How does $\frac {1} {a^n}$ compare to $\frac {1} {b^n}$ when $a>b$ and $n>0$?

Question

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.

Answer 1

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:

1. $0<b<a$;
2. $b<0<a$, $|b|<|a|$ and $n$ is even;
3. $b<0<a$, $|b|<|a|$ and $n$ is odd;
4. $b<0<a$, $|b|=|a|$ and $n$ is even;
5. $b<0<a$, $|b|=|a|$ and $n$ is odd;
6. $b<0<a$, $|b|>|a|$ and $n$ is even;
7. $b<0<a$, $|b|>|a|$ and $n$ is odd;
8. $b<a<0$ and $n$ is even;
9. $b<a<0$ and $n$ is odd.

What happens then?

1. $\Rightarrow\ {0<b^n<a^n}\ \Rightarrow\ {\frac{1}{b^n}>\frac{1}{a^n}>0}$.
2. $\Rightarrow\ {0<b^n<a^n}\ \Rightarrow\ {\frac{1}{b^n}>\frac{1}{a^n}>0}$.
3. $\Rightarrow\ {b^n<0<a^n}\ \Rightarrow\ {\frac{1}{b^n}<0<\frac{1}{a^n}}$.
4. $\Rightarrow\ {b^n=a^n>0}\ \Rightarrow\ {\frac{1}{b^n}=\frac{1}{a^n}>0}$.
5. $\Rightarrow\ {b^n<0<a^n}\ \Rightarrow\ {\frac{1}{b^n}<0<\frac{1}{a^n}}$.
6. $\Rightarrow\ {b^n>a^n>0}\ \Rightarrow\ {0<\frac{1}{b^n}<\frac{1}{a^n}}$.
7. $\Rightarrow\ {b^n<0<a^n}\ \Rightarrow\ {\frac{1}{b^n}<0<\frac{1}{a^n}}$.
8. $\Rightarrow\ {b^n>a^n>0}\ \Rightarrow\ {0<\frac{1}{b^n}<\frac{1}{a^n}}$.
9. $\Rightarrow\ {b^n<a^n<0}\ \Rightarrow\ {0>\frac{1}{b^n}>\frac{1}{a^n}}$.

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!