Finite Math Examples

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

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

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

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

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

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

${({(3 x^{2})}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

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

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Step 2.2.2.1.1

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

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Step 2.2.2.1.1.1

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

${(3 x^{2})}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 2.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(3 x^{2})}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(3 x^{2})}^{1} = 0^{3}$

Step 2.2.2.1.2

Simplify.

$3 x^{2} = 0^{3}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

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

$3 x^{2} = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Divide each term in $3 x^{2} = 0$ by $3$ and simplify.

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Step 2.3.1.1

Divide each term in $3 x^{2} = 0$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{0}{3}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $3$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{0}{3}$

Step 2.3.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{3}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $3$ .

$x^{2} = 0$

Step 2.3.2

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

$x = \pm \sqrt{0}$

Step 2.3.3

Simplify $\pm \sqrt{0}$ .

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Step 2.3.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 2.3.3.2

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

$x = \pm 0$

Step 2.3.3.3

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