Question

How do you simplify `(1/2+sqrt3i)/(1-sqrt2i)`?

Answer 1

`(1/2+sqrt3i)/(1-sqrt2i)=(1/(2sqrt2)-sqrt3)+(1/2+sqrt3/sqrt2)i`

Explanation:

To simpplify ratio of two complex numbers such as `(a+bi)/(c+di)`, just multiply numerator and denominator by the complex conjugate of denominator i.e. `c-di`.

Here, we have `(1/2+sqrt3i)/(1-sqrt2i)` and hence we should multipky numerator and denominator by `1+sqrt2i`

`:.(1/2+sqrt3i)/(1-sqrt2i)=((1/2+sqrt3i)(1+sqrt2i))/((1-sqrt2i)(1+sqrt2i))`

= `(1/2+(sqrt2i)/2+sqrt3i+sqrt6i^2)/(1^2-(sqrt2i)^2)`

= `(1/2+(sqrt2i)/2+sqrt3i+sqrt6(-1))/(1-2i^2)`

= `(1/2-sqrt6+(1/sqrt2i+sqrt3i))/(1-2(-1))`

= `(1-2sqrt6+(sqrt2+2sqrt3)i)/(2sqrt2)`

= `(1-2sqrt6)/(2sqrt2)+(sqrt2+2sqrt3)i/(2sqrt2)`

= `(1/(2sqrt2)-sqrt3)+(1/2+sqrt3/sqrt2)i`