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 .

Printable View

- May 29th 2009, 04:37 AMthereddevilssurds
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 . - May 29th 2009, 04:40 AMmr fantastic
$\displaystyle \sqrt{59-24\sqrt{6}} = p\sqrt{2}+q\sqrt{3}$

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.