Explanation:

Breaking \(\displaystyle{9}^{{{\frac{{{3}}}{{{2}}}}}}\) down into its component parts

\(\displaystyle{\frac{{{3}}}{{{2}}}}\) is the same as \(\displaystyle{3}\times{\frac{{{1}}}{{{2}}}}\) which is the same as \(\displaystyle{\frac{{{1}}}{{{2}}}}\times{3}\)

so \(\displaystyle{9}^{{{\frac{{{3}}}{{{2}}}}}}\) is the same as \(9^{\frac{1}{2}\times3}\)

Write this as: \(\displaystyle{\left({9}^{{{\frac{{{1}}}{{{2}}}}}}\right)}^{{3}}\)

Consider the part of \(\displaystyle{9}^{{{\frac{{{1}}}{{{2}}}}}}\)

This is the same as \(\displaystyle\sqrt{{{9}}}={3}\)

So \(\displaystyle{\left({9}^{{{\frac{{{1}}}{{{2}}}}}}\right)}^{{3}}={\left(\sqrt{{{9}}}\right)}^{{3}}\)

\(\displaystyle{3}^{{3}}={27}\)

However, it true to say that:

\(\displaystyle\sqrt{{{9}}}=\pm{3}\)

and \(\displaystyle{\left(\pm{3}\right)}^{{3}}=\pm{27}\)