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 \[729=\underline{3\times 3}\times \underline{3\times 3}\times \underline{3\times 3}\]
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 \[400=\underline{2\times 2}\times \underline{2\times 2}\times \underline{5\times 5}\]. 3 729 3 243 3 81 3 27 3 9 3
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 2 400 2 200 2 100 2 50 5 25 5
\[1764=\underline{2\times 2}\times \underline{3\ \times 3}\times \underline{7\times 7}\] 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 1764 2 882 3 441 3 147 7 49 7
\[4096=\underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\] 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 4096 2 2048 2 1024 2 512 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2
\[7744=\underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{11\times 11}\] 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 7744 2 3872 2 1936 2 968 2 484 2 242 11 121 11
\[9604=\underline{2\times 2}\times \underline{7\times 7}\times \underline{7\times 7}\] 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 2 9604 2 4802 7 2401 7 343 7 49 7
\[5929=\underline{7\times 7}=\underline{11\times 11}\] 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 7 5929 7 847 11 121 11
\[9216\,=\underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{2\times 2}\times \underline{3\times 3}\] 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 2 9216 2 4608 2 2304 2 1152 2 576 2 288 2 144 2 72 2 36 2 18 3 9 3
\[529=\underline{23\times 23}\] 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 23 529 23
\[8100=\underline{2\times 2}\times \underline{3\times 3}\times \underline{3\times 3}\times \underline{5\times 5}\] So, \[\sqrt{8100}=2\times 3\times 3\times 5=90\]. 2 8100 2 4050 3 2025 3 675 3 225 3 75 5 25 5
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