How Do You Evaluate (1+i)^(1-i) and Describe the Set {1^x} for Real x?

[cummings12332]

Homework Statement

Evaluate {(1+i)^(1-i)} and describe the set{1^x} when x is a real number, distinguish between the cases when x is rational and when x is rational. for now considering the complex number.

2. The attempt at a solution
i don't know how to start with,for firest part i just write it into e^((1-i)log(1+i)) then get the number with e to the power which inculding i , and the secound part , for 1=e^(i2npi) then 1^x is e^(2inxpi) then how to consider the case for rational and irrational here？？？？？

 

[jedishrfu]

start by breaking the factor into two factors (1+i)^1 * (1+i)^(-i)

 

[cummings12332]

start by breaking the factor into two factors (1+i)^1 * (1+i)^(-i)

i can do the first part now, many thanks ,but i don't understand the secound part of the question.should it be if 1=exp(2ikpi) then 1^x= exp(2ikpix) then consider the rational and irrational case on this form. but what's the differences , i have no idea

 

[haruspex]

should it be if 1=exp(2ikpi) then 1^x= exp(2ikpix) then consider the rational and irrational case on this form.

Yes. Think about whether different values for k can produce the same value.

 

[cummings12332]

Yes. Think about whether different values for k can produce the same value.

if rational, then 1^(p/q) is the q complex roots of 1, if x is irrational then 1^x=exp(2ikpix) then k can be chosen infinitely many values then there are infinite points

 

[haruspex]

if rational, then 1^(p/q) is the q complex roots of 1, if x is irrational then 1^x=exp(2ikpix) then k can be chosen infinitely many values then there are infinite points

That's it.