Expert: David Hemmer - 3/29/2007

Question
I need to substitute z = x + yi into z^5 = 1. Then expand on left side. Then equate the real parts to make an equation in x & y. The express |z| in terms of x and y to obtain an equation for y^2 and x^2. Then combine the two equations involving powers of x only. then get two exact positive solutions for x.

Answer
So do it! You know i^2=-1 so use the binomial theorem.

(x+yi)^5=x^5+5x^4yi-10x^3y^2-10ix^2y^3+5xy^4+iy^5

Thus you know the real and imaginary parts give:

x=x^5-10x^3y^2+5xy^4
y=5x^4y-10x^2y^3+y^5

Also the absolute value of z must be one so 1=x^2+y^2 to y^2=1-x^2.

Plug this in to the first equation and you will have an equation only in x. Good luck.