Precalculus Examples

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

Step 1

Set the denominator in $\frac{x}{\sqrt[3]{x} - 2}$ equal to $0$ to find where the expression is undefined.

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

Step 2

Solve for $x$ .

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Step 2.1

Add $2$ to both sides of the equation.

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

Step 2.2

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

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

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

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 2.3.2

Simplify the left side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

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

Step 2.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.3.2.1.1.2.1

Cancel the common factor.

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

Step 2.3.2.1.1.2.2

Rewrite the expression.

$x^{1} = 2^{3}$

Step 2.3.2.1.2

Simplify.

$x = 2^{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

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

$x = 8$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 8) \cup (8, \infty)$

Set-Builder Notation:

${x | x \neq 8}$

Step 4