Calculus Examples

$f (x) = \frac{3 x - 5}{\sqrt[5]{5 x - 15}}$

Step 1

Set the denominator in $\frac{3 x - 5}{\sqrt[5]{5 x - 15}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[5]{5 x - 15} = 0$

Step 2

Solve for $x$ .

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Step 2.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{5 x - 15}}^{5} = 0^{5}$

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 $\sqrt[5]{5 x - 15}$ as ${(5 x - 15)}^{\frac{1}{5}}$ .

${({(5 x - 15)}^{\frac{1}{5}})}^{5} = 0^{5}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(5 x - 15)}^{\frac{1}{5}})}^{5}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(5 x - 15)}^{\frac{1}{5}})}^{5}$ .

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Step 2.2.2.1.1.1

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

${(5 x - 15)}^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 2.2.2.1.1.2

Cancel the common factor of $5$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(5 x - 15)}^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(5 x - 15)}^{1} = 0^{5}$

Step 2.2.2.1.2

Simplify.

$5 x - 15 = 0^{5}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

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

$5 x - 15 = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Add $15$ to both sides of the equation.

$5 x = 15$

Step 2.3.2

Divide each term in $5 x = 15$ by $5$ and simplify.

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Step 2.3.2.1

Divide each term in $5 x = 15$ by $5$ .

$\frac{5 x}{5} = \frac{15}{5}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $5$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{15}{5}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{15}{5}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $15$ by $5$ .

$x = 3$

Step 3

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

Interval Notation:

$(- \infty, 3) \cup (3, \infty)$

Set-Builder Notation:

${x | x \neq 3}$

Step 4