Given exponential expression \(\displaystyle{10}^{{-{\frac{{{3}}}{{{2}}}}}}\)

By using

\(\displaystyle{a}^{{-{m}}}={\frac{{{1}}}{{{a}^{{m}}}}}\)

\(\displaystyle{a}^{{{m}{n}}}={\left({a}^{{m}}\right)}^{{n}}\)

Thus,

\(\displaystyle{10}^{{-{\frac{{{3}}}{{{2}}}}}}={\frac{{{1}}}{{{10}^{{{\frac{{{3}}}{{{2}}}}}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{\left({10}^{{{\frac{{{1}}}{{{2}}}}}}\right)}^{{3}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{\left(\sqrt{{{10}}}\right)}^{{3}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{\left(\sqrt{{{10}}}\right)}^{{2}}\times\sqrt{{{10}}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{10}\sqrt{{{10}}}}}}\)

By using

\(\displaystyle{a}^{{-{m}}}={\frac{{{1}}}{{{a}^{{m}}}}}\)

\(\displaystyle{a}^{{{m}{n}}}={\left({a}^{{m}}\right)}^{{n}}\)

Thus,

\(\displaystyle{10}^{{-{\frac{{{3}}}{{{2}}}}}}={\frac{{{1}}}{{{10}^{{{\frac{{{3}}}{{{2}}}}}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{\left({10}^{{{\frac{{{1}}}{{{2}}}}}}\right)}^{{3}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{\left(\sqrt{{{10}}}\right)}^{{3}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{\left(\sqrt{{{10}}}\right)}^{{2}}\times\sqrt{{{10}}}}}}\)

\(\displaystyle={\frac{{{1}}}{{{10}\sqrt{{{10}}}}}}\)