+0  
 
0
90
2
avatar

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.

 

Thank you so much!! <33

 Dec 12, 2020
 #1
avatar
0

We can write -4 in exponential notation as 4e^(pi*i), so the equation is z^4 = 4e^(pi*i).

 

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.

 Dec 12, 2020
 #2
avatar
0

Thank you! I actually figured the question out after I posted it and I got 

z = 1 + i

z = 1 - i

z = -1 - i

z = -1 + i

 

smiley

 Dec 12, 2020

29 Online Users

avatar
avatar