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Answer :
(i) We have prime factors of 252 as:
\(\qquad \begin{array}{c|lcr}
2 & 252\\
\hline
2 & 126\\
\hline
3 & 63\\
\hline
3 & 21\\
\hline
7 & 7\\
\hline
& 1\\
\end{array}
\)
\(252 = 2 \times 2 \times 3 \times 3 \times 7\)
Here, the prime factors are not in pair as 7 has no pair.
Thus, the smallest whole number by which the given number is to be divided to get a perfect square number is 7.
The new square number is \(252 ÷ 7 = 36\)
∴Square root of 36 is \(\sqrt{36} = 6\)
(ii) We have prime factors of 2925 as:
\(\qquad \begin{array}{c|lcr}
3 & 2925\\
\hline
3 & 975\\
\hline
5 & 325\\
\hline
5 & 65\\
\hline
13 & 13\\
\hline
& 1\\
\end{array}
\)
\(2925 = 3 \times 3 \times 5 \times 5 \times 13\)
Here, 13 has no pair.So, the smallest whole number by which 2925 is divided to get a square number is 13.
New square number =\( 2925 ÷ 13 = 225\)
∴ \(\sqrt{225} = 15\)
(iii)We have prime factors of 396 as:
\(\qquad \begin{array}{c|lcr}
2 & 396\\
\hline
2 & 198\\
\hline
3 & 99\\
\hline
3 & 33\\
\hline
11 & 11\\
\hline
& 1\\
\end{array}
\)
\(396 = 2 \times 2 \times 3 \times 3 \times 11\)
Here 11 is not in pair.So,the required smallest whole number by which 396 is divided to get a square number is 11.
New square number = \(396 ÷ 11 = 36\)
∴ \(\sqrt{36} = 6\)
(iv)We have prime factors of 2645 as:
\(\qquad \begin{array}{c|lcr}
5 & 2645\\
\hline
23 & 529\\
\hline
23 & 23\\
\hline
& 1\\
\end{array}
\)
\(2645 = 5 \times 23 \times 23\)
Here, 5 is not in pair.So, 5 is the required smallest whole number by which 2645 is divided to get a square number
New square number = \(2645 ÷ 5 = 529\)
∴ \(\sqrt{529} = 23\)
(v) We have prime factors of 2800 as:
\(\qquad \begin{array}{c|lcr}
2 & 2800\\
\hline
2 & 1400\\
\hline
2 & 700\\
\hline
2 & 350\\
\hline
5 & 175\\
\hline
5 & 35\\
\hline
7 & 7\\
\hline
& 1\\
\end{array}
\)
\(2800 = 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 7\)
Here, 7 is not in pair.So, 7 is the required smallest number by which 2800 is divided to get a square number.
New square number =\( 2800 ÷ 7 = 400\)
∴ \(\sqrt{400} = 20\)
(vi)We have prime factors of 1620 as:
\(\qquad \begin{array}{c|lcr}
2 & 1620\\
\hline
2 & 810\\
\hline
3 & 405\\
\hline
3 & 135\\
\hline
3 & 45\\
\hline
3 & 15\\
\hline
5 & 5\\
\hline
& 1\\
\end{array}
\)
\(1620 = 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 5\)
Here, 5 is not in pair.So, 5 is the required smallest prime number by which 1620 is divided to get a square number = \(1620 ÷ 5 = 324\)
∴ \(\sqrt{324} = 18\)