Precalculus Examples

$(\sqrt[3]{m^{4}}) (m^{- \frac{1}{2}})$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\sqrt[3]{m^{4}} \frac{1}{m^{\frac{1}{2}}}$

Step 1.2

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

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

Step 1.3

Anything raised to $1$ is the base itself.

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

Step 2

Set the radicand in $\sqrt{m}$ greater than or equal to $0$ to find where the expression is defined.

$m \geq 0$

Step 3

Set the denominator in $\frac{1}{\sqrt{m}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{m} = 0$

Step 4

Solve for $m$ .

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Step 4.1

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

${\sqrt{m}}^{2} = 0^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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Step 4.2.2.1.1.1

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

$m^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

$m^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

$m^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$m = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$m = 0$

Step 5

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${m | m > 0}$

Step 6