A) \[\frac{1}{5}{{\left[ \frac{{{n}^{2}}}{m{{6}^{10}}} \right]}^{1/3}}\]
B) \[\frac{1}{5}{{\left[ \frac{n}{{{m}^{2}}{{6}^{10}}} \right]}^{1/3}}\]
C) \[\frac{1}{5}{{\left[ \frac{{{n}^{2}}}{m{{6}^{-10}}} \right]}^{1/3}}\]
D) \[\frac{1}{5}{{\left[ \frac{{{n}^{2}}}{m{{6}^{10}}} \right]}^{2/3}}\]
Correct Answer: A
Solution :
(a)\[\frac{{{\left( 9{{m}^{2}} \right)}^{\frac{1}{3}}}}{{{6}^{4}}}\times \frac{{{\left( 4{{n}^{2}} \right)}^{\frac{1}{3}}}}{5m}=\frac{{{\left( {{3}^{2}}{{m}^{2}} \right)}^{1/3}}}{{{3}^{4}}\times {{2}^{4}}}\times \frac{{{\left( {{2}^{2}}{{n}^{2}} \right)}^{1/3}}}{5m}\] \[=\frac{{{\cancel{3}}^{\frac{2}{3}}}\times {{\cancel{m}}^{\frac{2}{3}}}\times {{\cancel{2}}^{\frac{2}{3}}}\times {{n}^{\frac{2}{3}}}}{\underset{{{3}^{\frac{10}{3}}}}{\mathop{\cancel{{{3}^{4}}}}}\,\times \underset{{{2}^{\frac{10}{3}}}}{\mathop{\cancel{{{2}^{4}}}}}\,\times 5\times \underset{{{m}^{1/3}}}{\mathop{\cancel{m}}}\,}\] \[=\frac{{{3}^{\left( \frac{2}{3}-4 \right)}}\times {{m}^{2/3-1}}\times {{2}^{\left( \frac{2}{3}-4 \right)}}\times {{n}^{2/3}}}{5}\] \[=\frac{{{6}^{\frac{-10}{3}}}{{m}^{\frac{-1}{3}}}{{n}^{\frac{2}{3}}}}{5}=\frac{1}{5}{{\left[ \frac{{{n}^{2}}}{m{{6}^{10}}} \right]}^{\frac{1}{3}}}\]
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