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 \(c≥0\)

\[a^b=c⟺a=\sqrt[b]{c}=c^{1/b}\]