Exercise
8
[20V] Prerequisites:[20T].Let \(๐ผ{\gt}0,๐ผโโ\) be fixed. We know that, for every natural \(nโฅ 1\), there exists an unique \(๐ฝ{\gt}0\) such that \(๐ฝ^ n=๐ผ\), and \(๐ฝ\) is denoted by the notation \(\sqrt[n]{๐ผ}\). (See e.g. Proposition 2.6.6 Chap.ย 2 Sec.ย 6 of the course notes [ 3 ] or Theorem 1.21 in [ 25 ] ). Given \(qโโ\), we write \(q=n/m\) with \(n,mโโค,mโฅ 1\), we define
\[ ๐ผ^{q}{\stackrel{.}{=}}\sqrt[m]{๐ผ^ n}\quad . \]
Show that this definition does not depend on the choice of representation \(q=n/m\); that
\[ ๐ผ^{q}={\big({\sqrt[m]{๐ผ}}\big)}^ n\quad ; \]
that for \(p,qโโ\)
\[ ๐ผ^{q}๐ผ^ p=๐ผ^{p+q}\quad ,\quad (๐ผ^ p)^ q=๐ผ^{(pq)}\quad ; \]
show that when \(๐ผ {\gt}1\) then \(pโฆ ๐ผ^ p\) is strictly monotonic increasing.