Suppose that we have $ x+(1/x)=1$, can we compute the expression $x^{1389}+(1/x)^{1389}$ from that? and how?
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2Hint: $x^3 = -1$. Pretty sure it's a duplicate question, though. – dxiv Dec 12 '16 at 06:37
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From the first equation, $x = (-1)^{1/3}$. Now, plug this into the second equation and see what you can say about $(-1)^{1/3\cdot1389} + \frac{1}{(-1)^{1/3\cdot1389}}$.
Hint: What is $-1$ raised to an odd power?
Erik M
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Hint: $x+ 1/x=1 \iff x^2-x+1=0 \implies x^3+1=(x+1)(x^2-x+1)=0$.
Therefore $x^3=-1$ so $x^{1389}=(x^3)^{463}=\cdots$
dxiv
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