SOLUTION: Z + 1/Z = 1 Z ^64 + 1/ Z^64 = ?

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Question 105128: Z + 1/Z = 1
Z ^64 + 1/ Z^64 = ?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
First off, I'm changing variables because 2 and Z look too much alike and it gets confusing trying to follow.
u+%2B+1%2Fu+=+1
Let's look at %28u%2B1%2Fu%29%5E2
%28u%2B1%2Fu%29%5E2=%28u%2B1%2Fu%29%28u%2B1%2Fu%29
Using the FOIL method
%28u%2B1%2Fu%29%5E2=u%5E2%2Bu%281%2Fu%29%2B%281%2Fu%29u%2B1%2Fu%5E2
%28u%2B1%2Fu%29%5E2=u%5E2%2B2%2B1%2Fu%5E2
But from your original equation, you know that
%28u%2B1%2Fu%29=1
highlight%28%28u%2B1%2Fu%29%29%5E2=u%5E2%2B2%2B1%2Fu%5E2
highlight%281%29%5E2=u%5E2%2B2%2B1%2Fu%5E2
1.u%5E2%2B1%2Fu%5E2=-1
Next, look at %28u%5E2%2B1%2Fu%5E2%29%5E2
%28u%5E2%2B1%2Fu%5E2%29%5E2=%28u%5E2%2B1%2Fu%5E2%29%28u%5E2%2B1%2Fu%5E2%29
Using the FOIL method
%28u%5E2%2B1%2Fu%5E2%29%5E2=u%5E4%2Bu%5E2%281%2Fu%5E2%29%2B%281%2Fu%5E2%29u%5E2%2B1%2Fu%5E4
%28u%5E2%2B1%2Fu%5E2%29%5E2=u%5E4%2B2%2B1%2Fu%5E4
From 1 you know,
1.u%5E2%2B1%2Fu%5E2=-1
highlight%28%28u%5E2%2B1%2Fu%5E2%29%29%5E2=u%5E4%2B2%2B1%2Fu%5E4
highlight%28%28-1%29%29%5E2=u%5E4%2B2%2B1%2Fu%5E4
1=u%5E4%2B2%2B1%2Fu%5E4
2.u%5E4%2B1%2Fu%5E4=-1
You see a pattern here.
Next, look at %28u%5E4%2B1%2Fu%5E4%29%5E2
%28u%5E4%2B1%2Fu%5E4%29%5E2=%28u%5E4%2B1%2Fu%5E4%29%28u%5E4%2B1%2Fu%5E4%29
%28u%5E4%2B1%2Fu%5E4%29%5E2=u%5E8%2Bu%5E4%281%2Fu%5E4%29%2B%281%2Fu%5E4%29u%5E4%2B1%2Fu%5E8
%28u%5E4%2B1%2Fu%5E4%29%5E2=u%5E8%2B2%2B1%2Fu%5E8
From 2 you know,
2.u%5E4%2B1%2Fu%5E4=-1
highlight%28%28u%5E4%2B1%2Fu%5E4%29%29%5E2=u%5E8%2B2%2B1%2Fu%5E8
highlight%28-1%29%5E2=u%5E8%2B2%2B1%2Fu%5E8
1=u%5E8%2B2%2B1%2Fu%5E8
3.u%5E8%2B1%2Fu%5E8=-1
.
.
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Let's fast forward.
I'll bet that you can get to this answer by continuing to follow the steps that we followed up until now to get the answer.
u%5E64%2B1%2Fu%5E64=-1