Calculus Examples

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

Step 1

Solve by substitution to find the intersection between the curves.

Tap for more steps...

Step 1.1

Replace all occurrences of $y$ with $2 x - 4$ in each equation.

Tap for more steps...

Step 1.1.1

Replace all occurrences of $y$ in $y^{2} = 4 x$ with $2 x - 4$ .

${(2 x - 4)}^{2} = 4 x$

$y = 2 x - 4$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Simplify ${(2 x - 4)}^{2}$ .

Tap for more steps...

Step 1.1.2.1.1

Rewrite ${(2 x - 4)}^{2}$ as $(2 x - 4) (2 x - 4)$ .

$(2 x - 4) (2 x - 4) = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.2

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

Tap for more steps...

Step 1.1.2.1.2.1

Apply the distributive property.

$2 x (2 x - 4) - 4 (2 x - 4) = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 4 - 4 (2 x - 4) = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

$2 x (2 x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.1.2.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot (2 x \cdot x) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 1.1.2.1.3.1.2.1

Move $x$ .

$2 \cdot (2 (x \cdot x)) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot (2 x^{2}) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

$2 \cdot (2 x^{2}) + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 4 - 4 (2 x) - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.3.1.4

Multiply $- 4$ by $2$ .

$4 x^{2} - 8 x - 4 (2 x) - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.3.1.5

Multiply $2$ by $- 4$ .

$4 x^{2} - 8 x - 8 x - 4 \cdot - 4 = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.3.1.6

Multiply $- 4$ by $- 4$ .

$4 x^{2} - 8 x - 8 x + 16 = 4 x$

$y = 2 x - 4$

$4 x^{2} - 8 x - 8 x + 16 = 4 x$

$y = 2 x - 4$

Step 1.1.2.1.3.2

Subtract $8 x$ from $- 8 x$ .

$4 x^{2} - 16 x + 16 = 4 x$

$y = 2 x - 4$

$4 x^{2} - 16 x + 16 = 4 x$

$y = 2 x - 4$

$4 x^{2} - 16 x + 16 = 4 x$

$y = 2 x - 4$

$4 x^{2} - 16 x + 16 = 4 x$

$y = 2 x - 4$

$4 x^{2} - 16 x + 16 = 4 x$

$y = 2 x - 4$

Step 1.2

Solve for $x$ in $4 x^{2} - 16 x + 16 = 4 x$ .

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 $4 x$ from both sides of the equation.

$4 x^{2} - 16 x + 16 - 4 x = 0$

$y = 2 x - 4$

Step 1.2.1.2

Subtract $4 x$ from $- 16 x$ .

$4 x^{2} - 20 x + 16 = 0$

$y = 2 x - 4$

$4 x^{2} - 20 x + 16 = 0$

$y = 2 x - 4$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor $4$ out of $4 x^{2} - 20 x + 16$ .

Tap for more steps...

Step 1.2.2.1.1

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 20 x + 16 = 0$

$y = 2 x - 4$

Step 1.2.2.1.2

Factor $4$ out of $- 20 x$ .

$4 (x^{2}) + 4 (- 5 x) + 16 = 0$

$y = 2 x - 4$

Step 1.2.2.1.3

Factor $4$ out of $16$ .

$4 x^{2} + 4 (- 5 x) + 4 \cdot 4 = 0$

$y = 2 x - 4$

Step 1.2.2.1.4

Factor $4$ out of $4 x^{2} + 4 (- 5 x)$ .

$4 (x^{2} - 5 x) + 4 \cdot 4 = 0$

$y = 2 x - 4$

Step 1.2.2.1.5

Factor $4$ out of $4 (x^{2} - 5 x) + 4 \cdot 4$ .

$4 (x^{2} - 5 x + 4) = 0$

$y = 2 x - 4$

$4 (x^{2} - 5 x + 4) = 0$

$y = 2 x - 4$

Step 1.2.2.2

Factor.

Tap for more steps...

Step 1.2.2.2.1

Factor $x^{2} - 5 x + 4$ using the AC method.

Tap for more steps...

Step 1.2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $- 5$ .

$- 4, - 1$

$y = 2 x - 4$

Step 1.2.2.2.1.2

Write the factored form using these integers.

