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4.Find the square roots of the following numbers by the prime factorisation Method. (i) 729 (ii) 400 (iii) 1764 (iv) 4096

(v) 7744

(vi) 9604

(vii) 5929

(viii) 9216

(ix) 529

(x) 8100

Answer :

(i) We have prime factors of 729 as:

\( \begin{array}{c|lcr} 3 & 729\\ \hline 3 & 243\\ \hline 3 & 81\\ \hline 3 & 27\\ \hline 3 & 9\\ \hline 3 & 3\\ \hline & 1 \end{array} \)

\(729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^2 \times 3^2 \times 3^2\)

∴\(\sqrt{729}= 3 \times 3 \times 3 = 27\)

(ii) We have prime factors of 400 as

\( \begin{array}{c|lcr} 2 & 400\\ \hline 2 & 200\\ \hline 2 & 100\\ \hline 2 & 50\\ \hline 5 & 25\\ \hline 5 & 5\\ \hline & 1 \end{array} \)

\(400 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 = 2^2 \times 2^2 \times 5^2\)

∴\(\sqrt{400} = 2 \times 2 \times 5 = 20\)

(iii) We have prime factors of 1764 as:

\( \begin{array}{c|lcr} 2 & 1764\\ \hline 2 & 882\\ \hline 3 & 441\\ \hline 3 & 147\\ \hline 7 & 49\\ \hline 7 & 7\\ \hline & 1 \end{array} \)

\(1764 = 2 \times 2 \times 3 \times 3 \times 7 \times 7 = 2^2 \times 3^2 \times 7^2\)

∴\(\sqrt{1764 }= 2 \times 3 \times 7 = 42\)

(iv) We have prime factors of 4096 as:

\( \begin{array}{c|lcr} 2 & 4096\\ \hline 2 & 2048\\ \hline 2 & 1024\\ \hline 2 & 512\\ \hline 2 &256\\ \hline 2 & 128\\ \hline 2 & 64\\ \hline 2 & 32\\ \hline 2 & 16\\ \hline 2 & 8\\ \hline 2 & 4\\ \hline 2 & 2\\ \hline & 1 \end{array} \)

\(4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\)

\(= 2^2 \times 2^2 \times 2^2 \times 2^2 \times 2^2 \times 2^2\)

∴\(\sqrt{4096} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\)

(v)We have prime factors of 7744 as:

\( \begin{array}{c|lcr} 2 & 7744\\ \hline 2 & 3872\\ \hline 2 & 1936\\ \hline 2 & 968\\ \hline 2 & 484\\ \hline 2 & 242\\ \hline 11 & 121\\ \hline 11 & 11\\ \hline & 1\\ \end{array} \)

\(7744 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 \times 11\)

\(= 2^2 \times 2^2 \times 2^2 \times 11^2\)

∴\(\sqrt{7744} = 2 \times 2 \times 2 \times 11 = 88\)

(vi) We have prime factors of 9604 as:

\( \begin{array}{c|lcr} 2 & 9604\\ \hline 2 & 4802\\ \hline 7 & 2401\\ \hline 7 & 343\\ \hline 7 & 49\\ \hline 7 & 7\\ \hline & 1 \end{array} \)

\(9604 = 2 \times 2 \times 7 \times 7 \times 7 \times 7 = 2^2 \times 7^2 \times 7^2\)

∴\(\sqrt{9604} = 2 \times 7 \times 7 = 98\)

(vii) We have prime factors of 5929 as:

\( \begin{array}{c|lcr} 7 & 5929\\ \hline 7 & 847\\ \hline 11 & 121\\ \hline 11 & 11\\ \hline & 1 \end{array} \)

\(5929 = 7 \times 7 \times 11 \times 11 = 7^2 \times 11^2\)

∴\(\sqrt{5929} = 7 \times 11 = 77\)

(viii) We have prime factors of 9216 as:

\( \begin{array}{c|lcr} 2 & 9216\\ \hline 2 & 4608\\ \hline 2 & 2304\\ \hline 2 & 1152\\ \hline 2 & 576\\ \hline 2 & 288\\ \hline 2 & 144\\ \hline 2 & 72\\ \hline 2 & 36\\ \hline 2 & 18\\ \hline 3 & 9\\ \hline 3 & 3\\ \hline & 1\\ \end{array} \)

\(9216 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3\)

\(= 2^2 \times 2^2 \times 2^2 \times 2^2 \times 2^2 \times 3^2\)

∴\(\sqrt{9216 }= 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 96\)

(ix) We have prime factors of 529 as:

\( \begin{array}{c|lcr} 23 & 529\\ \hline 23 & 23\\ \hline & 1 \end{array} \)

\(529 = 23 \times 23 = 23^2\)

∴\(\sqrt{529} = 23\)

(x)We have prime factors of 8100 as:

\( \begin{array}{c|lcr} 2 & 8100\\ \hline 2 & 4050\\ \hline 3 & 2025\\ \hline 3 & 675\\ \hline 3 & 225\\ \hline 3 & 75\\ \hline 5 & 25\\ \hline 5 & 5\\ \hline & 1\\ \end{array} \)

\(8100 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5 = 2^2 \times 3^2 \times 3^2 \times 5^2\)

∴\(\sqrt{8100} = 2 \times 3 \times 3 \times 5 = 90\)