<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] 
$$\sqrt{5+\sqrt{24}}=\sqrt { 5+2\sqrt { 6 }  } =\sqrt { 5+2\sqrt { 2 } \sqrt { 3 }  } =\sqrt { { \left( \sqrt { 2 } +\sqrt { 3 }  \right)  }^{ 2 } } =\sqrt { 2 } +\sqrt { 3 } $$

Another way $$ \sqrt { 5+\sqrt { 24 }  } =\sqrt { a } +\sqrt { b } \\ 5+\sqrt { 24 } =a+b+2\sqrt { ab } \\ \\ \sqrt { 24 } =2\sqrt { ab } \Rightarrow \sqrt { 6 } =\sqrt { ab } \\ \begin{cases} a+b=5 \\ \sqrt { 6 } =\sqrt { ab }  \end{cases}$$
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