$4 ((x - 4) (x - 1)) = 0$

$y = 2 x - 4$

$4 ((x - 4) (x - 1)) = 0$

$y = 2 x - 4$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$4 (x - 4) (x - 1) = 0$

$y = 2 x - 4$

$4 (x - 4) (x - 1) = 0$

$y = 2 x - 4$

$4 (x - 4) (x - 1) = 0$

$y = 2 x - 4$

Step 1.2.3

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

$x - 4 = 0$

$x - 1 = 0$

$y = 2 x - 4$

Step 1.2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

$y = 2 x - 4$

Step 1.2.4.2

Add $4$ to both sides of the equation.

$x = 4$

$y = 2 x - 4$

$x = 4$

$y = 2 x - 4$

Step 1.2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

$y = 2 x - 4$

Step 1.2.5.2

Add $1$ to both sides of the equation.

$x = 1$

$y = 2 x - 4$

$x = 1$

$y = 2 x - 4$

Step 1.2.6

The final solution is all the values that make $4 (x - 4) (x - 1) = 0$ true.

$x = 4, 1$

$y = 2 x - 4$

$x = 4, 1$

$y = 2 x - 4$

Step 1.3

Replace all occurrences of $x$ with $4$ in each equation.

Tap for more steps...

Step 1.3.1

Replace all occurrences of $x$ in $y = 2 x - 4$ with $4$ .

$y = 2 (4) - 4$

$x = 4$

Step 1.3.2

Simplify the right side.

Tap for more steps...

Step 1.3.2.1

Simplify $2 (4) - 4$ .

Tap for more steps...

Step 1.3.2.1.1

Multiply $2$ by $4$ .

$y = 8 - 4$

$x = 4$

Step 1.3.2.1.2

Subtract $4$ from $8$ .

$y = 4$

$x = 4$

$y = 4$

$x = 4$

$y = 4$

$x = 4$

$y = 4$

$x = 4$

Step 1.4

Replace all occurrences of $x$ with $1$ in each equation.

Tap for more steps...

Step 1.4.1

Replace all occurrences of $x$ in $y = 2 x - 4$ with $1$ .

$y = 2 (1) - 4$

$x = 1$

Step 1.4.2

Simplify the right side.

Tap for more steps...

Step 1.4.2.1

Simplify $2 (1) - 4$ .

Tap for more steps...

Step 1.4.2.1.1

Multiply $2$ by $1$ .

$y = 2 - 4$

$x = 1$

Step 1.4.2.1.2

Subtract $4$ from $2$ .

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

$y = - 2$

$x = 1$

Step 1.5

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

$(4, 4)$

$(1, - 2)$

$(4, 4)$

$(1, - 2)$

Step 2

Solve $y^{2} = 4 x$ in terms of $x$ .

Tap for more steps...

Step 2.1

Rewrite the equation as $4 x = y^{2}$ .

$4 x = y^{2}$

Step 2.2

Divide each term in $4 x = y^{2}$ by $4$ and simplify.

Tap for more steps...

Step 2.2.1

Divide each term in $4 x = y^{2}$ by $4$ .

$\frac{4 x}{4} = \frac{y^{2}}{4}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{y^{2}}{4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{y^{2}}{4}$

Step 3

Solve $y = 2 x - 4$ in terms of $x$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $2 x - 4 = y$ .

$2 x - 4 = y$

Step 3.2

Add $4$ to both sides of the equation.

$2 x = y + 4$

Step 3.3

Divide each term in $2 x = y + 4$ by $2$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $2 x = y + 4$ by $2$ .

$\frac{2 x}{2} = \frac{y}{2} + \frac{4}{2}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{y}{2} + \frac{4}{2}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{y}{2} + \frac{4}{2}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Divide $4$ by $2$ .

$x = \frac{y}{2} + 2$

Step 4

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.

$A r e a = \int_{- 2}^{4} \frac{y}{2} + 2 d y - \int_{- 2}^{4} \frac{y^{2}}{4} d y$

Step 5

Integrate to find the area between $- 2$ and $4$ .

Tap for more steps...

Step 5.1

Combine the integrals into a single integral.

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

Step 5.2

Multiply $- 1$ by $\frac{y^{2}}{4}$ .

