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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:
\( \begin{array}{c|lcr}
2 & 8100\\
\hline
2 & 4050\\
\hline
3 & 2025\\
\hline
3 & 675\\
\hline
3 & 225\\
\hline
3 & 75\\
\hline
5 & 25\\
\hline
5 & 5\\
\hline
& 1\\
\end{array}
\)
\(8100 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5 = 2^2 \times 3^2 \times 3^2 \times 5^2\)
∴\(\sqrt{8100} = 2 \times 3 \times 3 \times 5 = 90\)