If $\frac ab+\frac ba=1$, and if "a" and "b" are not equal to zero then what would be the value of $a^3-b^3$? It would be helpful if the answer is given in the form of steps leading to the value.
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2There are no real $a,b$ satisfying this equation. – vadim123 May 27 '13 at 14:57
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@user426, won't it be $a^3+b^3$? – lab bhattacharjee May 27 '13 at 14:59
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It is $a^3 - b^3$ – user426 May 27 '13 at 15:00
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Is there any identity for $a^3 - b^3$? – user426 May 27 '13 at 15:01
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@user426, yes $(a-b)^3+3ab(a-b)=(a-b)(a^2+ab+b^2)$ – lab bhattacharjee May 27 '13 at 15:01
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1@user426, I meant $$a^3-b^3=(a-b)^3+3ab(a-b)=(a-b)(a^2+b^2+ab)$$ – lab bhattacharjee May 27 '13 at 15:11
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Then from first expression we get $a^2+b^2 = ab$. Will this help in any way to get the answer? – user426 May 27 '13 at 15:19
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let us continue this discussion in chat – lab bhattacharjee May 27 '13 at 15:21
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@vadim123 I don't think so. The optional answer are 1, -1, 0 or 1/2. – user426 May 27 '13 at 15:30
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@labbhattacharjee I tried going in chat room but it is not loading. – user426 May 27 '13 at 15:32
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1If "The optional answer are 1, -1, 0 or 1/2" either $a/b+b/a=-1$ or it's $a^3+b^3$ – lab bhattacharjee May 27 '13 at 15:33
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@labbhattacharjee In case of that change in sign it does gives the answers from the options. Could be some typo error. – user426 May 27 '13 at 15:41
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1If $\frac ab+\frac ba=-1\implies a^2+b^2+ab=0$ and $a^3-b^3=(a-b)(a^2+ab+b^2)$ – lab bhattacharjee May 27 '13 at 15:47
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@labbhattacharjee I was typing up my answer and saw all your comments. I agree with you. Let me know if you want to post an answer, and I'd pull mine down. – Calvin Lin May 27 '13 at 15:48
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1@CalvinLin, no no. It's OK. Just joking. Concurrency issue is obvious here. – lab bhattacharjee May 27 '13 at 15:50
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I think only answers can be accepted. For the time being, you can safely focus on the basic algebraic formulae. – lab bhattacharjee May 27 '13 at 16:17
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@labbhattacharjee Thanks for help. – user426 May 27 '13 at 16:49
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@user426,my pleasure. But why did you un-accept the answer? – lab bhattacharjee May 27 '13 at 16:51
1 Answers
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Expanding on Lab's comments,
Making the assumption that $\frac{a}{b} + \frac{a}{b} = -1 $,
Hint: Clear denominators for the equation. It states that
$$a^2 + b^2 = -ab$$
Hint: $$a^3 -b^3 = (a-b)(a^2 + b^2 + ab) = ??$$
If we want $a^3 + b^3$ instead, we can proceed in a similar manner.
Calvin Lin
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