Rational Power is of Exponential Order Epsilon

Theorem

Let $r = \dfrac p q$ be a rational number, with $p, q \in \Z: q \ne 0, r > 0$.

Then:

$t \mapsto t^r$

is of exponential order $\epsilon$ for any $\epsilon > 0$ arbitrarily small in magnitude.

Proof

Write $t^r = t^{p/q}$, and set $t > 1$.

\(\ds t^{p/q}\)

\(<\)

\(\ds K e^{a t}\)

an Ansatz

\(\ds \impliedby \ \ \)

\(\ds t^p\)

\(<\)

\(\ds \paren {K e^{a t} }^q\)

Rational Power is Strictly Increasing

\(\ds \)

\(=\)

\(\ds K^q e^{q a t}\)

Exponential of Product

\(\ds \)

\(=\)

\(\ds K' e^{a' t}\)

$K^q = K', q a = a'$

Recall from Polynomial is of Exponential Order Epsilon, $t^p < K'e^{a't}$ for any $a' > 0$, arbitrarily small in magnitude.

Therefore the inequality $t^{p/q} < K e^{a t}$ has solutions of the same nature.

$\blacksquare$