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RADICE
RADICE
4
SQUARE
If a number is multiplied by itself, the product obtained is called the
square of that number.
Example :
62 = 6 x 6 = 36,
In words, 6 squared is equal to 36 or 36 is the
square of 6
Note : The square of a number is its second power. In the above
example the second power of 6=62 =36
SQUARE ROOT
The square root of a given number x is the number whose square is x. The
symbol ‘
’ is used to denote the square root.
Example : Square root of 36 is 6, as square of 6 is 36
Square of 6 is 62 = 36, Square root of 36 = 36 = 6.
Note :The square root of a number is its ½ th power. In the above example
6=(36)½
PERFECT SQUARE
Perfect square is the square of an integer.
Example : 9, 16, 49, etc. are perfect squares since
9 =3
16 = 4
49 = 7
Note : A perfect square always has a square root that has no decimal expansion.
RADICE
5
METHODS TO FIND THE SQUARE ROOT OF A NUMBER
The following table gives the square of some numbers which helps to find
out the square roots.
Number
Square
1
1
2
4
3
9
4
16
5
25
6
36
7
49
8
64
9
81
10
100
Number
625
{
52 = 5x5
= 25
{
7225
Square
20
400
30
900
40
1600
50
2500
60
3600
70
4900
80
6400
90
8100
100
10000
}
625
}
7225
From the above table we shall see that the square root of 625 lies between
the numbers 20 and 30
It is because 400 < 625 < 900.
Similarly the square root of 7225 lies between the numbers 80 and 90.
DIVISION METHOD
(To find the square root of perfect squares)
Illustration to find 54756 using division method
Step 1
Group the digits of the
number in pairs starting from
right to left.
5, 47, 56
RADICE
6
Step 2
Note the number in the left
most group.
5, 47, 56
Step 3
Use the symbol for division
method.
5, 47, 56
Find the largest perfect
square which is less than or
equal to the noted number in
step 2
22 = 4 < 5
Step 5
Note the square root of the
number mentioned in step 4
4=2
Step 6
Write the number noted in step
5 as the quotient and also as
the divisor. Note the
remainder.
Step 7
Bring down the second
group to the right of the
remainder.
Step 4
RADICE
2
2
5, 47, 56
4
1
2
2
5, 47, 56
4
1 47
7
2
Step 8
Double the quotient and
write it as the new divisor.
2
4
5, 47, 56
4
1 47
Double of 2
=2x2=4
2 3
Step 9
Put the largest possible digit
on the right of the new divisor
such that it becomes the next
digit in the quotient and note
the remainder.
2
43
5, 47, 56
4
1 47
1 29
18
43x3 = 129 < 147 we cannot take the
digits 4, 5, etc. since 44 x 4 = 176 > 147
2 3
2
Step 10
Bring down the next group to
the right of the remainder.
43
5, 47, 56
4
1 47
1 29
18 56
RADICE
8
2 3
Step 11
Repeat Step 8
(Double the quotient and
write it as the new divisor.)
2
43
46
5, 47, 56
4
1 47
1 29
18 56
Double of 23
= 2 x 23 = 46
Step 12
Repeat Step 9
(Put the largest possible digit
on the right of the new divisor
such that it becomes the next
digit in the quotient and note
the remainder.)
2 3 4
2
5, 47, 56
4
4 3 1 47
1 29
18 56
464
18 56
0
464x4 = 1856 we cannot take the
digits 5, 6, etc. since 465 x 5 > 1856
We cannot proceed further because in the last step the remainder is 0 and
there is no other groups to bring down.
Thus 54756 = 234
If the given number is not perfect then it won’t end with remainder zero.
Try yourself
Find the square root of 276676
Step 1
RADICE
Group the digits of the
number in pairs starting from
right to left.
9
Step 2
Note the number in the left
most group.
Step 3
Use the symbol for division
method.
27, 66, 76
Find the largest perfect
square which is less than or
equal to the noted number in
step 2
............ = ........... < 27
Step 4
Step 5
Note the square root of the
number mentioned in step 4
Step 6
Write the number noted in step
5 as the quotient and also as
the divisor. Note the
remainder.
Step 7
Bring down the second
group to the right of the
remainder
25 = ............
....
....
27, 66, 76
25
2
....
....
27, 66, 76
25
2 ....
RADICE
10
Step 8
Double the quotient and
write it as the new divisor.
....
....
Step 9
Put the largest possible digit
on the right of the new divisor
such that it becomes the next
digit in the quotient and note
the remainder.
....
10....
....
Step 10
Bring down the next group to
the right of the remainder.
10....
....
Step 11
Repeat Step 8
(Double the quotient and
write it as the new divisor.)
10....
104
Step 12
RADICE
Repeat Step 9
(Put the largest possible digit
on the right of the new divisor
such that it becomes the next
digit in the quotient and note
the remainder.)
....
10....
104......
....
27, 66, 76
25
2 66
.... ....
27, 66, 76
25
2 66
2 04
......
.... ....
27, 66, 76
25
2 66
2 04
62 ....
.... ....
27, 66, 76
25
2 66
2 04
62 76
.... .... .....
