Question 1: Find the square of each of the following numbers:

(i) 14 = 14      (ii) 137      (iii) \frac{4}{17}       (iv) 2 \frac{3}{4}       (v) 0.01       (vi) of 1.2       (vii) 0.17 
(viii) 4.6 

Answer:

(i) Square of 14 = 14 \times 14 = 196

(ii) Square of 137 = 137 \times 137 = 18769

(iii) Square of \frac{4}{17} = \frac{16}{289}

(iv) Square of 2 \frac{3}{4} = \frac{121}{16}

(v) Square 0.01 = 0.01 \times 0.01 = 0.0001

(vi) Square of 1.2 = 1.2 \times 1.2 = 1.44

(vii) Square of 0.17 =0.17 \times 0.17 = 0.0289

(viii) Square of 4.6 = 4.6 \times 4.6 = 21.16

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Question 2: Using prime factorization method, find which of the following are perfect square numbers:

(i) 196      (ii) 252      (iii) 324      (iv) 1225      (v) 2916      (vi) 3582      (vii) 4489

Note: A natural number is called a perfect square, if it is the square of some natural number

Answer:

(i) 196 = 14 \times 14 (hence perfect square)

(ii) 252 = 2 \times 2 \times 7 \times 3 \times 3 (not a perfect square)

(iii) 324 = 18 \times 18 (hence perfect square)

(iv) 1225 = 35 \times 35 (hence a perfect square)

(v) 2916 = 54 \times 54 (hence perfect square)

(vi) 3582 = 2 \times 3 \times 3 \times 199 (not a perfect square)

(vii) 4489 = 67 \times 67 (hence a perfect square)

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Question 3: Which of the following numbers are squares of even numbers?
Note: The Square of an even number is always an even number.

676 ,          1089 ,           5625 ,           729 ,           2304 ,           9216

Answer:

676 (square of 26 ) 2304 (square of 48 ) and 9216 (square of 96 ) are square of even numbers.

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Question 4: Using prime factorization method, find the square root of each of the following numbers:

(i) 441      (ii) 784      (iii) 3969      (vi) 4900     (v) 11025     (vi) 30625

Answer:

(i) 441 = 3 \times 7 \times 3 \times 7 . \therefore , square root of 441 = 3 \times 7 = 21

(ii) 784 = 4 \times 7 \times 4 \times 7 . Therefore, square root of 784 = 4 \times 7 = 28

(iii) 3969 = 7 \times 9 \times 7 \times 9 . Therefore, square root of 3969 = 7 \times 9 = 63

(vi) 4900 = 7 \times 10 \times 7 \times 10 . Therefore, square root of 4900 = 70

(v) 11025 = 3 \times 7 \times 5 \times 3 \times 7 \times 5 . Therefore, square root of 11025 = 3 \times 7 \times 5 = 105

(vi) 30625 = 5 \times 5 \times 7 \times 5 \times 5 \times 7 . Therefore, square root of 30625 = 5 \times 7 \times 5 = 175

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

Answer:

Let the number of students = x

Each student contributed x Rupees.

Therefore x^2 = 2601 or x = 51

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Question 6: Find the smallest number by which 588 be multiplied to get a perfect square number.

Answer:

588 = 2 \times 2 \times 7 \times 3 \times 7 . Therefore multiply by 3 to get a perfect square

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Question 7: Find the smallest number by which 2400 be multiplied to get a perfect square number. Find the square root of the resulting number.

Answer:

2400 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 . Therefore multiply by 6 . The square root would be 120

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Question 8: Find the smallest number by which 2592 be multiplied to get a perfect square number.

Answer:

(i) What is the perfect square number so obtained?

2592 = 2 \times 2 \times 2 \times 2 \times 2 \times 9 \times 9 .

Therefore smallest number to be multiplied to 2592 to get a perfect square is 2 . Perfect square number = 5184

(ii) What is the square root of the resulting number?

Square root of the resulting number is 72

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Question 9: Find the smallest number by which 1728 be divided to get a perfect square number.

Answer:

(i) What is the perfect square number so obtained?

1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 . Hence divide it by 3 . The number would be 576

(ii) Find the square root of this number.

Square root = 24

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Question 10: Find the smallest number by which 7776 be divided to get a perfect square number.

