Is this correct or completely wrong? I've bumped onto the problem described below and I couldn't tell if it is wrong. It looks like it could be correct but it makes no physical sense.
Thank you!
Marcelo!

I begin with
$$\frac{p_2}{p_1} = \left(\frac{T_1}{T_2}\right)^{\frac{n}{1-n}\cdot \frac{-1}{-1}} = \left(\frac{T_1}{T_2}\right)^\frac{-n}{n-1}$$
and since $a^{-1} = \frac{1}{a}$,
$$\frac{p_2}{p_1} = \left(\frac{T_2}{T_1}\right)^{\frac{n}{1-n}}$$
but this does not seem correct.
 A: In the final step, why have you taken -1 out of the denominator of the power as well? at the end it should be 
$$\frac{n}{n-1}$$ not $$\frac{n}{1-n}$$
I don't see the point downvoting things like this so much either. Bit confused why the answers have been downvoted too
A: In Physics, we often write
$P^{1-n}T^n = constant$. That is
$$P_1^{1-n}T_1^n = P_2^{1-n}T_2^n $$
or equivalently
$$P_1T_1^{n/(1-n)} = P_2T_2^{n/(1-n)}$$
or equivalently
$${P_2 \over P_1} = \left( {T_1 \over T_2} \right)^{n/(1-n)}$$ or equivalently
$${P_2 \over P_1} = \left( {T_2 \over T_1} \right)^{n/(n-1)}$$
For the last equivalence and the nub of your question, $$\left( {T_1 \over T_2} \right)^{n/(1-n)} = \left(\left( {T_2 \over T_1} \right)^{-1}\right)^{n/(1-n)} = \left( {T_2 \over T_1} \right)^{-n/(1-n)} = \left( {T_2 \over T_1} \right)^{n/(n-1)}$$
Hopefully that much is now clear.
This is the pressure-temperature relationship for an adiabatic process, where $n$ (also written as $\gamma$) is a physical constant for the system known as the heat capacity ratio. The relation $P^{1-n}T^n = constant$ differs from the $P/T = constant$ law you might be thinking of, because in the latter case, work (positive or negative) must be done to change the system that way.
