Question

Rationalise and simplify $\frac{\sqrt{1+4i}+\sqrt{1-4i}}{\sqrt{1+4i}-\sqrt{1-4i}}$.

Answer 1

$\frac{\sqrt{1+4i}+\sqrt{1-4i}}{\sqrt{1+4i}-\sqrt{1-4i}}$

$=\frac{\sqrt{1+4i}+\sqrt{1-4i}}{\sqrt{1+4i}-\sqrt{1-4i}}\times \frac{\sqrt{1+4i}+\sqrt{1-4i}}{\sqrt{1+4i}+\sqrt{1-4i}}$ .......Rationalising denominator

$=\frac{1+4i+1-4i+2\sqrt{1-16i^2}}{1+4i-1+4i}$ ....... $(a+b{)}^2=a^2+2ab+b^2$ and $a^2-b^2=(a-b)(a+b)$

$=\frac{2+2\sqrt{17}}{8i}$ ........ $\because i^2=-1$

$=\frac{1+\sqrt{17}}{4i}$

$=-i\frac{1+\sqrt{17}}{4}$