• 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.

(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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