• # question_answer 1)                 Which of the following numbers are not per feet cubes? (i) 216                                    (ii) 128                                   (iii) 1000                               (iv) 100 (v) 46656

By prime factorisation, $216=\underline{2\times 2\times 2}\times \underline{3\times 3\times 3}$                       | grouping the factors in triplets $={{2}^{3}}\ \times {{3}^{3}}$                                  | by laws of exponents $={{(2\times 3)}^{3}}$                                                 | by laws of exponents $={{6}^{3}}$ which is a perfect cube. Therefore, 216 is a perfect cube. (ii) 128    2 128 2 64 2 32 2 16 2 8 2 4 2 2 1
By prime factorisation, $128=\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\times 2$ | grouping the factors in triplets $={{2}^{3}}\times {{2}^{3}}\times 2$ In the above factorisation, 2 remains after grouping the 2's in triplets. Therefore, 128 is not a perfect cube. (iii) 1000  2 1000 2 500 2 250 5 125 5 25 5 5 1
By prime factorisation, $1000=\underline{2\times 2\times 2}\times \underline{5\times 5\times 5}$     | grouping the factors in triplets $={{2}^{3}}\times {{5}^{3}}$                                     | by laws of exponents $={{(2\times 5)}^{3}}={{10}^{3}},$                         | by laws of exponents which is a perfect cube. Therefore, 1000 is a perfect cube. (iv) 100    2 100 2 50 5 25 5 5 1
By prime factorisation $100=\underline{2\times 2}\times \underline{5\times 5}$ In the above factorisation, $2\times 2,\text{ }5\times 5$ remains when try to group the/factors in triplets. Therefore, 100 is not a perfect cube. (v) 46656               2 46656 2 23328 2 11664 2 5832 2 2916 2 1458 3 729 3 243 3 81 3 27 3 9 3 3 1
By prime factorisation, $46656=\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}$$\times \underline{3\times 3\times 3}\times \underline{3\times 3\times 3}$ | grouping the factors in triplets $={{2}^{3}}\times {{2}^{3}}\times {{3}^{3}}\times {{3}^{3}}$ $={{36}^{3}},$ which is a perfect cube. Therefore, 46656 is a perfect cube.