Calculus Examples

$y = - 100 x^{- \frac{1}{4}}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

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

$- 100 \frac{1}{x^{\frac{1}{4}}}$

Step 1.2

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

$- 100 \frac{1}{\sqrt[4]{x^{1}}}$

Step 1.3

Anything raised to $1$ is the base itself.

$- 100 \frac{1}{\sqrt[4]{x}}$

Step 2

Set the radicand in $\sqrt[4]{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 3

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

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

Step 4

Solve for $x$ .

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Step 4.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

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

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

${(x^{\frac{1}{4}})}^{4} = 0^{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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Step 4.2.2.1.1.1

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

$x^{\frac{1}{4} \cdot 4} = 0^{4}$

Step 4.2.2.1.1.2

Cancel the common factor of $4$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 0^{4}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{4}$

Step 4.2.2.1.2

Simplify.

$x = 0^{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$x = 0$

Step 5

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

Interval Notation:

$(0, \infty)$

Set-Builder Notation:

${x | x > 0}$

Step 6