+0  
 
0
63
2
avatar+32 

Find all complex numbers   such  that 

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

 May 18, 2021
edited by ch1ck3n  May 18, 2021
 #1
avatar+374 
+1

a calculator can do this fast recommend mathway for this problem, but the answer is positive or negative the fourth root of positive 4 or negative 4.

 May 18, 2021
 #2
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.

 May 18, 2021

21 Online Users

avatar
avatar
avatar
avatar