Finite Math Examples

$\frac{\sqrt{x}}{\sqrt{x + \sqrt{7}}}$

Step 1

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

$x \geq 0$

Step 2

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

$x + \sqrt{7} \geq 0$

Step 3

Subtract $\sqrt{7}$ from both sides of the inequality.

$x \geq - \sqrt{7}$

Step 4

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

$\sqrt{x + \sqrt{7}} = 0$

Step 5

Solve for $x$ .

Tap for more steps...

Step 5.1

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

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

Step 5.2

Simplify each side of the equation.

Tap for more steps...

Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + \sqrt{7}}$ as ${(x + \sqrt{7})}^{\frac{1}{2}}$ .

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

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

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

Tap for more steps...

Step 5.2.2.1.1

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

Tap for more steps...

Step 5.2.2.1.1.1

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

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

Step 5.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.2.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.2.1.1.2.2

Rewrite the expression.

${(x + \sqrt{7})}^{1} = 0^{2}$

Step 5.2.2.1.2

Simplify.

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

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

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

$x + \sqrt{7} = 0$

Step 5.3

Subtract $\sqrt{7}$ from both sides of the equation.

$x = - \sqrt{7}$

Step 6

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 7