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, \[\sqrt{8100}=2\times 3\times 3\times 5=90\].