Powers with rational and real exponent

Summary

• For $n∈N$ , $a^{1/n}$ is defined as the (principal) n th root of $a$ .
• For $n∈N$ , $a^{-1/n}$ is defined as $1/a^{1/n}$
• $8^{-1/3}$ is defined as $1/8^{1/3}=1/2$
• For $n,m∈Z$ , $a^{m/n}$ is defined as ${\left( a^{1/n} \right)}^m$
• $8^{2/3}={\left( 8^{1/3} \right)}^2=2^2=4$
• For $a,b∈R$ , $a^b$ is defined:
• $a^b$ is always defined if $a>0$ (then $a^b>0$ )
• $a^b$ is always defined if $b$ is an integer and $a≠0$
• The power rules are not in general valid for powers with rational and real exponents unless all bases are positive.
• For $a,b,c∈R$ with $a≥0, b>0$ and $c≥0$

$$a^b=c⟺a=\sqrt[b]{c}=c^{1/b}$$