Calculus Examples

$h (x) = {(x - 9)}^{- \frac{1}{6}}$

Step 1

Convert expressions with fractional exponents to radicals.

Tap for more steps...

Step 1.1

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

$\frac{1}{{(x - 9)}^{\frac{1}{6}}}$

Step 1.2

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

$\frac{1}{\sqrt[6]{{(x - 9)}^{1}}}$

Step 1.3

Anything raised to $1$ is the base itself.

$\frac{1}{\sqrt[6]{x - 9}}$

Step 2

Set the radicand in $\sqrt[6]{x - 9}$ greater than or equal to $0$ to find where the expression is defined.

$x - 9 \geq 0$

Step 3

Add $9$ to both sides of the inequality.

$x \geq 9$

Step 4

Set the denominator in $\frac{1}{\sqrt[6]{x - 9}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[6]{x - 9} = 0$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $6$ .

${\sqrt[6]{x - 9}}^{6} = 0^{6}$

Step 5.2

Simplify each side of the equation.

Tap for more steps...

Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x - 9}$ as ${(x - 9)}^{\frac{1}{6}}$ .

${({(x - 9)}^{\frac{1}{6}})}^{6} = 0^{6}$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Simplify ${({(x - 9)}^{\frac{1}{6}})}^{6}$ .

Tap for more steps...

Step 5.2.2.1.1

Multiply the exponents in ${({(x - 9)}^{\frac{1}{6}})}^{6}$ .

Tap for more steps...

Step 5.2.2.1.1.1

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

${(x - 9)}^{\frac{1}{6} \cdot 6} = 0^{6}$

Step 5.2.2.1.1.2

Cancel the common factor of $6$ .

Tap for more steps...

Step 5.2.2.1.1.2.1

Cancel the common factor.

${(x - 9)}^{\frac{1}{6} \cdot 6} = 0^{6}$

Step 5.2.2.1.1.2.2

Rewrite the expression.

${(x - 9)}^{1} = 0^{6}$

Step 5.2.2.1.2

Simplify.

$x - 9 = 0^{6}$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

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

$x - 9 = 0$

Step 5.3

Add $9$ to both sides of the equation.

$x = 9$

Step 6

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

Interval Notation:

$(9, \infty)$

Set-Builder Notation:

${x | x > 9}$

Step 7

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

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

Set-Builder Notation:

${y | y \neq 0}$

Step 8

Determine the domain and range.

Domain: $(9, \infty), {x | x > 9}$

Range: $(- \infty, 0) \cup (0, \infty), {y | y \neq 0}$

Step 9