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Answer :
(i)
\(\qquad \begin{array}{c|lcr}
&22\\
\hline
2 &\;\; \;\bar{5}\bar{25}\\
&-4\\
\hline
42 &\;\;\; 125\\
&\;\;\; 84\\
\hline
&\;\;\;41\\
\hline
\end{array}
\)
So we got remainder as 41.It represents that the square of 22 is less than 525.So, the next number is 23 and \(23^2 = 529\).
Hence, the number to be added = 529 – 525 = 4
∴Required perfect square = 529
Thus we have, \(\sqrt{529}=23\)
(ii)
\(\qquad \begin{array}{c|lcr}
&41\\
\hline
4 &\;\; \;\bar{17}\bar{50}\\
&-16\\
\hline
81 &\;\;\; 150\\
&\;\;\; 81\\
\hline
&\;\;\;69\\
\hline
\end{array}
\)
So we got remainder as 69.It represents that the square of 41 is less than 1750.So, the next number is 42 and \(42^2 = 1764\).
Hence, the number to be added = 1764– 1750 =14
∴Required perfect square = 1764
Thus we have, \(\sqrt{1764}=42\)
(iii)
\(\qquad \begin{array}{c|lcr}
&15\\
\hline
1 &\;\; \;\bar{2}\bar{52}\\
&-16\\
\hline
25 &\;\;\; 152\\
&\;\;\; 125\\
\hline
&\;\;\;27\\
\hline
\end{array}
\)
So we got remainder as 27.It represents that the square of 15 is less than 252.So, the next number is 16 and \(16^2 = 256\).
Hence, the number to be added = 256 – 252 = 4
∴Required perfect square = 256
Thus we have, \(\sqrt{256}=16\)
(iv)
\(\qquad \begin{array}{c|lcr}
&42\\
\hline
4 &\;\; \;\bar{18}\bar{25}\\
&-16\\
\hline
82 &\;\;\; 225\\
&\;\;\; 164\\
\hline
&\;\;\;61\\
\hline
\end{array}
\)
So we got remainder as 61.It represents that the square of 42 is less than 1825.So, the next number is 43 and \(43^2 = 1849\).
Hence, the number to be added = 1849 – 1825 = 24
∴New number = 1849
Thus we have,\(\sqrt{1849}=43\)
(v)
\(\qquad \begin{array}{c|lcr}
&80\\
\hline
8 &\;\; \;\bar{64}\bar{12}\\
&-64\\
\hline
160 &\;\;\; 12\\
&\;\;\; 0\\
\hline
&\;\;\;12\\
\hline
\end{array}
\)
So we got remainder as 12.It represents that the square of 80 is less than 6412.So, the next number is 81 and \(81^2 = 6561\).
Hence, the number to be added = 6561 – 6412 = 149
∴New number = 6561
Thus, we have \(\sqrt{6561}=81\)