Precalculus Examples

$\frac{\sqrt{2 x}}{\sqrt{x - 7}}$

Step 1

Set the radicand in $\sqrt{2 x}$ greater than or equal to $0$ to find where the expression is defined.

$2 x \geq 0$

Step 2

Divide each term in $2 x \geq 0$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x \geq 0$ by $2$ .

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \geq \frac{0}{2}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{0}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $2$ .

$x \geq 0$

Step 3

Set the radicand in $\sqrt{x - 7}$ greater than or equal to $0$ to find where the expression is defined.

$x - 7 \geq 0$

Step 4

Add $7$ to both sides of the inequality.

$x \geq 7$

Step 5

Set the denominator in $\frac{\sqrt{2 x}}{\sqrt{x - 7}}$ equal to $0$ to find where the expression is undefined.

$\sqrt{x - 7} = 0$

Step 6

Solve for $x$ .

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Step 6.1

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

${\sqrt{x - 7}}^{2} = 0^{2}$

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 $\sqrt{x - 7}$ as ${(x - 7)}^{\frac{1}{2}}$ .

${({(x - 7)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Simplify ${({(x - 7)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1

Multiply the exponents in ${({(x - 7)}^{\frac{1}{2}})}^{2}$ .

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Step 6.2.2.1.1.1

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

${(x - 7)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.2.1

Cancel the common factor.

${(x - 7)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 6.2.2.1.1.2.2

Rewrite the expression.

${(x - 7)}^{1} = 0^{2}$

Step 6.2.2.1.2

Simplify.

$x - 7 = 0^{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

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

$x - 7 = 0$

Step 6.3

Add $7$ to both sides of the equation.

$x = 7$

Step 7

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

Interval Notation:

$(7, \infty)$

Set-Builder Notation:

${x | x > 7}$

Step 8