}{2 \cdot 1}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.10.2.3

Rewrite the expression.

$\frac{1}{3} (216 - 3 (- (128 - \frac{72}{1}) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.10.2.4

Divide $72$ by $1$ .

$\frac{1}{3} (216 - 3 (- (128 - 1 \cdot 72) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.11

Multiply $- 1$ by $72$ .

$\frac{1}{3} (216 - 3 (- (128 - 72) + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.12

Subtract $72$ from $128$ .

$\frac{1}{3} (216 - 3 (- 1 \cdot 56 + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.13

Multiply $- 1$ by $56$ .

$\frac{1}{3} (216 - 3 (- 56 + (16 \cdot 16) - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.14

Multiply $16$ by $16$ .

$\frac{1}{3} (216 - 3 (- 56 + 256 - 16 \cdot 12 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.15

Multiply $- 16$ by $12$ .

$\frac{1}{3} (216 - 3 (- 56 + 256 - 192 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.16

Subtract $192$ from $256$ .

$\frac{1}{3} (216 - 3 (- 56 + 64 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.17

Add $- 56$ and $64$ .

$\frac{1}{3} (216 - 3 (8 + (\frac{24^{2}}{2} - 16 \cdot 24) - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.18

Raise $24$ to the power of $2$ .

$\frac{1}{3} (216 - 3 (8 + \frac{576}{2} - 16 \cdot 24 - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.19

Cancel the common factor of $576$ and $2$ .

Tap for more steps...

Step 3.20.2.5.19.1

Factor $2$ out of $576$ .

$\frac{1}{3} (216 - 3 (8 + \frac{2 \cdot 288}{2} - 16 \cdot 24 - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.19.2

Cancel the common factors.

Tap for more steps...

Step 3.20.2.5.19.2.1

Factor $2$ out of $2$ .

$\frac{1}{3} (216 - 3 (8 + \frac{2 \cdot 288}{2 (1)} - 16 \cdot 24 - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.19.2.2

Cancel the common factor.

$\frac{1}{3} (216 - 3 (8 + \frac{2 \cdot 288}{2 \cdot 1} - 16 \cdot 24 - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.19.2.3

Rewrite the expression.

$\frac{1}{3} (216 - 3 (8 + \frac{288}{1} - 16 \cdot 24 - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.19.2.4

Divide $288$ by $1$ .

$\frac{1}{3} (216 - 3 (8 + 288 - 16 \cdot 24 - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.20

Multiply $- 16$ by $24$ .

$\frac{1}{3} (216 - 3 (8 + 288 - 384 - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.21

Subtract $384$ from $288$ .

$\frac{1}{3} (216 - 3 (8 - 96 - (\frac{16^{2}}{2} - 16 \cdot 16)))$

Step 3.20.2.5.22

Raise $16$ to the power of $2$ .

$\frac{1}{3} (216 - 3 (8 - 96 - (\frac{256}{2} - 16 \cdot 16)))$

Step 3.20.2.5.23

Cancel the common factor of $256$ and $2$ .

Tap for more steps...

Step 3.20.2.5.23.1

Factor $2$ out of $256$ .

$\frac{1}{3} (216 - 3 (8 - 96 - (\frac{2 \cdot 128}{2} - 16 \cdot 16)))$

Step 3.20.2.5.23.2

Cancel the common factors.

Tap for more steps...

Step 3.20.2.5.23.2.1

Factor $2$ out of $2$ .

$\frac{1}{3} (216 - 3 (8 - 96 - (\frac{2 \cdot 128}{2 (1)} - 16 \cdot 16)))$

Step 3.20.2.5.23.2.2

Cancel the common factor.

$\frac{1}{3} (216 - 3 (8 - 96 - (\frac{2 \cdot 128}{2 \cdot 1} - 16 \cdot 16)))$

Step 3.20.2.5.23.2.3

Rewrite the expression.

$\frac{1}{3} (216 - 3 (8 - 96 - (\frac{128}{1} - 16 \cdot 16)))$

Step 3.20.2.5.23.2.4

Divide $128$ by $1$ .

$\frac{1}{3} (216 - 3 (8 - 96 - (128 - 16 \cdot 16)))$

Step 3.20.2.5.24

Multiply $- 16$ by $16$ .

$\frac{1}{3} (216 - 3 (8 - 96 - (128 - 256)))$

Step 3.20.2.5.25

Subtract $256$ from $128$ .

$\frac{1}{3} (216 - 3 (8 - 96 - - 128))$

Step 3.20.2.5.26

Multiply $- 1$ by $- 128$ .

$\frac{1}{3} (216 - 3 (8 - 96 + 128))$

Step 3.20.2.5.27

Add $- 96$ and $128$ .

$\frac{1}{3} (216 - 3 (8 + 32))$

Step 3.20.2.5.28

Add $8$ and $32$ .

$\frac{1}{3} (216 - 3 \cdot 40)$

Step 3.20.2.5.29

Multiply $- 3$ by $40$ .

$\frac{1}{3} (216 - 120)$

Step 3.20.2.5.30

Subtract $120$ from $216$ .

$\frac{1}{3} \cdot 96$

Step 3.20.2.5.31

Combine $\frac{1}{3}$ and $96$ .

$\frac{96}{3}$

Step 3.20.2.5.32

Cancel the common factor of $96$ and $3$ .

Tap for more steps...

Step 3.20.2.5.32.1

Factor $3$ out of $96$ .

$\frac{3 \cdot 32}{3}$

Step 3.20.2.5.32.2

Cancel the common factors.

Tap for more steps...

Step 3.20.2.5.32.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 32}{3 (1)}$

Step 3.20.2.5.32.2.2

Cancel the common factor.

$\frac{3 \cdot 32}{3 \cdot 1}$

Step 3.20.2.5.32.2.3

Rewrite the expression.

$\frac{32}{1}$

Step 3.20.2.5.32.2.4

Divide $32$ by $1$ .

$32$

Step 4