Question

If $a+8\sqrt{5}b=\frac{8+\sqrt{5}}{8-\sqrt{5}}+\frac{8-\sqrt{5}}{8+\sqrt{5}},$ find the values of a and b.

Answer 1

$\frac{8+\sqrt{5}}{8-\sqrt{5}}+\frac{8-\sqrt{5}}{8+\sqrt{5}}=a+8\sqrt{5}b$

$\Rightarrow \frac{(8+\sqrt{5}{)}^2+(8-\sqrt{5}{)}^2}{(8-\sqrt{5})(8+\sqrt{5})}=a+8\sqrt{5}b$

$\Rightarrow \frac{64+5+16\sqrt{5}+64+5-16\sqrt{5}}{64-5}$

$\Rightarrow \frac{138}{59}=a+8\sqrt{5}b$

So $a=\frac{138}{59}$ & $b=0$.