Simplify ( square root of a+1-2)/( square root of a+1+2)

$\frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} + 2}$

Step 1

Multiply $\frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} + 2}$ by $\frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} - 2}$ .

$\frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} + 2} \cdot \frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} - 2}$

Step 2

Combine fractions.

Multiply $\frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} + 2}$ by $\frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} - 2}$ .

$\frac{(\sqrt{a + 1} - 2) (\sqrt{a + 1} - 2)}{(\sqrt{a + 1} + 2) (\sqrt{a + 1} - 2)}$

Expand the denominator using the FOIL method.

$\frac{(\sqrt{a + 1} - 2) (\sqrt{a + 1} - 2)}{{\sqrt{a + 1}}^{2} + \sqrt{a + 1} \cdot - 2 + 2 \sqrt{a + 1} - 4}$

Simplify.

$\frac{(\sqrt{a + 1} - 2) (\sqrt{a + 1} - 2)}{a - 3}$

Step 3

Simplify the numerator.

Raise $\sqrt{a + 1} - 2$ to the power of $1$ .

$\frac{{(\sqrt{a + 1} - 2)}^{1} (\sqrt{a + 1} - 2)}{a - 3}$

Raise $\sqrt{a + 1} - 2$ to the power of $1$ .

$\frac{{(\sqrt{a + 1} - 2)}^{1} {(\sqrt{a + 1} - 2)}^{1}}{a - 3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(\sqrt{a + 1} - 2)}^{1 + 1}}{a - 3}$

Add $1$ and $1$ .

$\frac{{(\sqrt{a + 1} - 2)}^{2}}{a - 3}$