8th Class Mathematics Squares and Square Roots

  • question_answer 15)                 Find the square roots of the following numbers by the Prime Factorization Method:                              (i) 729                                    (ii) 400                                   (iii) 1764                               (iv) 4096          (v) 7744                                (vi) 9604                               (vii) 5929                              (viii) 9216           (ix) 529                                 (x) 8100.         

    Answer:

                    (i) 729                        The prime factorization of 729 is    \[729=3\times 3\times 3\times 3\times 3\times 3\]. By pairing the prime factors, we get \[729=\underline{3\times 3}\times \underline{3\times 3}\times \underline{3\times 3}\]                

    3 729
    3 243
    3 81
    3 27
    3 9
    3
                      So, \[\sqrt{729}\,=3\times 3\times 3=27\] (ii) 400 The prime factorisation of 400 is \[400=2\times 2\times 2\times 2\times 5\times 5\] . By the prime factors, we get \[400=\underline{2\times 2}\times \underline{2\times 2}\times \underline{5\times 5}\].
    2 400
    2 200
    2 100
    2 50
    5 25
    5
    Therefore, \[\sqrt{400}=2\times 2\times 5=20\]. (iii) 1764 The prime factorization of 1764 is \[1764=2\times 2\times 3\ \times 3\times 7\times 7\]. By pairing the prime factors, we get                
    2 1764
    2 882
    3 441
    3 147
    7 49
    7
    \[1764=\underline{2\times 2}\times \underline{3\ \times 3}\times \underline{7\times 7}\]                 So, \[\sqrt{1764}\,=2\times 3\times 7=42\].                 (iv) 4096 The prime factorization of 4096 is \[4096=2\times 2\times 2\times 2\times \] \[2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\].                 By pairing the prime factors, we get                
    2 4096
    2 2048
    2 1024
    2 512
    2 256
    2 128
    2 64
    2 32
    2 16
    2 8
    2 4
    2
    \[4096=\underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\] So, \[\sqrt{4096}\,=2\times 2\times 2\times 2\times 2\times 2=64\] (v) 7744 The prime factorization of 7744 is \[7744=2\times 2\times 2\times 2\times 2\times 2\times 11\times 11\]. By pairing the prime factors, we get                
    2 7744
    2 3872
    2 1936
    2 968
    2 484
    2 242
    11 121
    11
    \[7744=\underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{11\times 11}\] So, \[\sqrt{7144}\,=2\times 2\times 2\times 11=88\]. (vi) 9604 The prime factorization of 9604 is \[9604=2\times 2\times 7\times 7\times 7\times 7\] By pairing the prime factors, we get                
    2 9604
    2 4802
    7 2401
    7 343
    7 49
    7
    \[9604=\underline{2\times 2}\times \underline{7\times 7}\times \underline{7\times 7}\] So, \[\sqrt{9604}=2\times 7\times 7=98\] (vii) 5929 The prime factorization of 5929 is \[5929=7\times 7\times 11\times 11\].                 By pairing the prime factors, we get                
    7 5929
    7 847
    11 121
    11
                    \[5929=\underline{7\times 7}=\underline{11\times 11}\] So, \[\sqrt{5929}=7\times 11=77\]. (viii) 9216 The prime factorization of 9216 is \[9216=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\] By pairing the prime factors, we get
    2 9216
    2 4608
    2 2304
    2 1152
    2 576
    2 288
    2 144
    2 72
    2 36
    2 18
    3 9
    3
    \[9216\,=\underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{3\times 3}\] So, \[\sqrt{9216}\,=2\times 2\times 2\times 2\times 2\times 3=96\]. (ix) 529 The prime factorization of 529 is \[529=23\times 23\]. By pairing the prime factors, we get
    23 529
    23
    \[529=\underline{23\times 23}\] So, \[\sqrt{529}\,=23\] (x) 8100 The prime factorization of 8100 is  \[8100=2\times 2\times 3\times 3\times 3\times 3\times 5\times 5\]. By pairing the prime factors, we get
    2 8100
    2 4050
    3 2025
    3 675
    3 225
    3 75
    5 25
    5
    \[8100=\underline{2\times 2}\times \underline{3\times 3}\times \underline{3\times 3}\times \underline{5\times 5}\] So, \[\sqrt{8100}=2\times 3\times 3\times 5=90\].

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