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

 Oct 18, 2020
 #1
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\(z^{4}=-4 \rightarrow z^{2}=2\sqrt{-1} \rightarrow z=\sqrt{2i}\)

What can you do next?

 
 Oct 18, 2020
 #2
avatar+111124 
+1

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

 

It is a power of 4 so there are 4 answers.

first solution   

 

\(z^4=-4\\ z^4=4cis180\\ z=(4cis180)^{1/4}\\ z=\sqrt2(cis(180*\frac{1}{4}))\\ z=\sqrt2(cis(45))\\ z=\sqrt2(1+i)\\ \text{The other solutions will be }\\ z=\sqrt2(-1+i),\quad z=\sqrt2(-1-i)\quad z=\sqrt2(1-i) \)\(\)

 
 Oct 19, 2020
 #3
avatar+111124 
+1

Here is a long video that looks like it could be quite good...

That is if you want some insight.

https://www.youtube.com/watch?v=1Eu66E_MoQ8

 
 Oct 19, 2020

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