# Trying to show something

• Mar 15th 2006, 05:49 PM
OReilly
Trying to show something
Can someone tell me is this right? Its about rational exponents.

$\displaystyle a^{\frac{m}{n}} < a^{\frac{p}{q}}$ is true only if $\displaystyle \frac{m}{n} < \frac{p}{q}$ for $\displaystyle a>1$

I will show that this is true.

If we divide both sides with $\displaystyle a^{\frac{m}{n}}$ we get
$\displaystyle 1 < \frac{{a^{\frac{p}{q}} }}{{a^{\frac{m}{n}} }}$
which is $\displaystyle 1 < a^{\frac{p}{q} - \frac{m}{n}}$

$\displaystyle a^{\frac{p}{q} - \frac{m}{n}} > 1$ is true only if $\displaystyle \frac{p}{q} - \frac{m}{n} > 0$ so then it must be $\displaystyle \frac{p}{q} > \frac{m}{n}$ so then $\displaystyle a^{\frac{m}{n}} < a^{\frac{p}{q}}$ its true.

Is that ok?
• Apr 15th 2006, 09:17 AM
kit
It's been a while since I did proofs of this kind, but looking at what at you've got - it looks to be ok. Nice logical methodology etc.

One thing to point out though - if this is a written assignment, I would make your explanations clearer (such as a raised to the 0 = 1, hence the exponents have to be greater than 0).

Just a written explanation like this whilst going through the process makes it easier to read, and also to follow through logically.
• Apr 15th 2006, 01:32 PM
OReilly
Quote:

Originally Posted by kit
One thing to point out though - if this is a written assignment, I would make your explanations clearer (such as a raised to the 0 = 1, hence the exponents have to be greater than 0).

You are right, that must be written. My mistake.