Squares and Square Roots
Main Concepts and Results
• A natural number is called a perfect square if it is the square of some natural number.
i.e., if m =
, then m is a perfect square where m and n are natural numbers.
• Number obtained when a number is multiplied by itself is called the square of the
number
. •Squares of even numbers are even.
• Squares of odd numbers are odd
. • A perfect square can always be expressed as the product of pairs of prime factors.
• The unit digit of a perfect square can be only 0, 1, 4, 5, 6 or 9.
• The square of a number having:
1 or 9 at the units place ends in1.
2 or 8 at the units place ends in 4.
3 or 7 at the units place ends in 9.
4 or 6 at the units place ends in 6.
5 at the units place ends in 5.
• There are 2n natural numbers between the squares of numbers n and n+1.
• A number ending in odd numbers of zeroes is not a perfect square.
• The sum of first n odd natural numbers is given by
.
• Three natural numbers a, b, c are said to form a Pythagorean triplet if
• For every natural number m > 1; 2m,
–1 and
+
=
+ 1 form a pythagorean triplet.
.
• The square root of a number x is the number whose square is x. Positive square root of a
number x is denoted by √x.
• If a perfect square is of n digits, then its square root will have
) digit if n is odd.
digit if n is even or (
Exercise
1.
2.
3.
4.
What are square numbers? Give examples.
Is 32 a square number?
Find the perfect square numbers between (i) 30 and 40 (ii) 50 and 60
Can we say whether the following numbers are perfect squares? How do we know?
(i) 1057
(ii) 23453
(iii) 7928
(iv) 222222
(v) 1069
(vi) 2061
5. Write five numbers which you can decide by looking at their one’s digit that they are not
square numbers.
6. Write five numbers which you cannot decide just by looking at their unit’s digit (or one’s
place) whether they are square numbers or not.
7. Which of 123 , 77 , 82 , 161 ,109 would end with digit 1?
8. Which of the following numbers would have digit 6 at unit place?
9. State true or false?
(a) If a number has 1 or 9 in the unit’s place, then its square ends in 1.
(b) When a square number ends in 6, the number whose square it is, will have either 4 or 6
in unit’s place.
10. What will be the “one’s digit” in the square of the following numbers?
11. The square of which of the following numbers would be an odd number/an even number?
Why?
(i)
727
(ii)
158
(iii)
269
(iv)
1980
12. Which of the following is the square of an odd number?
(a) 256
(b) 361
(c) 144
(d) 400
13. What will be the number of zeros in the square of the following numbers?
(i)
60
(ii) 400
14. If we combine two consecutive triangular numbers, we get a square number. Explain.
15. The least number to be multiplied with 9 to make it a perfect cube is _______________.
16. There are 2n non perfect square numbers between the squares of the numbers n and (n +
1).True or false. If true explain.
17. How many natural numbers lie between 9 and10 ? Between 11 and 12 ?
18. How many non square numbers lie between the following pairs of numbers
19. If the number is a square number, it has to be the sum of successive odd numbers starting
from 1. Explain.
20. If a natural number cannot be expressed as a sum of successive odd natural numbers
starting with 1, then it is not a perfect square. State true or false?
21. Find whether each of the following numbers is a perfect square or not?
(i)
121
(ii) 55
(iii) 81
(iv) 49
(v) 69
22. What will be the unit digit of the squares of the following numbers?
23. The following numbers are obviously not perfect squares. Give reason.
24. The squares of which of the following would be odd numbers?
(i)
431
(ii)
2826
(iii)
7779
(iv)
25. Observe the following pattern and find the missing digits.
82004
26. Observe the following pattern and supply the missing numbers.
27. Using the given pattern, find the missing numbers.
28. Without adding, find the sum.
29. (i)
Express 49 as the sum of 7 odd numbers.
(ii)
Express 121 as the sum of 11 odd numbers.
30. How many numbers lie between squares of the following numbers?
31. Find the square of the following numbers without actual multiplication.
(i)
39
(ii) 42
(iii) 23
32. Find the squares of the following numbers containing 5 in unit’s place.
(i)
15
(ii) 95
(iii) 105
(iv) 205
33. What is Pythagorean triplet? Give examples.
34. Write a Pythagorean triplet whose smallest member is 8.
35. Find a Pythagorean triplet in which one member is 12.
36. Find the square of the following numbers.
37. Write a Pythagorean triplet whose one member is.
(i)
6
(ii)
14
(iii)
16
(iv
38. Area of a square is 144 cm2. What could be the side of the square?
39. What is the length of a diagonal of a square of side 8 cm? (Below figure)
) 18
40. In a right triangle the length of the hypotenuse and a side are respectively 5 cm and 3 cm.
(Figure given below). Can you find the third side?
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
What is the square root of 121?
What is the square root of 196?
Find the square root of 6400.