(i) What is the resulting number?    (ii) What is the square root of the number so obtained? 

Answer: 

(i) What is the resulting number?

7776 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 9 \times 9 . Hence divide this by 6 . The number would be 1296

(ii) What is the square root of the number so obtained?

Square root = 36

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Question 11: Find the least square number which is exactly divisible by each of the numbers 8, 9, 10 and 15 .

Answer:

8 = 2 \times 2 \times 2

9 = 3 \times 3

10 = 2 \times 5

15 = 3 \times 5

Therefore the number is 2 \times 2 \times 2 \times 5 \times 3   \times 3 = 360

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Question 12: Find the square root of each of the following by division method:

i) 961     ii) 5476     iii) 11449     iv) 225625     v) 4401604    vi) 9653449

Answer:

By the long division method, we have

i) 961

{3 \hspace{0.5cm}} | \overline{9} \hspace{0.5cm} \overline{61} \hspace{0.5cm} \overline{} \hspace{0.5cm}   ( 31  \\ {\hspace{0.7cm}| } \underline{ 9 \hspace{2.0cm} }  \\ 61 {\hspace{0.28cm}| } {\hspace{0.2cm} } \hspace{0.5cm}  61  \\   {\hspace{0.7cm}| } \underline{ {\hspace{0.2cm} } \hspace{0.5cm}  61 }  \\   {\hspace{0.7cm}| }  {\hspace{0.2cm} } \hspace{0.4cm}  0  \hspace{0.5cm}   

ii) 5476

{7 \hspace{0.7cm}} | \overline{54} \hspace{0.5cm} \overline{76} \hspace{0.5cm} \overline{} \hspace{0.5cm}   ( 74  \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{2.0cm} }  \\ 144 {\hspace{0.28cm}| } {\hspace{0.2cm} 5} \hspace{0.5cm}  76  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.2cm} 5} \hspace{0.5cm}  76  }  \\   {\hspace{0.9cm}| }  {\hspace{0.2cm} } \hspace{0.4cm}  0  \hspace{0.5cm}   

iii) 11449

{1 \hspace{0.7cm}} | \overline{1} \hspace{0.5cm} \overline{14} \hspace{0.5cm} \overline{49} \hspace{0.5cm} \overline{} \hspace{0.5cm}  ( 107  \\ {\hspace{0.9cm}| } \underline{ 1 \hspace{3.0cm} }  \\ 207 {\hspace{0.28cm}| } {\hspace{0.7cm} 14} \hspace{0.5cm}  49  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.7cm} 14} \hspace{0.5cm}  49 \hspace{1.5cm} }  \\   {\hspace{0.9cm}| }  {\hspace{0.7cm} } \hspace{0.7cm}  0  \hspace{0.5cm}   

iv) 225625

{4 \hspace{0.7cm}} | \overline{22} \hspace{0.5cm} \overline{56} \hspace{0.5cm} \overline{25} \hspace{0.5cm} \overline{} \hspace{0.5cm}  ( 475  \\ {\hspace{0.9cm}| } \underline{ 16 \hspace{3.0cm} }  \\ 87 {\hspace{0.48cm}| } {\hspace{0.3cm} 6} \hspace{0.5cm}  56  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 6} \hspace{0.5cm}  09 \hspace{1.5cm} }  \\ 945 {\hspace{0.28cm}| } {\hspace{0.95cm} 47} \hspace{0.5cm}  25 \hspace{0.5cm}  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 47} \hspace{0.5cm}  25  \hspace{0.5cm}  }  \\   {\hspace{0.9cm}| }  {\hspace{0.7cm} } \hspace{0.7cm}  0  \hspace{0.5cm}   

v) 4401604

{4 \hspace{1.0cm}} | \overline{4} \hspace{0.5cm} \overline{40} \hspace{0.5cm} \overline{16} \hspace{0.5cm} \overline{04} \hspace{0.5cm}  ( 2098  \\ {\hspace{1.2cm}| } \underline{ 4 \hspace{3.0cm} }  \\ 409 {\hspace{0.58cm}| } {\hspace{0.7cm} 40} \hspace{0.5cm}  16  \\   {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 36} \hspace{0.5cm}  81 \hspace{1.5cm} }  \\ 4188 {\hspace{0.38cm}| } {\hspace{0.7cm} 3} \hspace{0.5cm}  35 \hspace{0.5cm} 04  \\   {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 3} \hspace{0.5cm}  35  \hspace{0.5cm} 04 }  \\   {\hspace{1.2cm}| }  {\hspace{0.7cm} } \hspace{0.7cm}  0  \hspace{0.5cm}   

