\displaystyle \textbf{Question 1: }\text{Find the square of each of the following numbers:}
\displaystyle \text{(i) }14\qquad\text{(ii) }137\qquad\text{(iii) }\frac{4}{17}\qquad\text{(iv) }2\frac{3}{4}

\displaystyle \text{(v) }0.01\qquad\text{(vi) }1.2\qquad\text{(vii) }0.17\qquad\text{(viii) }4.6

\displaystyle \text{Answer:}
\displaystyle \text{(i) Square of }14=14\times14=196.

\displaystyle \text{(ii) Square of }137=137\times137=18769.

\displaystyle \text{(iii) Square of }\frac{4}{17} =\left(\frac{4}{17}\right)^2=\frac{16}{289}.

\displaystyle \text{(iv) Square of }2\frac{3}{4} =\left(\frac{11}{4}\right)^2=\frac{121}{16}.

\displaystyle \text{(v) Square of }0.01=0.01\times0.01=0.0001.

\displaystyle \text{(vi) Square of }1.2=1.2\times1.2=1.44.

\displaystyle \text{(vii) Square of }0.17=0.17\times0.17=0.0289.

\displaystyle \text{(viii) Square of }4.6=4.6\times4.6=21.16.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Using the prime factorization method, find which of the}
\displaystyle \text{following are perfect square numbers:}
\displaystyle \text{(i) }196\qquad\text{(ii) }252\qquad\text{(iii) }324\qquad\text{(iv) }1225

\displaystyle \text{(v) }2916\qquad\text{(vi) }3582\qquad\text{(vii) }4489

\displaystyle \text{Note: A natural number is called a perfect square if it is the square of}
\displaystyle \text{some natural number.}

\displaystyle \text{Answer:}
\displaystyle \text{(i) }196=2\times2\times7\times7=2^2\times7^2.
\displaystyle \text{All the prime factors occur in pairs.}
\displaystyle \therefore 196\text{ is a perfect square.}

\displaystyle \text{(ii) }252=2\times2\times3\times3\times7=2^2\times3^2\times7.
\displaystyle \text{The prime factor }7\text{ does not occur in a pair.}
\displaystyle \therefore 252\text{ is not a perfect square.}

\displaystyle \text{(iii) }324=2\times2\times3\times3\times3\times3=2^2\times3^4.
\displaystyle \text{All the prime factors occur in pairs.}
\displaystyle \therefore 324\text{ is a perfect square.}

\displaystyle \text{(iv) }1225=5\times5\times7\times7=5^2\times7^2.
\displaystyle \text{All the prime factors occur in pairs.}
\displaystyle \therefore 1225\text{ is a perfect square.}

\displaystyle \text{(v) }2916=2\times2\times3\times3\times3\times3\times3\times3
\displaystyle =2^2\times3^6.
\displaystyle \text{All the prime factors occur in pairs.}
\displaystyle \therefore 2916\text{ is a perfect square.}

\displaystyle \text{(vi) }3582=2\times3\times3\times199=2\times3^2\times199.
\displaystyle \text{The prime factors }2\text{ and }199\text{ do not occur in pairs.}
\displaystyle \therefore 3582\text{ is not a perfect square.}

\displaystyle \text{(vii) }4489=67\times67=67^2.
\displaystyle \text{The prime factor }67\text{ occurs in a pair.}
\displaystyle \therefore 4489\text{ is a perfect square.}
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Which of the following numbers are squares of even numbers?}
\displaystyle 676,\qquad1089,\qquad5625,\qquad729,\qquad2304,\qquad9216

\displaystyle \text{Note: The square of an even number is always an even number.}

\displaystyle \text{Answer:}
\displaystyle 676=26^2,\qquad2304=48^2,\qquad9216=96^2.
\displaystyle \therefore 676,\ 2304\text{ and }9216\text{ are squares of even numbers.}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Using the prime factorization method, find the square root}
\displaystyle \text{of each of the following numbers:}
\displaystyle \text{(i) }441\qquad\text{(ii) }784\qquad\text{(iii) }3969
\displaystyle \text{(iv) }4900\qquad\text{(v) }11025\qquad\text{(vi) }30625
\displaystyle \text{Answer:}
\displaystyle \text{(i) }441=3\times3\times7\times7=3^2\times7^2
\displaystyle \therefore \sqrt{441}=3\times7=21.

\displaystyle \text{(ii) }784=2\times2\times2\times2\times7\times7=2^4\times7^2
\displaystyle \therefore \sqrt{784}=2\times2\times7=28.

