• # question_answer 5)                 Find the cube root of each of the following numbers by prime factorisation method: (i) 64                                      (ii) 512                                   (iii) 10648                             (iv) 27000 (u) 15625                             (vi) 13824                             (vii) 110592                         (viii) 46656 (ix) 175616                          (x) 91125.

(i) 64

 2 64 2 32 2 16 2 8 2 4 2 2 1
Prime factorisation of 64 is $\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}$ | grouping the factors in triplets $={{2}^{3}}\times {{2}^{3}}={{(2\times 2)}^{3}}={{4}^{3}}$ | by laws of exponents Therefore, $\sqrt[3]{64}=4$. (ii) 512    2 512 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2 2 1
Prime factorisation of 512 is $\underline{2\times 2\times 2}\,\times \,\underline{2\times 2\times 2}\,\times \,\underline{2\times 2\times 2}$ | grouping the factors in triple $={{2}^{3}}\,\times {{2}^{3}}\times {{2}^{3}}={{(2\times 2\times 2)}^{3}}={{8}^{3}}$ | by laws of exponent Therefore, $\sqrt[3]{512}=8$. (iii) 10648  2 10648 2 6324 2 2662 11 1331 11 121 11 11 1
Prime factorisation of 10648 is $\underline{2\times 2\times 2}\times \underline{11\times 11\times 11}$ | grouping the factors in triplets $={{2}^{3}}\times {{11}^{3}}$     | by laws of exponents Therefore, $\sqrt[3]{10648}=2\times 11=22$. (iv) 27000  2 27000 2 13500 2 6750 3 3375 3 1125 3 375 5 125 5 25 5 5 1
Prime factorisation of 27000 is $\underline{2\times 2\times 2}\times \underline{3\times 3\times 3}\times \underline{5\times 5\times 5}$ | grouping the factors in triplets $={{2}^{3}}\times {{3}^{3}}\times {{5}^{3}}$ | by laws of exponents Therefore, $\sqrt[3]{27000}=2\times 3\times 5=30$. (v) 15625    5 15625 5 3125 5 625 5 125 5 25 5 5 1
Prime factorisation of 15625 is $\underline{5\times 5\times 5}\,\times \underline{5\times 5\times 5}$ | grouping the factors in triplets $={{5}^{3}}\times {{5}^{3}}={{(5\times 5)}^{3}}={{25}^{3}}$ | by laws of exponents Therefore, $\sqrt[3]{15625}\,=5\times 5=25$. (vi) 13824  2 13824 2 6912 2 3456 2 1728 2 864 2 432 2 216 2 108 2 54 3 27 3 9 3 3 1
Prime factorisation of 13824 is $\underline{2\times 2\times 2}\times \underline{2\times 2\times 2}\,\times \underline{2\times 2\times 2}$ $\times \underline{3\times 3\times 3}$ | grouping the factors in triplets $={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{3}^{3}}$ $={{(2\times 2\times 2\times 3)}^{3}}={{24}^{3}}$ | by laws of exponents Therefore, $\sqrt[3]{13824}=2\times 2\times 2\times 3=24$. (vii) 110592    2 110592 2 55296 2 27648 2 13824 2 6912 2 3456 2 1728 2 864 2 432 3 27 3 9 3 3 1
Prime factorisation of 46656 is $\,\underline{2\times 2\times 2}\,\times \underline{2\times 2\times 2}\times \underline{2\times 2\times 2}$ $\,\times \underline{2\times 2\times 2}\times \underline{3\times 3\times 3}$ | grouping the factors in triplets $={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{3}^{3}}$ $={{(2\times 2\times 2\times 2\times 3)}^{3}}={{48}^{3}}$ | by laws of exponents Therefore, $\sqrt[3]{110592}=2\times 2\times 2\times 2\times 3=48$. (viii) 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
Prime factorisation of 46656 is $\underline{2\times 2\times 2}\,\times \,\underline{2\times 2\times 2}\times \,\underline{3\times 3\times 3}$$\times \,\underline{3\times 3\times 3}$ |grouping the factors in triplets $={{2}^{3}}\times {{2}^{3}}\times {{3}^{3}}\times {{3}^{3}}$ $={{(2\times 2\times 3\times 3)}^{3}}={{36}^{3}}$ | by laws of exponents                 Therefore, $\sqrt[3]{46656}=2\times 2\times 3\times 3=36$.                 (ix) 175616  2 175616 2 87808 2 43904 2 21952 2 10976 2 5488 2 2744 2 1372 2 686 7 343 7 49 7 7 1
Prime factorisation of 175616 is $\underline{2\times 2\times 2}\,\times \underline{2\times 2\times 2}\,\times \underline{2\times 2\times 2}$$\times \underline{7\times 7\times 7}$ |grouping the factors in triplets $={{2}^{3}}\times {{2}^{3}}\times {{2}^{3}}\times {{7}^{3}}$ $={{(2\times 2\times 2\times 7)}^{3}}={{56}^{3}}$ |by laws of exponents Therefore, $\sqrt[3]{175616}=2\times 2\times 2\times 7=56$. (x) 91125  3 91125 3 30375 3 10125 3 3375 3 1125 3 375 5 125 5 25 5 5 1
Prime factorisation of 91125 is $\underline{3\times 3\times 3}\,\times \underline{3\times 3\times 3}\,\times \underline{5\times 5\times 5}$ |grouping the factors in triplets $={{3}^{3}}\times {{3}^{3}}\times {{5}^{3}}$ $={{(3\times 3\times 5)}^{3}}{{45}^{3}}$ |by laws of exponents Therefore, $\sqrt[3]{91125}\,=3\times 3\times 5=45$.

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