Simplify (√a + 1)/(a − 1).

Step-by-step solution

1. 1

   Spot the hidden difference of squares. For $a\ge 0$ we have $a=\left(\sqrt a\right)^{2}$ and $1=1^{2}$, so the denominator factors:

   $a-1=\left(\sqrt a\right)^{2}-1^{2}=\left(\sqrt a-1\right)\left(\sqrt a+1\right)$

   This is the key move — without it, nothing in the fraction looks cancellable.

2. 2

   Rewrite the fraction with the factored denominator.

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

3. 3

   Cancel the common factor. Since $a\ge 0$, $\sqrt a+1\ge 1>0$, so the cancellation is always legitimate — no extra exclusion is introduced:

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

4. 4

   Record the domain. The original expression needs $a\ge 0$ (for the square root) and $a\neq 1$ (denominator nonzero); the simplified form carries the same two restrictions, since $\sqrt a-1=0$ exactly when $a=1$.

   $\boxed{\dfrac{1}{\sqrt a-1}\qquad(a\ge 0,\ a\neq 1)}$

5. 5

   Check with a value. At $a=9$: the original is $\dfrac{3+1}{9-1}=\dfrac48=0.5$, and the simplified form gives $\dfrac{1}{3-1}=0.5$ ✓. At $a=0.25$: original $=\dfrac{0.5+1}{0.25-1}=\dfrac{1.5}{-0.75}=-2$, simplified $=\dfrac{1}{0.5-1}=-2$ ✓ — including the negative branch $0\le a<1$.