Question

How do you write the expression `(sqrt(2) - i)^6` in the standard form a + bi?

Answer 1

`-23+i10sqrt(2)~=-23+i*14.142136`

Explanation:

Calling
`C=sqrt(2)-i`
We can express `C` as `|C| /_ phi`
With
`|C|=sqrt((sqrt(2))^2+1^2)=sqrt(2+1)=sqrt(3)`
And, since `-90^@< phi<0`
`phi=-tan^(-1)(1/sqrt(2))=-35.26439^@`
=> `C=sqrt(3)" " /_-35.26439^@`

Then

`C^6=(sqrt(3))^6" "/_6*(-35.26439^@)`
`C^6=27" "/_-211.58634^@`
`C^6=27cos(-211.58634^@)+i*27sin(-211.58634^@)`
`C^6=-23+i14.142136`
If you like, reminding that `sqrt(2)=1.4142136`
`C^6=-23+i10sqrt(2)`