1. ## surds

Express $\sqrt{59-24\sqrt{6}}$ in the form of $p\sqrt{2}+q\sqrt{3}$ , where p and q are integers

Thanks .

2. Originally Posted by thereddevils
Express $\sqrt{59-24\sqrt{6}}$ in the form of $p\sqrt{2}+q\sqrt{3}$ , where p and q are integers

Thanks .
$\sqrt{59-24\sqrt{6}} = p\sqrt{2}+q\sqrt{3}$

Square both sides:

$59 - 24 \sqrt{6} = 2p^2 + 3q^2 + 2pq \sqrt{6}$.

Therefore:

$59 = 2p^2 + 3q^2$ .... (1)

$-24 = 2pq \Rightarrow -12 = pq$ .... (2)

Solve equations (1) and (2) simultaneously.