If x = 5 + 3 5 - 3 and y = 5 - 3 5 + 3 then ( x + y ) equals ?

Answer: Option A

$$\begin{array}{rl}& x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\\ & \Rightarrow x=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\times \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}+\sqrt{3}}\\ & \Rightarrow x=\frac{{\left(\sqrt{5}+\sqrt{3}\right)}^2}{2}\\ & \text{Similarly}\\ & y=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}\\ & \Rightarrow y=\frac{{\left(\sqrt{5}-\sqrt{3}\right)}^2}{2}\\ & \text{Now, }x+y\\ & =\frac{5+3+2\sqrt{15}+5+3-2\sqrt{15}}{2}\\ & =\frac{16}{2}\\ & =8\end{array}$$