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Section 1.2 : Rational Exponents
10. Simplify the following expression and write the answer with only positive exponents.
\[{\left( {\frac{{{m^{\frac{1}{2}}}\,{n^{ - \frac{1}{3}}}}}{{{n^{\frac{2}{3}}}\,{m^{ - \frac{7}{4}}}}}} \right)^{ - \frac{1}{6}}}\] Show SolutionThere isn’t really a lot to do here other than to use the exponent properties from the previous section to do the simplification.
\[{\left( {\frac{{{m^{\frac{1}{2}}}\,{n^{ - \frac{1}{3}}}}}{{{n^{\frac{2}{3}}}\,{m^{ - \frac{7}{4}}}}}} \right)^{ - \frac{1}{6}}} = {\left( {\frac{{{m^{\frac{1}{2}}}\,{m^{\frac{7}{4}}}}}{{{n^{\frac{2}{3}}}\,{n^{\frac{1}{3}}}}}} \right)^{ - \,\,\frac{1}{6}}}\, = {\left( {\frac{{{m^{\frac{9}{4}}}}}{{{n^1}}}} \right)^{ - \,\,\frac{1}{6}}} = {\left( {\frac{n}{{{m^{\frac{9}{4}}}}}} \right)^{\frac{1}{6}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{{n^{\frac{1}{6}}}}}{{\,{m^{\frac{3}{8}}}}}}}\]