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Find all X such that X⁴=-4

 Apr 12, 2022
 #1
avatar+1384 
+1

Rewrite as follows: 

 

\({x^2}^2 = -4\)


Take the square root of both sides: 

 

\(x^2 = 2i\)

 

Take the square root again: 

 

\(\color{brown}\boxed{x = 1 + i}\)

 Apr 13, 2022
 #2
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0

There are three other cases left

Guest Apr 13, 2022
 #3
avatar+1384 
+1

The remaining 3 are very similar to \(1 + i\). You can find them yourself by taking the inverse(s) of 1 or both terms. 

BuilderBoi  Apr 13, 2022
 #4
avatar+23183 
0

After you find the first root  1 + i

you can picture finding the other roots by making (around the origin)

a 90o rotation, giving you  -1 + i

a 180o rotation, giving you  -1 - i

a 270o rotation, giving you  1 - i

 

A 360o rotation will take you back to your original root.

 

There will be four fourth roots, separated by 90o [ 360o / 4  =  90o ]

 

There will be three third roots, separated by 120o

 

There will be five fifth roots, separated by 72o

 

Etc.

 Apr 13, 2022

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