A) \[{{\left( \frac{2}{3} \right)}^{-4}}\]
B) \[\frac{32}{3}\]
C) \[\frac{243}{16}\]
D) \[\frac{243}{32}\]
Correct Answer: D
Solution :
We have, \[\frac{{{\left[ \frac{2}{3} \right]}^{3}}\times {{\left[ \frac{2}{3} \right]}^{-2}}\times {{\left[ {{\left( \frac{1}{2} \right)}^{2}} \right]}^{-2}}\times \frac{1}{24}}{{{\left( \frac{2}{3} \right)}^{-5}}\times {{\left( \frac{3}{2} \right)}^{-12}}}\] \[=\frac{{{\left( \frac{2}{3} \right)}^{3-2}}\times {{\left( \frac{1}{2} \right)}^{-4}}\times \frac{1}{24}}{{{\left( \frac{2}{3} \right)}^{12-5}}}=\frac{\frac{2}{3}\times {{2}^{4}}\times \frac{1}{24}}{{{\left( \frac{2}{3} \right)}^{7}}}\] \[=\frac{{{2}^{5}}}{3}\times \frac{1}{{{2}^{3}}\times 3}\times \frac{{{3}^{7}}}{{{2}^{7}}}=\frac{{{3}^{7-2}}}{{{2}^{7+3-5}}}=\frac{{{3}^{5}}}{{{2}^{5}}}=\frac{243}{32}\]
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