Precalculus Examples

\frac{x}{\sqrt{x - 6}}

$\frac{x}{\sqrt{x - 6}}$

Step 1

Set the radicand in

\sqrt{x - 6}

$\sqrt{x - 6}$ greater than or equal to

0

$0$ to find where the expression is defined.

x - 6 \geq 0

$x - 6 \geq 0$

Step 2

Add

6

$6$ to both sides of the inequality.

x \geq 6

$x \geq 6$

Step 3

Set the denominator in

\frac{x}{\sqrt{x - 6}}

$\frac{x}{\sqrt{x - 6}}$ equal to

0

$0$ to find where the expression is undefined.

\sqrt{x - 6} = 0

$\sqrt{x - 6} = 0$

Step 4

Solve for

x

$x$ .

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Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

{\sqrt{x - 6}}^{2} = 0^{2}

${\sqrt{x - 6}}^{2} = 0^{2}$

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}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{x - 6}

$\sqrt{x - 6}$ as

{(x - 6)}^{\frac{1}{2}}

${(x - 6)}^{\frac{1}{2}}$ .

{({(x - 6)}^{\frac{1}{2}})}^{2} = 0^{2}

${({(x - 6)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify

{({(x - 6)}^{\frac{1}{2}})}^{2}

${({(x - 6)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in

{({(x - 6)}^{\frac{1}{2}})}^{2}

${({(x - 6)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

{(x - 6)}^{\frac{1}{2} \cdot 2} = 0^{2}

${(x - 6)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of

2

$2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(x - 6)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(x - 6)}^{1} = 0^{2}$

${(x - 6)}^{1} = 0^{2}$

${(x - 6)}^{1} = 0^{2}$

Step 4.2.2.1.2

Simplify.

$x - 6 = 0^{2}$

$x - 6 = 0^{2}$

$x - 6 = 0^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raising

$0$ to any positive power yields

$0$ .

$x - 6 = 0$

$x - 6 = 0$

$x - 6 = 0$

Step 4.3

Add

$6$ to both sides of the equation.

$x = 6$

$x = 6$

Step 5

The domain is all values of

$x$ that make the expression defined.

Interval Notation:

$(6, \infty)$

Set-Builder Notation:

${x | x > 6}$

Step 6

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

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