Simplify (a - b)/(a + b + 2 root(ab)) times (root a + root b)/(root a - root b).

Step-by-step solution

1. 1

   Change variables in your head to $\sqrt a$ and $\sqrt b$. Every piece of this expression is built from $u=\sqrt a$ and $v=\sqrt b$. Rewriting it as

   $\frac{u^2-v^2}{u^2+v^2+2uv}\cdot\frac{u+v}{u-v}$

   turns two unfamiliar radical expressions into two of the most standard identities in algebra.

2. 2

   Factor the numerator as a difference of squares.

   $a-b=(\sqrt a)^2-(\sqrt b)^2=(\sqrt a-\sqrt b)(\sqrt a+\sqrt b)$

3. 3

   Recognise the denominator as a perfect square. The pattern $u^2+2uv+v^2$ gives

   $a+b+2\sqrt{ab}=(\sqrt a+\sqrt b)^2$

   The cross term $2\sqrt{ab}=2\sqrt a\sqrt b$ is exactly the $2uv$ that completes the square - spotting this is the crux of the problem.

4. 4

   Substitute both factorisations.

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

5. 5

   Cancel systematically. The factor $\sqrt a-\sqrt b$ appears once on top and once on the bottom; $\sqrt a+\sqrt b$ appears twice on top and twice on the bottom. Everything cancels, leaving

   $1$

   The cancellations are legitimate precisely because $a\neq b$ (so $\sqrt a-\sqrt b\neq0$) and $a,b>0$ (so $\sqrt a+\sqrt b>0$).

6. 6

   Verify with numbers. At $a=4,b=9$: $\tfrac{-5}{4+9+12}\cdot\tfrac{2+3}{2-3}=\tfrac{-5}{25}\cdot(-5)=1$ ✓. At $a=2.5,b=7.3$ a direct decimal evaluation also returns $1.000000$ ✓.