Question

Given that a does not equal b is it ever possible to have square root a + square root b = square root a+b? Someone please help.

Answer 1

Yes, it's possible.

Suppose $\sqrt a+\sqrt b$ and $\sqrt{a+b}$ are equal. Take squares on both sides.

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

$\left(\sqrt a\right)^2 + 2\sqrt a\sqrt b + \left(\sqrt b\right)^2 = a + b$

$a + 2\sqrt a \sqrt + b = a + b$

$2\sqrt a \sqrt b = 0$

This can only be true if either $\sqrt a=0$ or $\sqrt b=0$, or equivalently if $a=0$ or $b=0$.

Now choose $a=0$ and $b$ any other positive number. Then

$\sqrt 0 + \sqrt b = \sqrt b$

and

$\sqrt{0+b} = \sqrt b$

are clearly the same.