$\int_{- 2}^{4} \frac{y}{2} + 2 - \frac{y^{2}}{4} d y$

Step 5.3

Split the single integral into multiple integrals.

$\int_{- 2}^{4} \frac{y}{2} d y + \int_{- 2}^{4} 2 d y + \int_{- 2}^{4} - \frac{y^{2}}{4} d y$

Step 5.4

Since $\frac{1}{2}$ is constant with respect to $y$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- 2}^{4} y d y + \int_{- 2}^{4} 2 d y + \int_{- 2}^{4} - \frac{y^{2}}{4} d y$

Step 5.5

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

$\frac{1}{2} \frac{1}{2} y^{2}]_{- 2}^{4} + \int_{- 2}^{4} 2 d y + \int_{- 2}^{4} - \frac{y^{2}}{4} d y$

Step 5.6

Apply the constant rule.

$\frac{1}{2} \frac{1}{2} y^{2}]_{- 2}^{4} + 2 y]_{- 2}^{4} + \int_{- 2}^{4} - \frac{y^{2}}{4} d y$

Step 5.7

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

$\frac{1}{2} \frac{1}{2} y^{2}]_{- 2}^{4} + 2 y]_{- 2}^{4} - \int_{- 2}^{4} \frac{y^{2}}{4} d y$

Step 5.8

Since $\frac{1}{4}$ is constant with respect to $y$ , move $\frac{1}{4}$ out of the integral.

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

Step 5.9

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

$\frac{1}{2} \frac{1}{2} y^{2}]_{- 2}^{4} + 2 y]_{- 2}^{4} - \frac{1}{4} \frac{1}{3} y^{3}]_{- 2}^{4}$

Step 5.10

Substitute and simplify.

Tap for more steps...

Step 5.10.1

Evaluate $\frac{1}{2} y^{2}$ at $4$ and at $- 2$ .

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

Step 5.10.2

Evaluate $2 y$ at $4$ and at $- 2$ .

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

Step 5.10.3

Evaluate $\frac{1}{3} y^{3}$ at $4$ and at $- 2$ .

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

Step 5.10.4

Simplify.

Tap for more steps...

Step 5.10.4.1

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

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

Step 5.10.4.2

Combine $\frac{1}{2}$ and $16$ .

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

Step 5.10.4.3

Cancel the common factor of $16$ and $2$ .

Tap for more steps...

Step 5.10.4.3.1

Factor $2$ out of $16$ .

$\frac{1}{2} (\frac{2 \cdot 8}{2} - \frac{1}{2} {(- 2)}^{2}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.3.2

Cancel the common factors.

Tap for more steps...

Step 5.10.4.3.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} (\frac{2 \cdot 8}{2 (1)} - \frac{1}{2} {(- 2)}^{2}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.3.2.2

Cancel the common factor.

$\frac{1}{2} (\frac{2 \cdot 8}{2 \cdot 1} - \frac{1}{2} {(- 2)}^{2}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.3.2.3

Rewrite the expression.

$\frac{1}{2} (\frac{8}{1} - \frac{1}{2} {(- 2)}^{2}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.3.2.4

Divide $8$ by $1$ .

$\frac{1}{2} (8 - \frac{1}{2} {(- 2)}^{2}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.4

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

$\frac{1}{2} (8 - \frac{1}{2} \cdot 4) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.5

Multiply $4$ by $- 1$ .

$\frac{1}{2} (8 - 4 (\frac{1}{2})) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.6

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

$\frac{1}{2} (8 + \frac{- 4}{2}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.7

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

Tap for more steps...

Step 5.10.4.7.1

Factor $2$ out of $- 4$ .

$\frac{1}{2} (8 + \frac{2 \cdot - 2}{2}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.7.2

Cancel the common factors.

Tap for more steps...

Step 5.10.4.7.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} (8 + \frac{2 \cdot - 2}{2 (1)}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.7.2.2

Cancel the common factor.

$\frac{1}{2} (8 + \frac{2 \cdot - 2}{2 \cdot 1}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.7.2.3

Rewrite the expression.

$\frac{1}{2} (8 + \frac{- 2}{1}) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.7.2.4

Divide $- 2$ by $1$ .