vi) 9653449

{3 \hspace{1.0cm}} | \overline{9} \hspace{0.5cm} \overline{65} \hspace{0.5cm} \overline{34} \hspace{0.5cm} \overline{49} \hspace{0.5cm}  ( 3107  \\ {\hspace{1.2cm}| } \underline{ 9 \hspace{3.0cm} }  \\ 61 {\hspace{0.78cm}| } {\hspace{0.7cm} 65} \hspace{0.5cm}  \\   {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 61} \hspace{0.5cm}   \hspace{1.5cm} }  \\ 6207 {\hspace{0.38cm}| } {\hspace{0.7cm} 3} \hspace{0.5cm}  34 \hspace{0.5cm} 49  \\   {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 4} \hspace{0.5cm}  34  \hspace{0.5cm} 49 }  \\   {\hspace{1.2cm}| }  {\hspace{0.7cm} } \hspace{0.7cm}  0  \hspace{0.5cm}   

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Question 13: The area of a square field is 77841 sq. meters. Find its perimeter.

Answer:

Area = side \times side = 77841 = 279 meter

Perimeter = 4 side = 4  279 = 1116 sq. meters

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Question 14: Find the least number which must be subtracted from 7581 to obtain a perfect square. Find this perfect square and its square root.

Answer:

{8 \hspace{0.5cm}} | \overline{75} \hspace{0.5cm} \overline{81} \hspace{0.5cm} \overline{} \hspace{0.5cm}   ( 31  \\ {\hspace{0.7cm}| } \underline{ 64 \hspace{2.0cm} }  \\ 167 {\hspace{0.08cm}| } {\hspace{0.2cm} 11} \hspace{0.5cm}  81  \\   {\hspace{0.7cm}| } \underline{ {\hspace{0.2cm} 11} \hspace{0.5cm}  69 }  \\   {\hspace{0.7cm}| }  {\hspace{0.2cm} } \hspace{0.4cm}  12  \hspace{0.5cm}   

Subtract 12 from 7581 to obtain a perfect square. The number would be 7569 and the square root would be 87 .

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Question 15: Find the least number which must be subtracted from 43379 to obtain a perfect Find this perfect square and its square root.

Answer:

{2 \hspace{0.7cm}} | \overline{4} \hspace{0.5cm} \overline{33} \hspace{0.5cm} \overline{79} \hspace{0.5cm} \overline{} \hspace{0.5cm}  ( 208  \\ {\hspace{0.9cm}| } \underline{ 4 \hspace{3.0cm} }  \\ 207 {\hspace{0.28cm}| } {\hspace{0.7cm} 33} \hspace{0.5cm}  79  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.7cm} 32} \hspace{0.5cm}  64 \hspace{1.5cm} }  \\   {\hspace{0.9cm}| }  {\hspace{0.7cm} 1} \hspace{0.7cm}  15  \hspace{0.5cm}   

Subtract 115 from 43379 to obtain perfect square.

Question 16: Find the least number which must be added to 6203 to obtain a perfect square. Find the perfect square and its square

Answer:

{7 \hspace{0.7cm}} | \overline{62} \hspace{0.5cm} \overline{03} \hspace{0.5cm} \overline{} \hspace{0.5cm}   ( 78  \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{2.0cm} }  \\ 148 {\hspace{0.28cm}| } {\hspace{0.1cm} 13} \hspace{0.5cm}  03  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.1cm} 11} \hspace{0.5cm}  84  }  \\   {\hspace{0.9cm}| }  {\hspace{0.2cm} 1 } \hspace{0.4cm}  19  \hspace{0.5cm}   

Therefore 78^2 < 7203 < 79^2

79^2 = 6241

Therefore add (6241-6203) = 38 to 6203 to obtain a perfect square (6241) .

Its square root would be 79 .

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Question 17: Find the least number which must be added to 506900 to make it a perfect square. Find this perfect square and its square root.