\displaystyle \text{(iii) }3969=3\times3\times3\times3\times7\times7=3^4\times7^2
\displaystyle \therefore \sqrt{3969}=3\times3\times7=63.

\displaystyle \text{(iv) }4900=2\times2\times5\times5\times7\times7=2^2\times5^2\times7^2
\displaystyle \therefore \sqrt{4900}=2\times5\times7=70.

\displaystyle \text{(v) }11025=3\times3\times5\times5\times7\times7=3^2\times5^2\times7^2
\displaystyle \therefore \sqrt{11025}=3\times5\times7=105.

\displaystyle \text{(vi) }30625=5\times5\times5\times5\times7\times7=5^4\times7^2
\displaystyle \therefore \sqrt{30625}=5\times5\times7=175.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{The students of a class arranged a picnic. Each student}
\displaystyle \text{contributed as many rupees as the number of students in the class. If the}
\displaystyle \text{total contribution is Rs. }2601,\text{ find the strength of the class.}

\displaystyle \text{Answer:}
\displaystyle \text{Let the number of students in the class be }x.
\displaystyle \therefore \text{Each student contributed Rs. }x.
\displaystyle \text{Total contribution}=x\times x=x^2.
\displaystyle \therefore x^2=2601
\displaystyle x=\sqrt{2601}=51.
\displaystyle \therefore \text{The strength of the class is }51\text{ students.}
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Find the smallest number by which }588\text{ should be}
\displaystyle \text{multiplied to get a perfect square number.}

\displaystyle \text{Answer:}
\displaystyle 588=2\times2\times3\times7\times7=2^2\times3\times7^2.
\displaystyle \text{The prime factor }3\text{ does not occur in a pair.}
\displaystyle \therefore \text{The smallest number by which }588\text{ should be multiplied is }3.
\displaystyle 588\times3=1764=42^2,\text{ which is a perfect square.}
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Find the smallest number by which }2400\text{ should be}
\displaystyle \text{multiplied to get a perfect square number. Find the square root of the}
\displaystyle \text{resulting number.}

\displaystyle \text{Answer:}
\displaystyle 2400=2\times2\times2\times2\times2\times3\times5\times5
\displaystyle =2^5\times3\times5^2.
\displaystyle \text{The prime factors }2\text{ and }3\text{ do not occur in pairs.}
\displaystyle \therefore \text{The smallest number by which }2400\text{ should be multiplied is}
\displaystyle 2\times3=6.
\displaystyle 2400\times6=14400.
\displaystyle \sqrt{14400}=120.
\displaystyle \therefore \text{The required number is }6\text{ and the square root is }120.
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Find the smallest number by which }2592\text{ should be}
\displaystyle \text{multiplied to get a perfect square number.}
\displaystyle \text{(i) What is the perfect square number so obtained?}

\displaystyle \text{(ii) What is the square root of the resulting number?}

\displaystyle \text{Answer:}
\displaystyle 2592=2\times2\times2\times2\times2\times3\times3\times3\times3
\displaystyle =2^5\times3^4.
\displaystyle \text{The prime factor }2\text{ occurs an odd number of times.}
\displaystyle \therefore \text{The smallest number by which }2592\text{ should be multiplied is }2.

\displaystyle \text{(i) Perfect square number}=2592\times2=5184.
\displaystyle \therefore \text{The perfect square number obtained is }5184.

\displaystyle \text{(ii) }\sqrt{5184}=72.
\displaystyle \therefore \text{The square root of the resulting number is }72.
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Find the smallest number by which }1728\text{ should be}
\displaystyle \text{divided to get a perfect square number.}
\displaystyle \text{(i) What is the perfect square number so obtained?}

\displaystyle \text{(ii) Find the square root of this number.}

\displaystyle \text{Answer:}
\displaystyle 1728=2\times2\times2\times2\times2\times2\times3\times3\times3
\displaystyle =2^6\times3^3.
\displaystyle \text{The prime factor }3\text{ occurs an odd number of times.}
\displaystyle \therefore \text{The smallest number by which }1728\text{ should be divided is }3.

\displaystyle \text{(i) Perfect square number}=\frac{1728}{3}=576.
\displaystyle \therefore \text{The perfect square number obtained is }576.

