<s>[INST] How does $\frac {1} {a^n}$ compare to $\frac {1} {b^n}$ when $a>b$ and $n>0$? To put it briefly, my question is : suppose $a>b$ and $n>0$, how does $\frac {1} {a^n}$ compare to  $\frac {1} {b^n}$ ? 
I have considered various cases without arriving at finding a general rule. 
In view of deriving  the order relation between $\frac {1}{a^n}$ and $\frac {1}{b^n}$ in each case, I use this rule : let a given ordering relation ( greater than/ less than)  hold between $N$ and $M$, then 


*

*if $N$ and $M$ have the same sign, their ( multiplicative)  inverses have the reverse order 

*if $N$ and $M$ have opposite signs, then, the ( multiplicative ) inverses preserve the order. 
I apply this rule to the $n$th power of $a$ and of $b$, previously ordered in each case. 
My "strategy" was as follows: (1) first determining the order relation of the $n$th powers, and then (1) deriving from this the order relation of the inverses of the $n$th powers. But finally, what I end up with is a mess. 
I managed to find a sort of rule for the $n$th powers, but not for their inverses. The rule for $n$th powers was as follows : 
"In case a> b , and n > 0 , then $n$th-powers conserve the order, that is ,      $a^n > b^n$, except when $n$ is even and  either  (1) $a$ and $b$ are both negative  , or (2) $a$ and $b$ have different signs  and $a$ is smaller than $b$ in absolute value." 
If there a way to find a general rule for the cases distinguished below. 

 [/INST] If $n>0$ is an integer, this is $n\in\mathbb{Z}^+$, and $a>b$ are any two real numbers, then we have even nine different cases:


*

*$0<b<a$;

*$b<0<a$, $|b|<|a|$ and $n$ is even;

*$b<0<a$, $|b|<|a|$ and $n$ is odd;

*$b<0<a$, $|b|=|a|$ and $n$ is even;

*$b<0<a$, $|b|=|a|$ and $n$ is odd;

*$b<0<a$, $|b|>|a|$ and $n$ is even;

*$b<0<a$, $|b|>|a|$ and $n$ is odd;

*$b<a<0$ and $n$ is even;

*$b<a<0$ and $n$ is odd.


What happens then?


*

*$\Rightarrow\ {0<b^n<a^n}\ \Rightarrow\ {\frac{1}{b^n}>\frac{1}{a^n}>0}$.

*$\Rightarrow\ {0<b^n<a^n}\ \Rightarrow\ {\frac{1}{b^n}>\frac{1}{a^n}>0}$.

*$\Rightarrow\ {b^n<0<a^n}\ \Rightarrow\ {\frac{1}{b^n}<0<\frac{1}{a^n}}$.

*$\Rightarrow\ {b^n=a^n>0}\ \Rightarrow\ {\frac{1}{b^n}=\frac{1}{a^n}>0}$.

*$\Rightarrow\ {b^n<0<a^n}\ \Rightarrow\ {\frac{1}{b^n}<0<\frac{1}{a^n}}$.

*$\Rightarrow\ {b^n>a^n>0}\ \Rightarrow\ {0<\frac{1}{b^n}<\frac{1}{a^n}}$.

*$\Rightarrow\ {b^n<0<a^n}\ \Rightarrow\ {\frac{1}{b^n}<0<\frac{1}{a^n}}$.

*$\Rightarrow\ {b^n>a^n>0}\ \Rightarrow\ {0<\frac{1}{b^n}<\frac{1}{a^n}}$.

*$\Rightarrow\ {b^n<a^n<0}\ \Rightarrow\ {0>\frac{1}{b^n}>\frac{1}{a^n}}$.


So be careful because


*

*$n$ even or odd matters only if at least one between $a$ and $b$ is smaller than $0$;

*the signs of $a$ and $b$ matter and, if they are different ($b<0<a$), the absolute values also matter!

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