Answer:

{7 \hspace{0.7cm}} | \overline{50} \hspace{0.5cm} \overline{69} \hspace{0.5cm} \overline{00} \hspace{0.5cm} \overline{} \hspace{0.5cm}  ( 711  \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{3.0cm} }  \\ 141 {\hspace{0.28cm}| } {\hspace{0.3cm} 1} \hspace{0.5cm}  69  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 1} \hspace{0.5cm}  41 \hspace{1.5cm} }  \\ 1421 {\hspace{0.08cm}| } {\hspace{0.95cm} 28} \hspace{0.5cm}  00 \hspace{0.5cm}  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 14} \hspace{0.5cm}  28  \hspace{0.5cm}  }

Therefore 711^2 < 506900 < 712^2

712^2 = 506944

Therefore add 44 to 506900 to make it a perfect square of 712 .

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Question 18:  Find the greatest number of six digits, which is a perfect square. Find the square root of this number.

Answer:

{9 \hspace{1.0cm}} | \overline{99} \hspace{0.5cm} \overline{99} \hspace{0.5cm} \overline{99} \hspace{0.5cm} \overline{} \hspace{0.5cm}  ( 999  \\ {\hspace{1.2cm}| } \underline{ 81 \hspace{3.0cm} }  \\ 189 {\hspace{0.55cm}| } {\hspace{0.1cm} 18} \hspace{0.5cm} 99  \\   {\hspace{1.2cm}| } \underline{ {\hspace{0.1cm} 17} \hspace{0.5cm} 01  \hspace{1.5cm} }  \\ 1989 {\hspace{0.38cm}| } {\hspace{0.1cm} 1} \hspace{0.5cm}  98 \hspace{0.5cm} 99  \\   {\hspace{1.2cm}| } \underline{ {\hspace{0.1cm} 1} \hspace{0.5cm}  79  \hspace{0.5cm} 01 }  \\   {\hspace{1.2cm}| }  {\hspace{0.7cm} 19} \hspace{0.7cm}  98  \hspace{0.5cm}   

Subtract 1998    from 999999    to make a perfect square. The number is 998001   .

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Question 19: Find the least number of four digits which is a perfect square.

Answer:

{3 \hspace{0.7cm}} | \overline{10} \hspace{0.5cm} \overline{00} \hspace{0.5cm} \overline{} \hspace{0.5cm}   ( 31  \\ {\hspace{0.9cm}| } \underline{ 9 \hspace{2.0cm} }  \\ 61 {\hspace{0.48cm}| } {\hspace{0.1cm} 1} \hspace{0.5cm}  00  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.1cm} } \hspace{0.7cm}  61  }  \\   {\hspace{0.9cm}| }  {\hspace{0.2cm}  } \hspace{0.5cm}  39  \hspace{0.5cm}   

Therefore 31^2 < 1000 < 32^2

32^2 = 1024

Therefore add 24 to 1000 to get the least number of four digits which is a perfect square which is 1024 .

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Question 20: Find the least number by which 69192 must be (i) decreased (ii) increased (iii) multiplied (iv) divided to make it a perfect.

Answer:

{2 \hspace{0.7cm}} | \overline{6} \hspace{0.5cm} \overline{91} \hspace{0.5cm} \overline{92} \hspace{0.5cm} \overline{} \hspace{0.5cm}  ( 263  \\ {\hspace{0.9cm}| } \underline{ 4 \hspace{3.0cm} }  \\ 46 {\hspace{0.48cm}| } {\hspace{0.3cm} 2} \hspace{0.5cm}  91  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 2} \hspace{0.5cm}  76 \hspace{1.5cm} }  \\ 52 {\hspace{0.48cm}| } {\hspace{0.95cm} 15} \hspace{0.5cm}  92 \hspace{0.5cm}  \\   {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 15} \hspace{0.5cm}  69  \hspace{0.5cm}  }  \\   {\hspace{0.9cm}| }  {\hspace{0.7cm} } \hspace{0.7cm}  23  \hspace{0.5cm}   

Subtract 23 from 69192 to make it a perfect square.

263^2 < 69191 < 2642, 264^2 = 69696 . Therefore add 504 to 69192 to make it a perfect square.

69192 = 2 \times 2 \times 2 \times 3 \times 3 \times 31 \times 31 . Therefore multiply by 2 to make it a perfect square

Or divide it by 2 to make it a perfect square.