Question

Find the value of {sqrt(i) + sqrt(-i)} ?

Answer 1

`sqrt(i) + sqrt(-i)` is `sqrt2i` or `-sqrt2i`

Explanation:

Let `sqrt(i)=x`, then `sqrt(-i)=sqrt(i^3)=isqrt(i)=ix`

then `sqrt(i) + sqrt(-i)=x(1+i)`

Let `sqrti=a+bi`, then squaring we get

`i=(a^2-b^2)+2abi`

i.e. `a^2-b^2=0` and `2ab=1`

meaning `a^2+b^2=sqrt((a^2-b^2)^2+4a^2b^2)=sqrt1=1`

Hence `a^2=1/2` and `b^2=1/2` i.e. `a=1/sqrt2` and `b=1/sqrt2` or `a=-1/sqrt2` and `b=-1/sqrt2`

and `sqrti=1/sqrt2+1/sqrt2i` or `-1/sqrt2-1/sqrt2i`

and `sqrti+sqrt(-i)=(1/sqrt2+1/sqrt2i)(1+i)`

= `sqrt2i`

or `sqrti+sqrt(-i)=(-1/sqrt2-1/sqrt2i)(1+i)`

= `-sqrt2i`