<s>[INST] 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?
 [/INST] $$\sqrt { 7+4\sqrt { 3 }  } =\sqrt { 7+2\cdot 2\sqrt { 3 }  } =\sqrt { { \left( \sqrt { 3 }  \right)  }^{ 2 }+{ 2 }^{ 2 }+4\sqrt { 3 }  } =\sqrt { { \left( \sqrt { 3 } +2 \right)  }^{ 2 } } =\sqrt { 3 } +2\\ \\ $$
</s>