Algebra Examples

$y = \frac{x^{2} + 8 x + 12}{\sqrt[4]{x + 5}}$

Step 1

Set the radicand in $\sqrt[4]{x + 5}$ greater than or equal to $0$ to find where the expression is defined.

$x + 5 \geq 0$

Step 2

Subtract $5$ from both sides of the inequality.

$x \geq - 5$

Step 3

Set the denominator in $\frac{x^{2} + 8 x + 12}{\sqrt[4]{x + 5}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[4]{x + 5} = 0$

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 $4$ .

${\sqrt[4]{x + 5}}^{4} = 0^{4}$

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 $\sqrt[4]{x + 5}$ as ${(x + 5)}^{\frac{1}{4}}$ .

${({(x + 5)}^{\frac{1}{4}})}^{4} = 0^{4}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(x + 5)}^{\frac{1}{4}})}^{4}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(x + 5)}^{\frac{1}{4}})}^{4}$ .

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Step 4.2.2.1.1.1

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

${(x + 5)}^{\frac{1}{4} \cdot 4} = 0^{4}$

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.

${(x + 5)}^{\frac{1}{4} \cdot 4} = 0^{4}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(x + 5)}^{1} = 0^{4}$

Step 4.2.2.1.2

Simplify.

$x + 5 = 0^{4}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

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

$x + 5 = 0$

Step 4.3

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 5

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

Interval Notation:

$(- 5, \infty)$

Set-Builder Notation:

${x | x > - 5}$

Step 6