COmplex Number (1 Viewer)

[EvoRevolution]

Without using any series expansions, prove (√3+i)^n + (√3 - i)^n is real.

 

[jet]

You need to retype that equation to make sense. You're square roots are either in the wrong place or missing an argument.

 

[EvoRevolution]

opps soz

 

[lolokay]

just put each in polar (mod/arg) form

 

[jet]

sqrt(3) + i = cos(π/6) + isin(π/6)

Similarly, sqrt(3) - i = cos(-π/6) + isin(-π/6)

Hence (sqrt(3) + i)ⁿ + (3 - i)ⁿ =
cos(nπ/6) + isin(nπ/6) + cos(-nπ/6) + isin(-nπ/6)
=cos(nπ/6) + isin(nπ/6) + cos(nπ/6) - isin(nπ/6)
= 2cos(nπ/6)

Hence the expression is real.

 

[EvoRevolution]

ohh i understand thanks for the help simple lol