EDB — 20V

[20V] Prerequisites:[20T]↺↻.Let $𝛼>0,𝛼\in ℝ$ be fixed. We know that, for every natural $n\ge 1$, there exists an unique $𝛽>0$ such that ${𝛽}^n=𝛼$, and $𝛽$ is denoted by the notation $\sqrt[n]{𝛼}$. (See e.g. Proposition 2.6.6 Chap. 2 Sec. 6 of the course notes [ 3 ] or Theorem 1.21 in [ 25 ] ). Given $q\in \mathbb{Q}$, we write $q=n/m$ with $n,m\in \mathbb{Z},m\ge 1$, we define

Show that this definition does not depend on the choice of representation $q=n/m$; that

that for $p,q\in \mathbb{Q}$

show that when $𝛼>1$ then $p\mapsto {𝛼}^p$ is strictly monotonic increasing.