\displaystyle \text{(ii) }\sqrt{576}=24.
\displaystyle \therefore \text{The square root of the resulting number is }24.
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Find the smallest number by which }7776\text{ should be}
\displaystyle \text{divided to get a perfect square number.}
\displaystyle \text{(i) What is the resulting number?}

\displaystyle \text{(ii) What is the square root of the number so obtained?}

\displaystyle \text{Answer:}
\displaystyle 7776=2\times2\times2\times2\times2\times3\times3\times3\times3\times3
\displaystyle =2^5\times3^5.
\displaystyle \text{The prime factors }2\text{ and }3\text{ occur an odd number of times.}
\displaystyle \therefore \text{The smallest number by which }7776\text{ should be divided is}
\displaystyle 2\times3=6.

\displaystyle \text{(i) Resulting number}=\frac{7776}{6}=1296.
\displaystyle \therefore \text{The resulting perfect square number is }1296.

\displaystyle \text{(ii) }\sqrt{1296}=36.
\displaystyle \therefore \text{The square root of the resulting number is }36.
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Find the least square number which is exactly divisible}
\displaystyle \text{by each of the numbers }8,\ 9,\ 10\text{ and }15.
\displaystyle \text{Answer:}
\displaystyle 8=2^3,\qquad 9=3^2,\qquad 10=2\times5,\qquad 15=3\times5.
\displaystyle \therefore \text{L.C.M.}=2^3\times3^2\times5=360.
\displaystyle 360=2^3\times3^2\times5.
\displaystyle \text{For a perfect square, the power of each prime factor must be even.}
\displaystyle \text{Therefore, }360\text{ must be multiplied by }2\times5=10.
\displaystyle \therefore \text{Least square number}=360\times10=3600.
\displaystyle 3600=60^2.
\displaystyle \therefore \text{The least square number exactly divisible by }8,\ 9,\ 10\text{ and }15
\displaystyle \text{is }3600.
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{Find the square root of each of the following by the}
\displaystyle \text{division method:}
\displaystyle \text{(i) }961\qquad\text{(ii) }5476\qquad\text{(iii) }11449

\displaystyle \text{(iv) }225625\qquad\text{(v) }4401604\qquad\text{(vi) }9653449

\displaystyle \text{Answer:}
\displaystyle \text{By the long division method, we have:}

\displaystyle \text{(i) }\sqrt{961}
\displaystyle \text{Pair the digits as }\overline{9}\ \overline{61}.
\displaystyle 3^2=9,\qquad 9-9=0.
\displaystyle \text{Bring down }61.\text{ Double }3\text{ to get }6.
\displaystyle 61\times1=61.
\displaystyle \therefore \sqrt{961}=31.

\displaystyle \text{(ii) }\sqrt{5476}
\displaystyle \text{Pair the digits as }\overline{54}\ \overline{76}.
\displaystyle 7^2=49,\qquad54-49=5.
\displaystyle \text{Bring down }76\text{ to get }576.\text{ Double }7\text{ to get }14.
\displaystyle 144\times4=576.
\displaystyle \therefore \sqrt{5476}=74.

\displaystyle \text{(iii) }\sqrt{11449}
\displaystyle \text{Pair the digits as }\overline{1}\ \overline{14}\ \overline{49}.
\displaystyle 1^2=1,\qquad1-1=0.
\displaystyle \text{Bring down }14.\text{ Double }1\text{ to get }2.
\displaystyle 20\times0=0,\qquad\text{so the next digit of the quotient is }0.
\displaystyle \text{Bring down }49\text{ to get }1449.\text{ Double }10\text{ to get }20.
\displaystyle 207\times7=1449.
\displaystyle \therefore \sqrt{11449}=107.

\displaystyle \text{(iv) }\sqrt{225625}
\displaystyle \text{Pair the digits as }\overline{22}\ \overline{56}\ \overline{25}.
\displaystyle 4^2=16,\qquad22-16=6.
\displaystyle \text{Bring down }56\text{ to get }656.\text{ Double }4\text{ to get }8.
\displaystyle 87\times7=609,\qquad656-609=47.
\displaystyle \text{Bring down }25\text{ to get }4725.\text{ Double }47\text{ to get }94.
\displaystyle 945\times5=4725.
\displaystyle \therefore \sqrt{225625}=475.

\displaystyle \text{(v) }\sqrt{4401604}
\displaystyle \text{Pair the digits as }\overline{4}\ \overline{40}\ \overline{16}\ \overline{04}.
\displaystyle 2^2=4,\qquad4-4=0.
\displaystyle \text{Bring down }40.\text{ Double }2\text{ to get }4.
\displaystyle 40\times0=0,\qquad\text{so the next digit of the quotient is }0.
\displaystyle \text{Bring down }16\text{ to get }4016.\text{ Double }20\text{ to get }40.
\displaystyle 409\times9=3681,\qquad4016-3681=335.
\displaystyle \text{Bring down }04\text{ to get }33504.\text{ Double }209\text{ to get }418.
\displaystyle 4188\times8=33504.
\displaystyle \therefore \sqrt{4401604}=2098.