Is 90 a perfect square?
Is 2352 a perfect square? If not, find the smallest multiple of 2352 which is a perfect square.
Find the square root of the new number.
Find the smallest number by which 9408 must be divided so that the quotient is a perfect
square. Find the square root of the quotient.
Find the smallest square number which is divisible by each of the numbers 6, 9 and 15.
What could be the possible ‘one’s’ digits of the square root of each of the following
numbers?
(i)
9801
(ii) 99856
(iii) 998001
(iv) 657666025
Without doing any calculation, find the numbers which are surely not perfect squares.
(i)
153
(ii) 257
(iii) 408
(iv) 441
Find the square roots of 100 and 169 by the method of repeated subtraction.
Find the square roots of the following numbers by the Prime Factorisation Method.
52. For each of the following numbers, find the smallest whole number by which it should be
multiplied so as to get a perfect square number. Also find the square root of the square
number so obtained.
53. For each of the following numbers, find the smallest whole number by which it should be
divided so as to get a perfect square. Also find the square root of the square number so
obtained.
54. The students of Class VIII of a school donated Rs 2401 in all, for Prime Minister’s National
Relief Fund. Each student donated as many rupees as the number of students in the class.
Find the number of students in the class.
55. During a mass drill exercise, 6250 students of different schools are arranged in rows such
that the number of students in each row is equal to the number of rows. In doing so, the
instructor finds out that 9 children are left out. Find the number of children in each row of
the square.
56. Find the least number that must be added to 1500 so as to get a perfect square. Also find
the square root of the perfect square.
57. 2025 plants are to be planted in a garden in such a way that each row contains as many
plants as the number of rows. Find the number of rows and the number of plants in each
row.
58. Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.
59. Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.
60. Without calculating square roots, find the number of digits in the square root of the
following numbers.
(i)
25600
(ii) 100000000
(iii) 36864
61. Find the square root of:
(i) 729
(ii) 1296
62. Find the least number that must be subtracted from 5607 so as to get a perfect square. Also
find the square root of the perfect square.
63. Find the greatest 4-digit number which is a perfect square.
64. Find the least number that must be added to 1300 so as to get a perfect square. Also find
the square root of the perfect square.
65. Find the square root of 12.25 and 17.64.
66. Area of a square plot is 2304 . Find the side of the square plot.
67. There are 2401 students in a school. P.T. teacher wants them to stand in rows and columns
such that the number of rows is equal to the number of columns. Find the number of rows.
68. Estimate the value of the following to the nearest whole number.
69. Find the square root of each of the following numbers by Division method.
70. Find the number of digits in the square root of each of the following numbers (without any
calculation).
71. Find the square root of the following decimal numbers.
72. Find the least number which must be subtracted from each of the following numbers so as
to get a perfect square. Also find the square root of the perfect square so obtained.
73. Find the least number which must be added to each of the following numbers so as to get a
perfect square. Also find the square root of the perfect square so obtained.
74. Find the length of the side of a square whose area is 441
.
75. In a right triangle ABC, ∠B = 90°.
(a) If AB = 6 cm, BC = 8 cm, find AC
(b) If AC = 13 cm, BC = 5 cm, find AB
76. A gardener has 1000 plants. He wants to plant these in such a way that the number of rows
and the number of columns remain same. Find the minimum number of plants he needs
more for this.
77. There are 500 children in a school. For a P.T. drill they have to stand in such a manner that
the number of rows is equal to number of columns. How many children would be left out in
this arrangement?
78. State whether the statements are true (T) or false (F).
(I)
The square of 0.4 is 0.16.
(II)
The cube root of 729 is 8
(III)
There are 21 natural numbers between 102 and 112.
(IV)
The sum of first 7 odd natural numbers is 49.
(V)
The square root of a perfect square of n digits will have n/2 digits if n is even.
79. Express 36 as a sum of successive odd natural numbers.
80. Check whether 90 is a perfect square or not by using prime factorisation.
81. Using distributive law, find the square of 43.
82. Write a Pythagorean triplet whose smallest number is 6.
83. The area of a rectangular field whose length is twice its breadth is 2450 m2. Find the
perimeter of the field.
84. During a mass drill exercise, 6250 students of different schools are arranged in rows such
that the number of students in each row is equal to the number of rows. In doing so, the
instructor finds out that 9 children are left out. Find the number of children in each row of
the square.
85. Find the smallest number by which 1620 must be divided to get a perfect square.
86. The square root of 24025 will have _________ digits.
87. The square of 5.5 is _________.
88. The square root of 5.3 × 5.3 is _________.
89. 1
= _________
.
90. The sum of first six odd natural numbers is _________.
91. The digit at the ones place of 572 is _________.
92. The sides of a right triangle whose hypotenuse is 17cm are _________ and _________.
93. √1.96 = _________.
