I have the to prove the following statements:
$$\forall a,b >0 \space\forall r > 0,r\in\mathbb{Q}:a<b \iff a^r<b^r $$ $$\forall a,0<a<1,\forall r,s\in\mathbb{Q},r<s: a^r>a^s$$
I have tried various things, for example by simply multiplying $a<b$ with various terms on both sides, but I can't seem to find an answer to this. The problem is that I am not allowed to take the derivative or do anything more advanced than using the ordering axioms and exponentiation rules (for example $a^r*a^s = a^{r+s}$)
I know that if $r$ was a natural number, I could prove it by factorizing the term $b^r-a^r$, but I cannot do that for rational numbers. Is there any other way of proving this?