Express $\displaystyle \sqrt{59-24\sqrt{6}}$ in the form of $\displaystyle p\sqrt{2}+q\sqrt{3}$ , where p and q are integers
Thanks .
$\displaystyle \sqrt{59-24\sqrt{6}} = p\sqrt{2}+q\sqrt{3}$
Square both sides:
$\displaystyle 59 - 24 \sqrt{6} = 2p^2 + 3q^2 + 2pq \sqrt{6}$.
Therefore:
$\displaystyle 59 = 2p^2 + 3q^2$ .... (1)
$\displaystyle -24 = 2pq \Rightarrow -12 = pq$ .... (2)
Solve equations (1) and (2) simultaneously.