question_answer 1)

Identify the greater number, wherever possible, in each of the following? (i) \[{{4}^{3}}\]or \[{{3}^{4}}\]                    (ii) \[{{5}^{3}}\]or \[{{3}^{5}}\]                   (iii) \[{{2}^{8}}\]or \[{{8}^{2}}\]                       (iv) \[{{100}^{2}}\]or \[{{2}^{100}}\]

Answer:

                (i) \[{{\mathbf{4}}^{\mathbf{3}}}\]or \[{{\mathbf{3}}^{\mathbf{4}}}\]                 \[{{4}^{3}}=4\times 4\times 4=64\]                 \[{{3}^{4}}=3\times 3\times 3\times 3=81\] \[\because \]     \[81>64\] \[\therefore \]  \[{{3}^{4}}>{{4}^{3}}\]                 (ii) \[{{\mathbf{5}}^{\mathbf{3}}}\]or \[{{\mathbf{3}}^{\mathbf{5}}}\]                 \[{{5}^{3}}=5\times 5\times 5=125\]                 \[{{3}^{5}}=3\times 3\times 3\times 3\times 3=81\] \[\because \]     \[125>81\] \[\therefore \]  \[{{5}^{3}}>{{3}^{5}}\]                 (iii) \[{{\mathbf{2}}^{\mathbf{8}}}\]or \[{{\mathbf{8}}^{\mathbf{2}}}\]                 \[{{2}^{8}}=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=256\]    \[{{8}^{2}}=8\times 8=64\] \[\because \]     \[256>64\] \[\therefore \]  \[{{2}^{8}}>{{8}^{2}}\].                 (iv) \[\mathbf{10}{{\mathbf{0}}^{\mathbf{2}}}\]or \[{{\mathbf{2}}^{\mathbf{100}}}\]                 \[{{100}^{2}}=100\times 100=10000\]                 \[{{2}^{100}}={{2}^{10}}\times {{2}^{10}}\times {{2}^{10}}\times {{2}^{10}}\times {{2}^{10}}\times {{2}^{10}}\times {{2}^{10}}\times {{2}^{10}}\]\[\times {{2}^{10}}\times {{2}^{10}}\]                 \[=1024\times 1024\times 1024\times 1024\times 1024\times 1024\times 1024\]\[\times 1024\times 1024\times 1024\] \[\therefore \]  \[{{2}^{100}}>{{100}^{2}}\]                 (v) \[{{\mathbf{2}}^{\mathbf{10}}}\]or \[\mathbf{1}{{\mathbf{0}}^{\mathbf{2}}}\]                 \[{{2}^{10}}=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=1024\]                 \[{{10}^{2}}=10\times 10=100\]. \[\because \]     \[1024>100\] \[\therefore \]  \[{{2}^{10}}>{{10}^{2}}.\]