Question

If $\frac{7+\sqrt{5}}{7-\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}=a+\frac{7\sqrt{5}}{11}$b then find value of a and b.

Answer 1

$a+\frac{7\sqrt{5}}{11}b=\frac{7+\sqrt{5}}{7-\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}=\frac{(7+\sqrt{5}{)}^2-(7-\sqrt{5}{)}^2}{(7+\sqrt{5})(7-\sqrt{5})}=\frac{49+5+14\sqrt{5}-49-5+14\sqrt{5}}{49-5}=\frac{28\sqrt{5}}{44}=\frac{7\sqrt{5}}{11}$

$a+\frac{7\sqrt{5}}{11}b=0+\frac{7\sqrt{5}}{11}\left(1\right)$

$⟹a=0,b=1$