$\begin{array}{l} \sqrt {5 + 2\sqrt 6 }  = \sqrt a  + \sqrt b  \\ 5 + 2\sqrt 6  = a + 2\sqrt {ab}  + b \\ 5 + 2\sqrt 6  = a + b + 2\sqrt {ab}  \\ \left\{ \begin{array}{l} a + b = 5 \\ ab = 6 \\ \end{array} \right. \\ \left\{ \begin{array}{l} a + b = 5 \\ a = \frac{6}{b} \\ \end{array} \right. \\ \left\{ \begin{array}{l} \frac{6}{b} + b = 5 \\ a = \frac{6}{b} \\ \end{array} \right. \\ \left\{ \begin{array}{l} 6 + b^2  = 5b \\ a = \frac{6}{b} \\ \end{array} \right. \\ \left\{ \begin{array}{l} b^2  - 5b + 6 = 0 \\ a = \frac{6}{b} \\ \end{array} \right. \\ \left\{ \begin{array}{l} \left( {b - 2} \right)\left( {b - 3} \right) = 0 \\ a = \frac{6}{b} \\ \end{array} \right. \\ \left\{ \begin{array}{l} b = 2 \vee b = 3 \\ a = \frac{6}{b} \\ \end{array} \right. \\ \left\{ \begin{array}{l} b = 2 \\ a = 3 \\ \end{array} \right. \vee \left\{ \begin{array}{l} b = 3 \\ a = 2 \\ \end{array} \right. \\ \sqrt {5 + 2\sqrt 6 }  = \sqrt 2  + \sqrt 3  \\ \end{array}$

Met de formule!?

$\begin{array}{l} \sqrt {5 + 2\sqrt 6 }  = a + b\sqrt c  \\ a = \sqrt 2  \\ b = \frac{1}{2}\sqrt 2  \\ c = 6 \\ \sqrt {5 + 2\sqrt 6 }  = \sqrt 2  + \frac{1}{2}\sqrt 2 \sqrt 6  = \sqrt 2  + \sqrt 3  \\ \end{array}$