| I. \[{{({{3}^{2}})}^{3}}\] |
| II. \[{{({{8}^{4}})}^{\frac{1}{2}}}\] |
| III. \[{{({{3}^{1/3}})}^{9}}\] |
A)
I II III > < =
B)
I II III < = >
C)
I II III < > >
D)
I II III < > >
Correct Answer: D
Solution :
(I) \[{{({{3}^{2}})}^{3}}={{(3)}^{6}},\,{{3}^{{{3}^{3}}}}={{(3)}^{8}}\] \[\because \] \[{{3}^{6}}<{{3}^{6}}\Rightarrow {{\left( {{3}^{2}} \right)}^{3}}<{{3}^{{{2}^{3}}}}\] (II) \[{{\left( {{8}^{4}} \right)}^{\frac{1}{2}}}={{8}^{2}}=64,\,{{2}^{4}}=16\] \[\because \] \[64>16\,\,\,\,\Rightarrow \,\,{{\left( {{8}^{4}} \right)}^{\frac{1}{2}}}>{{2}^{4}}\] (III) \[{{\left( {{3}^{\frac{1}{3}}} \right)}^{9}}={{3}^{3}}=27,\,\,{{\left( {{81}^{\frac{1}{4}}} \right)}^{2}}={{(81)}^{\frac{1}{2}}}={{\left( {{9}^{2}} \right)}^{\frac{1}{2}}}=9\] \[\because \] \[27>9\Rightarrow {{\left( {{3}^{\frac{1}{3}}} \right)}^{9}}>{{\left( {{81}^{\frac{1}{4}}} \right)}^{2}}\]
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