Algebra Examples

$f (x) = \frac{1}{2} (\sqrt[3]{24 - 8 x}) + 1$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{1}{2} \cdot (\sqrt[3]{24 - 8 x}) + 1$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $\frac{1}{2} \cdot \sqrt[3]{24 - 8 x} + 1 = 0$ .

$\frac{1}{2} \cdot \sqrt[3]{24 - 8 x} + 1 = 0$

Step 1.2.2

Subtract $1$ from both sides of the equation.

$\frac{1}{2} \cdot \sqrt[3]{24 - 8 x} = - 1$

Step 1.2.3

To remove the radical on the left side of the equation, cube both sides of the equation.

${(\frac{1}{2} \cdot \sqrt[3]{24 - 8 x})}^{3} = {(- 1)}^{3}$

Step 1.2.4

Simplify each side of the equation.

Tap for more steps...

Step 1.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{24 - 8 x}$ as ${(24 - 8 x)}^{\frac{1}{3}}$ .

${(\frac{1}{2} \cdot {(24 - 8 x)}^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 1.2.4.2

Simplify the left side.

Tap for more steps...

Step 1.2.4.2.1

Simplify ${(\frac{1}{2} \cdot {(24 - 8 x)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 1.2.4.2.1.1

Combine fractions.

Tap for more steps...

Step 1.2.4.2.1.1.1

Combine $\frac{1}{2}$ and ${(24 - 8 x)}^{\frac{1}{3}}$ .

${(\frac{{(24 - 8 x)}^{\frac{1}{3}}}{2})}^{3} = {(- 1)}^{3}$

Step 1.2.4.2.1.1.2

Apply the product rule to $\frac{{(24 - 8 x)}^{\frac{1}{3}}}{2}$ .

$\frac{{({(24 - 8 x)}^{\frac{1}{3}})}^{3}}{2^{3}} = {(- 1)}^{3}$

Step 1.2.4.2.1.2

Simplify the numerator.

Tap for more steps...

Step 1.2.4.2.1.2.1

Multiply the exponents in ${({(24 - 8 x)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 1.2.4.2.1.2.1.1

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

$\frac{{(24 - 8 x)}^{\frac{1}{3} \cdot 3}}{2^{3}} = {(- 1)}^{3}$

Step 1.2.4.2.1.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.4.2.1.2.1.2.1

Cancel the common factor.

$\frac{{(24 - 8 x)}^{\frac{1}{3} \cdot 3}}{2^{3}} = {(- 1)}^{3}$

Step 1.2.4.2.1.2.1.2.2

Rewrite the expression.

$\frac{{(24 - 8 x)}^{1}}{2^{3}} = {(- 1)}^{3}$

Step 1.2.4.2.1.2.2

Simplify.

$\frac{24 - 8 x}{2^{3}} = {(- 1)}^{3}$

Step 1.2.4.2.1.2.3

Factor $8$ out of $24 - 8 x$ .

Tap for more steps...

Step 1.2.4.2.1.2.3.1

Factor $8$ out of $24$ .

$\frac{8 (3) - 8 x}{2^{3}} = {(- 1)}^{3}$

Step 1.2.4.2.1.2.3.2

Factor $8$ out of $- 8 x$ .

$\frac{8 (3) + 8 (- x)}{2^{3}} = {(- 1)}^{3}$

Step 1.2.4.2.1.2.3.3

Factor $8$ out of $8 (3) + 8 (- x)$ .

$\frac{8 (3 - x)}{2^{3}} = {(- 1)}^{3}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 1.2.4.2.1.3.1

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

$\frac{8 (3 - x)}{8} = {(- 1)}^{3}$

Step 1.2.4.2.1.3.2

Cancel the common factor of $8$ .

Tap for more steps...

Step 1.2.4.2.1.3.2.1

Cancel the common factor.

$\frac{8 (3 - x)}{8} = {(- 1)}^{3}$

Step 1.2.4.2.1.3.2.2

Divide $3 - x$ by $1$ .

$3 - x = {(- 1)}^{3}$

Step 1.2.4.3

Simplify the right side.

Tap for more steps...

Step 1.2.4.3.1

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

$3 - x = - 1$

Step 1.2.5

Solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 1.2.5.1.1

Subtract $3$ from both sides of the equation.

$- x = - 1 - 3$

Step 1.2.5.1.2

Subtract $3$ from $- 1$ .

$- x = - 4$

Step 1.2.5.2

Divide each term in $- x = - 4$ by $- 1$ and simplify.

Tap for more steps...

Step 1.2.5.2.1

Divide each term in $- x = - 4$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 4}{- 1}$

Step 1.2.5.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 4}{- 1}$

Step 1.2.5.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 1}$

Step 1.2.5.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.3.1

Divide $- 4$ by $- 1$ .

$x = 4$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(4, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{1}{2} \cdot (\sqrt[3]{24 - 8 \cdot 0}) + 1$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Multiply $\frac{1}{2}$ by $\sqrt[3]{24 - 8 \cdot 0}$ .

$y = \frac{1}{2} \cdot \sqrt[3]{24 - 8 \cdot 0} + 1$

Step 2.2.2

Remove parentheses.

$y = \frac{1}{2} \cdot (\sqrt[3]{24 - 8 \cdot 0}) + 1$

Step 2.2.3

Simplify each term.

Tap for more steps...

Step 2.2.3.1

Multiply $- 8$ by $0$ .

$y = \frac{1}{2} \cdot \sqrt[3]{24 + 0} + 1$

Step 2.2.3.2

Add $24$ and $0$ .

$y = \frac{1}{2} \cdot \sqrt[3]{24} + 1$

Step 2.2.3.3

Rewrite $24$ as $2^{3} \cdot 3$ .

Tap for more steps...

Step 2.2.3.3.1

Factor $8$ out of $24$ .

$y = \frac{1}{2} \cdot \sqrt[3]{8 (3)} + 1$

Step 2.2.3.3.2

Rewrite $8$ as $2^{3}$ .

$y = \frac{1}{2} \cdot \sqrt[3]{2^{3} \cdot 3} + 1$

Step 2.2.3.4

Pull terms out from under the radical.

$y = \frac{1}{2} \cdot (2 \sqrt[3]{3}) + 1$

Step 2.2.3.5

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.3.5.1

Factor $2$ out of $2 \sqrt[3]{3}$ .

$y = \frac{1}{2} \cdot (2 (\sqrt[3]{3})) + 1$

Step 2.2.3.5.2

Cancel the common factor.

$y = \frac{1}{2} \cdot (2 \sqrt[3]{3}) + 1$

Step 2.2.3.5.3

Rewrite the expression.

$y = \sqrt[3]{3} + 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \sqrt[3]{3} + 1)$

Step 3

List the intersections.

x-intercept(s): $(4, 0)$

y-intercept(s): $(0, \sqrt[3]{3} + 1)$

Step 4