Question

The value of $\sqrt{\text{i}}\text{+}\sqrt{\text{- i}}$ is:

• A. $\sqrt{\text{2}}$
• B. $\pm \frac{1+i}{\sqrt{2}}\pm \frac{1-i}{\sqrt{2}}$
• C. 0
• D. None of these

Answer 1

The correct option is A $\sqrt{\text{2}}$

we have to find $\sqrt{i}+\sqrt{-i}$

$\sqrt{i}=\sqrt{e^{i\pi /2}}=(e^{i\pi /2}{)}^{1/2}=e^{i\pi /4}$

$=\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}$

$=\frac{1+i}{\sqrt{2}}$

and $\sqrt{-i}=\sqrt{e^{-i\pi /2}}=e^{-i\pi /4}=\frac{(1-i)}{\sqrt{2}}$

Now $\sqrt{i}+\sqrt{-i}=\frac{1+i}{\sqrt{2}}+\frac{1-i}{\sqrt{2}}$

$=\frac{2\sqrt{2}}{2}$

$=\sqrt{2}$