Algebra Examples

$f (x) = \sqrt[3]{2 (x + 1)} + 4$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = \sqrt[3]{2 (x + 1)} + 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\sqrt[3]{2 (x + 1)} + 4 = 0$ .

$\sqrt[3]{2 (x + 1)} + 4 = 0$

Step 1.2.2

Subtract $4$ from both sides of the equation.

$\sqrt[3]{2 (x + 1)} = - 4$

Step 1.2.3

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

${\sqrt[3]{2 (x + 1)}}^{3} = {(- 4)}^{3}$

Step 1.2.4

Simplify each side of the equation.

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Simplify ${({(2 (x + 1))}^{\frac{1}{3}})}^{3}$ .

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Step 1.2.4.2.1.1

Multiply the exponents in ${({(2 (x + 1))}^{\frac{1}{3}})}^{3}$ .

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Step 1.2.4.2.1.1.1

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

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

Step 1.2.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.1.2.1

Cancel the common factor.

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

Step 1.2.4.2.1.1.2.2

Rewrite the expression.

${(2 (x + 1))}^{1} = {(- 4)}^{3}$

Step 1.2.4.2.1.2

Apply the distributive property.

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

Step 1.2.4.2.1.3

Simplify.

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Step 1.2.4.2.1.3.1

Multiply $2$ by $1$ .

${(2 x + 2)}^{1} = {(- 4)}^{3}$

Step 1.2.4.2.1.3.2

Simplify.

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

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

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

$2 x + 2 = - 64$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

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

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Step 1.2.5.1.1

Subtract $2$ from both sides of the equation.

$2 x = - 64 - 2$

Step 1.2.5.1.2

Subtract $2$ from $- 64$ .

$2 x = - 66$

Step 1.2.5.2

Divide each term in $2 x = - 66$ by $2$ and simplify.

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Step 1.2.5.2.1

Divide each term in $2 x = - 66$ by $2$ .

$\frac{2 x}{2} = \frac{- 66}{2}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 66}{2}$

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 66}{2}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Divide $- 66$ by $2$ .

$x = - 33$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 33, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = \sqrt[3]{2 ((0) + 1)} + 4$

Step 2.2

Simplify each term.

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Step 2.2.1

Add $0$ and $1$ .

$y = \sqrt[3]{2 \cdot 1} + 4$

Step 2.2.2

Multiply $2$ by $1$ .

$y = \sqrt[3]{2} + 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \sqrt[3]{2} + 4)$

Step 3

List the intersections.

x-intercept(s): $(- 33, 0)$

y-intercept(s): $(0, \sqrt[3]{2} + 4)$

Step 4