Precalculus Examples

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

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 $x$ .

$0 = \sqrt[3]{x} + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$\sqrt[3]{x} + 2 = 0$

Step 1.2.2

Subtract $2$ from both sides of the equation.

$\sqrt[3]{x} = - 2$

Step 1.2.3

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

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

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

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

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Step 1.2.4.2.1.1

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

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Step 1.2.4.2.1.1.1

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

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

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.

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

Step 1.2.4.2.1.1.2.2

Rewrite the expression.

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

Step 1.2.4.2.1.2

Simplify.

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

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

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

$x = - 8$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 8, 0)$

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 $y$ .

$y = \sqrt[3]{0} + 2$

Step 2.2

Simplify $\sqrt[3]{0} + 2$ .

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Step 2.2.1

Remove parentheses.

$y = \sqrt[3]{0} + 2$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Rewrite $0$ as $0^{3}$ .

$y = \sqrt[3]{0^{3}} + 2$

Step 2.2.2.2

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

$y = 0 + 2$

Step 2.2.3

Add $0$ and $2$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(- 8, 0)$

y-intercept(s): $(0, 2)$

Step 4