<s>[INST] How to simplify this expression I'm trying to find a way to simplify $\sqrt{5+\sqrt{24}}$. I know that this expression is equivalent to $\sqrt{2}+\sqrt{3}$ because they are both roots of the equation: $x^4-10x^2+1$ (and the decimal equivalents are the same using a calculator). My question is , how does one get from $\sqrt{5+\sqrt{24}}$ to $\sqrt{2}+\sqrt{3}$ algebraically?
 [/INST] If you don't want guess-and-check or hope-you-see-it approaches, suppose that $a+\sqrt{b} = \sqrt{5+\sqrt{24}}$ for some $a,b$. (Note that $a$ could be a square root.. it often works out where one of the two terms is an integer and the other is a root - which is why I wrote it in this way.) Squaring both expressions gives us
$$a^2+b + 2a\sqrt{b} = 5+\sqrt{24}.$$
Thus $a^2 + b = 5$ and $2a\sqrt{b} = \sqrt{24}$ or $4a^2b = 24.$ Thus $a^2 = 5-b$ and substituting into the other expression gives
$$4(5-b)b = 24 \Longrightarrow (5-b)b = 6 \Longrightarrow b^2-5b+6 = 0.$$
You will get two solutions to this. I will let you take it from here.
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