8th Class Mathematics Cubes and Cube Roots

  • 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.

    Answer:

                    (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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