\displaystyle \text{(vi) }\sqrt{9653449}
\displaystyle \text{Pair the digits as }\overline{9}\ \overline{65}\ \overline{34}\ \overline{49}.
\displaystyle 3^2=9,\qquad9-9=0.
\displaystyle \text{Bring down }65.\text{ Double }3\text{ to get }6.
\displaystyle 61\times1=61,\qquad65-61=4.
\displaystyle \text{Bring down }34\text{ to get }434.\text{ Double }31\text{ to get }62.
\displaystyle 620\times0=0,\qquad\text{so the next digit of the quotient is }0.
\displaystyle \text{Bring down }49\text{ to get }43449.\text{ Double }310\text{ to get }620.
\displaystyle 6207\times7=43449.
\displaystyle \therefore \sqrt{9653449}=3107.
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{The area of a square field is }77841\text{ sq. metres.}
\displaystyle \text{Find its perimeter.}
\displaystyle \text{Answer:}
\displaystyle \text{Area of the square}=(\text{side})^2=77841\text{ m}^2.
\displaystyle \therefore \text{Side}=\sqrt{77841}=279\text{ m}.
\displaystyle \text{Perimeter}=4\times\text{side}=4\times279=1116\text{ m}.
\displaystyle \therefore \text{The perimeter of the square field is }1116\text{ m}.
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{Find the least number which must be subtracted from }7581
\displaystyle \text{to obtain a perfect square. Find this perfect square and its square root.}

\displaystyle \text{Answer:}
\displaystyle \text{By the long division method, pair the digits as }\overline{75}\ \overline{81}.
\displaystyle 8^2=64,\qquad75-64=11.
\displaystyle \text{Bring down }81\text{ to get }1181.\text{ Double }8\text{ to get }16.
\displaystyle 167\times7=1169,\qquad1181-1169=12.
\displaystyle \therefore \text{The least number to be subtracted is }12.
\displaystyle 7581-12=7569.
\displaystyle 7569=87^2.
\displaystyle \therefore \text{The required perfect square is }7569\text{ and its square root is }87.
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{Find the least number which must be subtracted from }43379
\displaystyle \text{to obtain a perfect square. Find this perfect square and its square root.}

\displaystyle \text{Answer:}
\displaystyle \text{By the long division method, pair the digits as }\overline{4}\ \overline{33}\ \overline{79}.
\displaystyle 2^2=4,\qquad4-4=0.
\displaystyle \text{Bring down }33.\text{ Double }2\text{ to get }4.
\displaystyle 40\times0=0,\qquad\text{so the next digit of the quotient is }0.
\displaystyle \text{Bring down }79\text{ to get }3379.\text{ Double }20\text{ to get }40.
\displaystyle 408\times8=3264,\qquad3379-3264=115.
\displaystyle \therefore \text{The least number to be subtracted is }115.
\displaystyle 43379-115=43264.
\displaystyle 43264=208^2.
\displaystyle \therefore \text{The required perfect square is }43264\text{ and its square root is }208.
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{Find the least number which must be added to }6203\text{ to}
\displaystyle \text{obtain a perfect square. Find the perfect square and its square root.}
\displaystyle \text{Answer:}
\displaystyle \text{By the long division method, pair the digits as }\overline{62}\ \overline{03}.
\displaystyle 7^2=49,\qquad 62-49=13.
\displaystyle \text{Bring down }03\text{ to get }1303.\text{ Double }7\text{ to get }14.
\displaystyle 148\times8=1184,\qquad 1303-1184=119.
\displaystyle \therefore 78^2=6084<6203<6241=79^2.
\displaystyle \text{Therefore, the next perfect square after }6203\text{ is }79^2=6241.
\displaystyle \therefore \text{Least number to be added}=6241-6203=38.
\displaystyle \therefore \text{The required perfect square is }6241\text{ and its square root is }79.
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{Find the least number which must be added to }506900
\displaystyle \text{to make it a perfect square. Find this perfect square and its square root.}

