How do you simplify this square root of sum: $\sqrt{7+4\sqrt3}$? I came around this expression when solving a problem.
$$\sqrt{7+4\sqrt{3}}$$
WolframAlpha says it equals $2+\sqrt{3}$. We can confirm it like this
$$\left(2+\sqrt{3}\right)^2  \;=\;  4+4\sqrt{3} + 3  \;=\;  7 + 4\sqrt{3}.$$
However, the only way I can think of how to simplify that expression in hand is guessing. Is there a better way of calculating square root of a sum like that one?

Use conjugates; start from $(a\pm b\sqrt{3})^2=7\pm 4\sqrt{3}$. Multiplying these equations gives $a^2-3b^2=\pm1$, whereas averaging them gives $a^2+3b^2=7$. If we want a rational $b$, the $\pm1$ should be $1$ so $b=1\implies a=2$.
