Question

Find the values of a and b:
$\frac{7+\sqrt{5}}{7-\sqrt{5}}-\frac{7-\sqrt{5}}{7+\sqrt{5}}=a+7\sqrt{5}b$

Answer 1

$\Rightarrow \frac{(7+\sqrt{5}{)}^2-(7-\sqrt{5}{)}^2}{(7-\sqrt{5})(7+\sqrt{5})}$ $[\because (a+\sqrt{b}\left)\right(a-\sqrt{b})=a^2-b^2]$

$\Rightarrow \frac{49+14\sqrt{5}+\sqrt{5}-49+14\sqrt{5}-5}{7^2-5}$ $\left[\begin{array}{c}(a+b{)}^2=a^2+2ab+b^2\\ (a-b{)}^2=a^2-2ab+b^2\\ (a-b)(a+b)=a^2-b^2\end{array}\right]$

$\Rightarrow \frac{28\sqrt{5}}{44}$

$\Rightarrow \frac{7}{11}\sqrt{5}=a+7b\sqrt{5}$

$\therefore a=0;b=\frac{1}{11}$.