\displaystyle \text{Answer:}
\displaystyle \text{By the long division method, pair the digits as }\overline{50}\ \overline{69}\ \overline{00}.
\displaystyle 7^2=49,\qquad50-49=1.
\displaystyle \text{Bring down }69\text{ to get }169.\text{ Double }7\text{ to get }14.
\displaystyle 141\times1=141,\qquad169-141=28.
\displaystyle \text{Bring down }00\text{ to get }2800.\text{ Double }71\text{ to get }142.
\displaystyle 1421\times1=1421,\qquad2800-1421=1379.
\displaystyle \therefore 711^2=505521<506900<506944=712^2.
\displaystyle \text{The next perfect square after }506900\text{ is }712^2=506944.
\displaystyle \therefore \text{The least number to be added}=506944-506900=44.
\displaystyle \therefore \text{The required perfect square is }506944\text{ and its square root is }712.
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{Find the greatest number of six digits which is a perfect}
\displaystyle \text{square. Find the square root of this number.}
\displaystyle \text{Answer:}
\displaystyle \text{The greatest six-digit number is }999999.
\displaystyle \text{By the long division method, pair the digits as }\overline{99}\ \overline{99}\ \overline{99}.
\displaystyle 9^2=81,\qquad 99-81=18.
\displaystyle \text{Bring down }99\text{ to get }1899.\text{ Double }9\text{ to get }18.
\displaystyle 189\times9=1701,\qquad 1899-1701=198.
\displaystyle \text{Bring down }99\text{ to get }19899.\text{ Double }99\text{ to get }198.
\displaystyle 1989\times9=17901,\qquad 19899-17901=1998.
\displaystyle \therefore 999^2=998001<999999<1000^2=1000000.
\displaystyle \therefore \text{The greatest six-digit perfect square is }998001.
\displaystyle \sqrt{998001}=999.
\displaystyle \therefore \text{The required number is }998001\text{ and its square root is }999.
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{Find the least number of four digits which is a perfect}
\displaystyle \text{square.}

\displaystyle \text{Answer:}
\displaystyle \text{The least four-digit number is }1000.
\displaystyle \text{By the long division method, pair the digits as }\overline{10}\ \overline{00}.
\displaystyle 3^2=9,\qquad10-9=1.
\displaystyle \text{Bring down }00\text{ to get }100.\text{ Double }3\text{ to get }6.
\displaystyle 61\times1=61,\qquad100-61=39.
\displaystyle \therefore 31^2=961<1000<1024=32^2.
\displaystyle \text{The next perfect square after }1000\text{ is }32^2=1024.
\displaystyle \therefore \text{The least four-digit number which is a perfect square is }1024.
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{Find the least number by which }69192\text{ must be}
\displaystyle \text{(i) decreased, (ii) increased, (iii) multiplied and (iv) divided to}
\displaystyle \text{make it a perfect square.}
\displaystyle \text{Answer:}
\displaystyle \text{By the long division method, pair the digits as }\overline{6}\ \overline{91}\ \overline{92}.
\displaystyle 2^2=4,\qquad 6-4=2.
\displaystyle \text{Bring down }91\text{ to get }291.\text{ Double }2\text{ to get }4.
\displaystyle 46\times6=276,\qquad 291-276=15.
\displaystyle \text{Bring down }92\text{ to get }1592.\text{ Double }26\text{ to get }52.
\displaystyle 523\times3=1569,\qquad 1592-1569=23.
\displaystyle \text{(i) }69192-23=69169=263^2.
\displaystyle \therefore \text{The least number by which }69192\text{ must be decreased is }23.

\displaystyle \text{(ii) }263^2<69192<264^2.
\displaystyle 264^2=69696.
\displaystyle 69696-69192=504.
\displaystyle \therefore \text{The least number by which }69192\text{ must be increased is }504.

\displaystyle \text{Now, }69192=2\times2\times2\times3\times3\times31\times31
\displaystyle =2^3\times3^2\times31^2.
\displaystyle \text{(iii) The prime factor }2\text{ has an odd power.}
\displaystyle \text{Therefore, multiply by }2\text{ to make the power of }2\text{ even.}
\displaystyle 69192\times2=2^4\times3^2\times31^2=(2^2\times3\times31)^2
\displaystyle =372^2=138384.
\displaystyle \therefore \text{The least number by which }69192\text{ must be multiplied is }2.

\displaystyle \text{(iv) Dividing by }2\text{ also makes the power of }2\text{ even.}
\displaystyle \frac{69192}{2}=2^2\times3^2\times31^2=(2\times3\times31)^2
\displaystyle =186^2=34596.
\displaystyle \therefore \text{The least number by which }69192\text{ must be divided is }2.
\displaystyle \\

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.