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?