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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 multiplied to get a perfect square number is 7.
The new square number is \(252 \times 7 = 1764\)
∴Square root of 1764 is
\(\sqrt{1764} = 2 \times 3 \times 7 = 42\)
(ii)We have prime factors of180 as:
\( \begin{array}{c|lcr}
2 & 180\\
\hline
2 & 90\\
\hline
5 & 45\\
\hline
3 & 9\\
\hline
3 & 3\\
\hline
& 1\\
\end{array}
\)
\(180 = 2 \times 2 \times 3 \times 3 \times 5\)
Here, 5 has no pair.So the least number to be multiplied to get perfect square is 5
New square number = \(180 \times 5 = 900\)
The square root of 900 is
∴\(\sqrt{900} = 2 \times 3 \times 5 = 30\)
(iii) We have prime factors of 1008 as:
\( \begin{array}{c|lcr}
2 & 1008\\
\hline
2 & 504\\
\hline
2 & 252\\
\hline
2 & 126\\
\hline
3 & 63\\
\hline
3 & 21\\
\hline
7 & 7\\
\hline
& 1\\
\end{array}
\)
\(1008 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7\)
Here, 7 has no pair.So the least number to be multiplied to get perfect square is 7
New square number = \(1008 \times 7 = 7056\)
Square root of 7056 is
∴\(\sqrt{7056}= 2 \times 2 \times 3 \times 7 = 84\)
(iv) We have prime factors of 2028 as:
\( \begin{array}{c|lcr}
2 & 2028\\
\hline
2 & 1014\\
\hline
3& 507\\
\hline
13 & 169\\
\hline
13 & 13\\
\hline
& 1\\
\end{array}
\)
\(2028 = 2 \times 2 \times 3 \times 13 \times 13\)
Here, 3 is not in pair.So the least number to be multiplied to get perfect square is 3
New square number =\(2028 \times 3 = 6084\)
Square root of 6084 is
∴\(\sqrt{6084 }= 2 \times 13 \times 3 = 78\)
(v)We have prime factors of 1458 as:
\( \begin{array}{c|lcr}
2 & 1458\\
\hline
2 & 729\\
\hline
3 & 243\\
\hline
3 & 81\\
\hline
3 & 27\\
\hline
3 & 9\\
\hline
3 & 3\\
\hline
& 1\\
\end{array}
\)
\(1458 = 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3\)
Here, 2 is not in pair.So the least number to be multiplied to get perfect square is 2
New square number =\( 1458 \times 2 = 2916\)
Square root of 2916 is
∴\(\sqrt{2916 }= 3 \times 3 \times 3 \times 2 = 54\)
(vi) We have prime factors of 768 as:
\( \begin{array}{c|lcr}
2 & 768\\
\hline
2 & 384\\
\hline
2 & 192\\
\hline
2 & 96\\
\hline
2 & 48\\
\hline
2 & 24\\
\hline
2 & 12\\
\hline
2 & 6\\
\hline
3 & 3\\
\hline
& 1\\
\end{array}
\)
\(768 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3\)
Here, 3 is not in pair.So the least number to be multiplied to get perfect square is 3
New square number = \(768 \times 3 = 2304\)
Square root of 2304 is
∴\(\sqrt{2304} = 2 \times 2 \times 2 \times 2 \times 3 = 48\)