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

 Jun 19, 2021
 #1
avatar+11835 
+2

\(z^4 = -4. \) Find all complex numbers z.

 

Hello Guest!

 

\(z^4+4=0\\ z=-1-i\\ z=-1+i\\ z=1-i\\ z=1+i\)

laugh  !

 Jun 19, 2021
 #2
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-1

How many more times are we going to see this ****** question ?

 Jun 19, 2021
 #3
avatar+114012 
+2

here is a clip that might help

 

https://www.youtube.com/watch?v=ZwT1ET0j0w8

 Jun 19, 2021
 #4
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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.

 Jun 19, 2021

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