So a while back I learned that $(a+b)^2 = a^2 + 2ab + b^2$
So you can probably see where that's going, I just want to see what the reverse of that is.
What I've tried is this (spoiler alert, it literally goes nowhere)
$$ \sqrt{a+b} = \sqrt{(\sqrt{a^2} + \sqrt{b^2)}}\\ a + 2\sqrt{ab} + b = (\sqrt{a} + \sqrt{b})^2\\ \sqrt{(\sqrt{a} + \sqrt{b})^2 - 2\sqrt{ab}} = \sqrt{a+b}$$
So yeah, I end up with what I started with. Which is funny yet simultaneously sad... Anyway, I feel like i'm sort of on the right track but I just can't figure it out.
I feel that, because you can work out what $(a + b)^2$ is, in terms of a and b itself, the same should be true for the square root of $a + b$.
You can't do it. If $a$ is much larger than $b$ you can write $$\sqrt {a+b}=\sqrt a\sqrt {1+\frac ba}\approx \left(1+\frac b{2a}\right)\sqrt a$$ which can be useful. This shows how the larger of $a,b$ dominates the square root.