Precalculus Examples

$\frac{\sqrt{4 x - 16}}{\sqrt[4]{{(x - 4)}^{3}}}$

Step 1

Set the radicand in $\sqrt{4 x - 16}$ greater than or equal to $0$ to find where the expression is defined.

$4 x - 16 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $16$ to both sides of the inequality.

$4 x \geq 16$

Step 2.2

Divide each term in $4 x \geq 16$ by $4$ and simplify.

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Step 2.2.1

Divide each term in $4 x \geq 16$ by $4$ .

$\frac{4 x}{4} \geq \frac{16}{4}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $4$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \geq \frac{16}{4}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{16}{4}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $16$ by $4$ .

$x \geq 4$

Step 3

Set the radicand in $\sqrt[4]{{(x - 4)}^{3}}$ greater than or equal to $0$ to find where the expression is defined.

${(x - 4)}^{3} \geq 0$

Step 4

Solve for $x$ .

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Step 4.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{{(x - 4)}^{3}} \geq \sqrt[3]{0}$

Step 4.2

Simplify the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Pull terms out from under the radical.

$x - 4 \geq \sqrt[3]{0}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 4.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x - 4 \geq \sqrt[3]{0^{3}}$

Step 4.2.2.1.2

Pull terms out from under the radical.

$x - 4 \geq 0$

Step 4.3

Add $4$ to both sides of the inequality.

$x \geq 4$

Step 5

Set the denominator in $\frac{\sqrt{4 x - 16}}{\sqrt[4]{{(x - 4)}^{3}}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[4]{{(x - 4)}^{3}} = 0$

Step 6

Solve for $x$ .

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Step 6.1

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

${\sqrt[4]{{(x - 4)}^{3}}}^{4} = 0^{4}$

Step 6.2

Simplify each side of the equation.

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Step 6.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{{(x - 4)}^{3}}$ as ${(x - 4)}^{\frac{3}{4}}$ .

${({(x - 4)}^{\frac{3}{4}})}^{4} = 0^{4}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Multiply the exponents in ${({(x - 4)}^{\frac{3}{4}})}^{4}$ .

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Step 6.2.2.1.1

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

${(x - 4)}^{\frac{3}{4} \cdot 4} = 0^{4}$

Step 6.2.2.1.2

Cancel the common factor of $4$ .

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Step 6.2.2.1.2.1

Cancel the common factor.

${(x - 4)}^{\frac{3}{4} \cdot 4} = 0^{4}$

Step 6.2.2.1.2.2

Rewrite the expression.

${(x - 4)}^{3} = 0^{4}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

${(x - 4)}^{3} = 0$

Step 6.3

Solve for $x$ .

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Step 6.3.1

Set the $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 6.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7

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

Interval Notation:

$(4, \infty)$

Set-Builder Notation:

${x | x > 4}$

Step 8