Question

Simplify:
$\frac{3+\sqrt{5}}{3-\sqrt{5}}+\frac{3-\sqrt{5}}{3+\sqrt{5}}$

Answer 1

Taking $LCM$ of the denominator,

$\frac{\left[\right(\sqrt{5}+3\left)\right(\sqrt{5}+3)+(-\sqrt{5}+3\left)\right(-\sqrt{5}+3\left)\right]}{\left[\right(-\sqrt{5}+3)\times (\sqrt{5}+3\left)\right]}$

$=\frac{\left[\right(5+3\sqrt{5}+3\sqrt{5}+9)+(5+-3\sqrt{5}-3\sqrt{5}+9\left)\right]}{-(\sqrt{5}{)}^2+3^2}$

$=\frac{\left[\right(14+6\sqrt{5})+(14+-6\sqrt{5})}{-5+9}$

$=\frac{14+14}{4}=\frac{28}{4}=7$ .

Therefore, correct answer is $7$.