<s>[INST] Confusing question on exponents and algebra. I was going through some sample papers of math, and I found this question which I cant solve:
If $abc=1$, find $1/(1+a+b^{-1})+1/(1+b+c^{-1})+1/(1+c+a^{-1})$.
Please help me with this.... I have spent almost 3 hours on this question...
Thanks for the help.
 [/INST] $$\frac 1{1+a+b^{-1}}=\frac b{b+ab+1}=\frac b{1+b+c^{-1}}$$
The last equation follows because, $ab=1/c$.
In a similar manner,
$$\frac 1{1+c+a^{-1}}=\frac a{a+ac+1}=\frac a{a+b^{-1}+1}=\frac{ab}{1+b+c^{1}}$$
So, putting this altogether gives
$$\begin{align}
& \frac 1{1+a+b^{-1}}+\frac 1{1+b+c^{-1}}+\frac 1{1+c+a^{-1}} \\
= & \frac b{1+b+c^{-1}}+\frac 1{1+b+c^{-1}}+\frac{ab}{1+b+c^{-1}}\\
= & \frac{1+b+c^{-1}}{1+b+c^{-1}}=1.
\end{align}$$
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