94. The least number by which 125 be multiplied to make it a perfect square is _____________
95. State whether the statements are true (T) or false (F).
96. The square of 86 will have 6 at the units place.
97. The sum of two perfect squares is a perfect square.
98. The product of two perfect squares is a perfect square.
99. There is no square number between 50 and 60.
100.
The square root of 1521 is 31.
101.
The square root of 0.9 is 0.3.
102.
The square of every natural number is always greater than the number itself.
103.
There are 200 natural numbers between 100 and 101 .
104.
1000 is a perfect square.
105.
Square root of a number x is denoted by √x.
106.
Find the area of a square field if its perimeter is 96m.
107.
How many square metres of carpet will be required for a square room of side 6.5m
to be carpeted?
108.
Find the side of a square whose area is equal to the area of a rectangle with sides
6.4m and 2.5m.
109.
A hall has a capacity of 2704 seats. If the number of rows is equal to the number of
seats in each row, then find the number of seats in each row.
110.
A General wishes to draw up his 7500 soldiers in the form of a square. After
arranging, he found out that some of them are left out. How many soldiers were left out?
111.
8649 students were sitting in a lecture room in such a manner that there were as
many students in the row as there were rows in the lecture room. How many students were
there in each row of the lecture room?
112.
Rahul walks 12m north from his house and turns west to walk 35m to reach his
friend’s house. While returning, he walks diagonally from his friend’s house to reach back to
his house. What distance did he walk while returning?
113.
A 5.5m long ladder is leaned against a wall. The ladder reaches the wall to a height
of 4.4m. Find the distance between the wall and the foot of the ladder.
114.
A king wanted to reward his advisor, a wise man of the kingdom. So he asked the
wise man to name his own reward. The wise man thanked the king but said that he would
ask only for some gold coins each day for a month. The coins were to be counted out in a
pattern of one coin for the first day, 3 coins for the second day, 5 coins for the third day and
so on for 30 days. Without making calculations, find how many coins will the advisor get in
that month?
115.
Find the smallest square number divisible by each one of the numbers 8, 9 and 10.
116.
Find the square root of 324 by the method of repeated subtraction.
117.
A perfect square number has four digits, none of which is zero. The digits from left
to right have values that are: even, even, odd, even. Find the number.
118.
The area of a rectangular field whose length is twice its breadth is 2450 m2. Find the
perimeter of the field.
119.
The perimeters of two squares are 40 and 96 metres respectively. Find the
perimeter of another square equal in area to the sum of the first two squares.
120.
A ladder 10m long rests against a vertical wall. If the foot of the ladder is 6m away
from the wall and the ladder just reaches the top of the wall, how high is the wall?
Multiple choice Questions:
1) Which of the following is the square of an odd number?
(a) 256
(b) 361
(c) 144
(d) 400
2) Which of the following will have 1 at its units place?
3) How many natural numbers lie between 18 and 19 ?
(a) 30
(b) 37
(c) 35
(d) 36
4) Which of the following is not a perfect square?
(a) 361
(b) 1156
(c) 1128
(d) 1681
5) A perfect square can never have the following digit at ones place.
(a) 1
(b) 6
(c) 5
(d) 3
6) The value of
(a) 14
7) Given that
(b) 15
(c) 16
(a) 82.5
(b) 0.75
(c) 8.25
8) 196 is the square of
(a) 11
(b) 12
(c) 14
9) Which of the following is a square of an even number?
(d) 17
(d) 75.05
(d) 16
(a) 144
(b) 169
(c) 441
(d) 625
10) A number ending in 9 will have the units place of its square as
(a) 3
(b) 9
(c) 1
(d) 6
11) Which of the following will have 4 at the units place?
12) How many natural numbers lie between 5 and 6 ?
(a) 9
(b) 10
(c) 11
(d) 12
13) Which of the following cannot be a perfect square?
(a)
(b) 529
(c) 198
(d) All of the above
14) A square board has an area of 144 square units. How long is each side of the board?
(a) 11 units
(b) 12 units
(c) 13 units
(d) 14 units
15) Which letter best represents the location of √25 on a number line?
16) If one member of a Pythagorean triplet is 2m, then the other two members are
17) The sum of successive odd numbers 1, 3, 5, 7, 9, 11, 13 and 15 is
(a) 81
(b) 64
(c) 49
(d) 36
18) The sum of first n odd natural numbers is
19) The hypotenuse of a right triangle with its legs of lengths 3x x 4x is
(a) 5x
(b) 7x
(c) 16x
20) The next two numbers in the number pattern 1, 4, 9, 16, 25 ... are
(a) 35, 48
(b) 36, 49
(c) 36, 48
21) If m is the square of a natural number n, then n is
(d) 25x
(d) 35,49
22) A perfect square number having n digits where n is even will have square root with
23) The value of
24) Given that,