Find all complex numbers z such that z^4=-4

Note: All solutions should be expressed in the form a+bi, where a and b are real numbers.

Chocolatehere May 9, 2021

#1**0 **

By Hamilton's Theorem, the solutions are z = 4^{1/4}*e^(pi*i/4), 4^{1/4}*e^(pi*i/4 + pi/4), 4^{1/4}*e^(pi*i/4 + 2*pi/4), and 4^{1/4}*e^(pi*i/4 + 3*pi/4). Since 4^{1/4} = sqrt(2) and e^(pi*i/4) = (1 + i)/sqrt(2), the first solution is 1 + i. Then the other roots work out as

4^{1/4}*e^(pi*i/4 + pi/4) = 1 - i,

4^{1/4}*e^(pi*i/4 + 2*pi/4) = -1 - i, and

4^{1/4}*e^(pi*i/4 + 3*pi/4) = -1 + i.

Guest May 9, 2021

#2**+1 **

Hey there, Chocolatehere!

FYI, this question has already been answered here: https://web2.0calc.com/questions/find-all-complex-numbers-z-such-that-z-4-4-note-all

So that's that.

Now have some chocolate. 🍫

Hope this helped! :)

( ﾟдﾟ)つ Bye

TaliaArticula May 10, 2021

#3**+2 **

**Find all complex numbers z such that z^4=-4**

here is another answer: https://web2.0calc.com/questions/please-help_20319#r2

heureka May 10, 2021