Calculus Examples

$\frac{1}{\sqrt[3]{x^{4}}}$

Step 1

Set the denominator in $\frac{1}{\sqrt[3]{x^{4}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x^{4}} = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

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

${\sqrt[3]{x^{4}}}^{3} = 0^{3}$

Step 2.2

Simplify each side of the equation.

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{4}}$ as $x^{\frac{4}{3}}$ .

${(x^{\frac{4}{3}})}^{3} = 0^{3}$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Multiply the exponents in ${(x^{\frac{4}{3}})}^{3}$ .

Tap for more steps...

Step 2.2.2.1.1

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

$x^{\frac{4}{3} \cdot 3} = 0^{3}$

Step 2.2.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.2.1.2.1

Cancel the common factor.

$x^{\frac{4}{3} \cdot 3} = 0^{3}$

Step 2.2.2.1.2.2

Rewrite the expression.

$x^{4} = 0^{3}$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$x^{4} = 0$

Step 2.3

Solve for $x$ .

Tap for more steps...

Step 2.3.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{0}$

Step 2.3.2

Simplify $\pm \sqrt[4]{0}$ .

Tap for more steps...

Step 2.3.2.1

Rewrite $0$ as $0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 2.3.2.2

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

$x = \pm 0$

Step 2.3.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 3

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4