$\frac{1}{2} (8 - 2) + 2 \cdot 4 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.8

Subtract $2$ from $8$ .

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

Step 5.10.4.9

Combine $\frac{1}{2}$ and $6$ .

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

Step 5.10.4.10

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 5.10.4.10.1

Factor $2$ out of $6$ .

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

Step 5.10.4.10.2

Cancel the common factors.

Tap for more steps...

Step 5.10.4.10.2.1

Factor $2$ out of $2$ .

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

Step 5.10.4.10.2.2

Cancel the common factor.

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

Step 5.10.4.10.2.3

Rewrite the expression.

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

Step 5.10.4.10.2.4

Divide $3$ by $1$ .

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

Step 5.10.4.11

Multiply $2$ by $4$ .

$3 + 8 - 2 \cdot - 2 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.12

Multiply $- 2$ by $- 2$ .

$3 + 8 + 4 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.13

Add $8$ and $4$ .

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

Step 5.10.4.14

Add $3$ and $12$ .

$15 - \frac{1}{4} ((\frac{1}{3} \cdot 4^{3}) - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.15

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

$15 - \frac{1}{4} (\frac{1}{3} \cdot 64 - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.16

Combine $\frac{1}{3}$ and $64$ .

$15 - \frac{1}{4} (\frac{64}{3} - \frac{1}{3} {(- 2)}^{3})$

Step 5.10.4.17

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

$15 - \frac{1}{4} (\frac{64}{3} - \frac{1}{3} \cdot - 8)$

Step 5.10.4.18

Multiply $- 8$ by $- 1$ .

$15 - \frac{1}{4} (\frac{64}{3} + 8 (\frac{1}{3}))$

Step 5.10.4.19

Combine $8$ and $\frac{1}{3}$ .

$15 - \frac{1}{4} (\frac{64}{3} + \frac{8}{3})$

Step 5.10.4.20

Combine the numerators over the common denominator.

$15 - \frac{1}{4} \cdot \frac{64 + 8}{3}$

Step 5.10.4.21

Add $64$ and $8$ .

$15 - \frac{1}{4} \cdot \frac{72}{3}$

Step 5.10.4.22

Cancel the common factor of $72$ and $3$ .

Tap for more steps...

Step 5.10.4.22.1

Factor $3$ out of $72$ .

$15 - \frac{1}{4} \cdot \frac{3 \cdot 24}{3}$

Step 5.10.4.22.2

Cancel the common factors.

Tap for more steps...

Step 5.10.4.22.2.1

Factor $3$ out of $3$ .

$15 - \frac{1}{4} \cdot \frac{3 \cdot 24}{3 (1)}$

Step 5.10.4.22.2.2

Cancel the common factor.

$15 - \frac{1}{4} \cdot \frac{3 \cdot 24}{3 \cdot 1}$

Step 5.10.4.22.2.3

Rewrite the expression.

$15 - \frac{1}{4} \cdot \frac{24}{1}$

Step 5.10.4.22.2.4

Divide $24$ by $1$ .

$15 - \frac{1}{4} \cdot 24$

Step 5.10.4.23

Multiply $24$ by $- 1$ .

$15 - 24 (\frac{1}{4})$

Step 5.10.4.24

Combine $- 24$ and $\frac{1}{4}$ .

$15 + \frac{- 24}{4}$

Step 5.10.4.25

Cancel the common factor of $- 24$ and $4$ .

Tap for more steps...

Step 5.10.4.25.1

Factor $4$ out of $- 24$ .

$15 + \frac{4 \cdot - 6}{4}$

Step 5.10.4.25.2

Cancel the common factors.

Tap for more steps...

Step 5.10.4.25.2.1

Factor $4$ out of $4$ .

$15 + \frac{4 \cdot - 6}{4 (1)}$

Step 5.10.4.25.2.2

Cancel the common factor.

$15 + \frac{4 \cdot - 6}{4 \cdot 1}$

Step 5.10.4.25.2.3

Rewrite the expression.

$15 + \frac{- 6}{1}$

Step 5.10.4.25.2.4

Divide $- 6$ by $1$ .

$15 - 6$

Step 5.10.4.26

Subtract $6$ from $15$ .

$9$

Step 6