+0  
 
+1
43
1
avatar

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

 

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

 

Thanks in advance ot anyone who helps!

 

I started by getting the equation to a form ax+bx+c=0, but I got stuck there, please help!!!
 

 Sep 12, 2020
 #1
avatar
-1

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.

 Sep 12, 2020

38 Online Users

avatar
avatar