Question

Find the complex number w=(-√2 -i√2)^0.5 in exponential form and remain the π in your answers.?

I would really appreciate all steps in details please.Thank you v much.

Answer 1

`omega1=sqrt2e^(5ipi/8)`
and
`omega2=sqrt2e^(ipi/8)`

Explanation:

Given:
`omega=(-sqrt2-isqrt2)^0.5`
`omega=z^0.5`
`z=-sqrt2-sqrt2i`
Re(z)=`-sqrt2`
Im(z)=`-sqrt2`
Amplitude(z)`=sqrt((-sqrt2)^2+-(sqrt2)^2`
`=sqrt(2+2)`
`=sqrt4`
Amplitude(z)`=2`
Argument(z)`=tan^-1((-sqrt2)/(-sqrt2))`
`=tan^-1(1)`
Argument(z)`=pi+pi/4`
Thus,

z=Amplitude(z)e^i(argument(z))
`z=2e^(i(pi+pi/4))`
`z=2e^(i(5pi/4))`

Now,
`omega=z^0.5`
`z^0.5=(2e^(5ipi/4))^0.5`
By the law of indices
`(2e^(ipi/4))^0.5=(2)(e^(ipi/4))^0.5`
`=(2)^0.5(e^(5ipi/4))^0.5`
`=(2)^0.5(e^0.5(5ipi/4))`
`=(2)^0.5(e^(5ipi/8))`
`omega=sqrt2e^(5ipi/8)`