Question

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

Answer 1

We have,

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

$\frac{{(7+\sqrt{5})}^2-{(7-\sqrt{5})}^2}{(7+\sqrt{5})(7-\sqrt{5})}=a+b\sqrt{5}$

$a+b\sqrt{5}=\frac{(49+5+14\sqrt{5})-(49+5-14\sqrt{5})}{(49-5)}$

$a+b\sqrt{5}=\frac{28\sqrt{5}}{44}$

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

Therefore,

$a=0,b=\frac{7}{11}$

Hence, this is the answer.