27, 66, 76
25
2 66
2 04
62 76
62 76
0
11
DIVISION METHOD
(To find the square root of decimal numbers)
Illustration to find
10485.76 using division method
Step 1
Group the integral part of the
given decimal number in pairs
from right to left.
1, 04, 85 . 76
Step 2
Group the decimal part of the
given decimal number in pairs
from left to right.
1, 04, 85 . 76,
Step 3
Note the number in the left
most group.
1, 04, 85 . 76,
Step 4
Use the symbol for division
method.
1, 04, 85 . 76,
Step 5
Find the largest perfect
square which is less than or
equal to the noted number in
step 3
Step 6
Note the square root of the
number mentioned in step 4
Step 7
Write the number noted in step
6 as the quotient & also as the
divisor. Note the remainder.
12 = 1
1 =1
1
1
1, 04, 85. 76
1
0
RADICE
12
1
Step 8
Bring down the next group to
the right of the remainder.
1
1, 04, 85. 76
1
0, 04
1
Step 9
Double the quotient and write
it as the new divisor.
1
double of 1
=1x2=2
Step 10
2
Put the largest possible digit
1
on the right of the new divisor
such that it becomes the next
20
digit in the quotient and note
the remainder.
20x0 = 0 < 4 we cannot
take the digits 1, 2, etc.
since 21 x 1 = 21 > 4
1, 04, 85. 76
1
0, 04
10
1, 04, 85. 76,
1
0, 04
00
4
10
Step 11
RADICE
Repeat Step 8
(Bring down the next group to
the right of the remainder.)
1
20
1, 04, 85. 76,
1
0, 04
00
4 85
13
10
Step 12
Repeat Step 9
(Double the quotient and write
it as the new divisor.)
1
20
20
1, 04, 85. 76,
1
0, 04
00
4 85
Double of 10
= 2 x 10 = 20
10 2
Step 13
Repeat Step 10
(Put the largest possible digit
on the right of the new divisor
such that it becomes the next
digit in the quotient and note
the remainder.)
1
20
202
1, 04, 85. 76,
1
0, 04
00
4 85
4 04
81
202 x 2 = 404 we
cannot take the digits 3, 4, etc.
since 203 x 3 = 609 > 485
10 2 .
Step 14
Place a decimal point after the
quotient of the above step
since the next group comes
after the decimal point.
1
20
202
1, 04, 85. 76,
1
0, 04
00
4 85
4 04
81
RADICE
14
10 2 .
1
Step 15
Repeat step 8
(Bring down the next group to
the right of the remainder.)
20
202
1, 04, 85. 76,
1
0, 04
00
4 85
4 04
81 76
10 2 .
1
Step 16
Repeat step 9
(Double the quotient and write
it as the new divisor.)
20
202
Double of 102
= 2 x 102 = 204
204
1, 04, 85. 76,
1
0, 04
00
4 85
4 04
81 76
10 2 .4
Step 17
Repeat step 10
(Put the largest possible digit
on the right of the new divisor
such that it becomes the next
digit in the quotient and note
the remainder.)
1
20
202
2044
1, 04, 85. 76,
1
0, 04
00
4 85
4 04
81 76
81 76
0
We cannot proceed further because in the last step the remainder is 0 and
there is no other groups to bring down.
Thus
RADICE
10485.76
= 102.4
15
Try yourself
Find the square root of 605.16
Step 1
Group the integral part of the
given decimal number in pairs
from right to left.
Step 2
Group the decimal part of the
given decimal number in pairs
from left to right.
Step 3
Note the number in the left
most group.
Step 4
Use the symbol for division
method.
Step 5
Find the largest perfect
square which is less than or
equal to the noted number in
step 3
Step 6
Note the square root of the
number mentioned in step 4
Step 7
Write the number noted in step
6 as the quotient & also as the
divisor. Note the remainder.
6, 05, . 16
..... = ...... < 6
... = ...
..
....
6, 05, . 16
...
2
RADICE
16
..
Step 8
Bring down the next group to
the right of the remainder.
Step 9
Double the quotient and write
it as the new divisor.
....
....
....
Step 10
Put the largest possible digit
on the right of the new divisor
such that it becomes the next
digit in the quotient and note
the remainder.
6, 05, . 16
...
2 ...
..
6, 05, . 16
...
2 ...
.. ....
....
.... ....
6, 05, . 16
...
2 05
1 76
........
decimal point
Step 11
Place a decimal point after the
quotient of the above step
since the next group comes
after the decimal point.
.. .... .
....
.... ....
6, 05, . 16
...
2 05
1 76
........
.. .... .
....
Step 12
RADICE
Repeat Step 8
(Bring down the next group to
the right of the remainder.)
.... ....
6, 05, . 16
...
2 05
1 76
........ .....
17
.. .... .
....
Step 13
Repeat Step 9
(Double the quotient and write
it as the new divisor.)
6, 05, . 16
...
2 05
1 76
29 16
.... ....
....
Step 14
Repeat Step 10
(Put the largest possible digit
on the right of the new divisor
such that it becomes the next
digit in the quotient and note
the remainder.)
.. .... . ....
6, 05, . 16
...
2 05
1 76
29 16
29 16
0
....
.... ....
.... ....
PRIME FACTORISATION METHOD
Illustration to find
Step 1
900 using Prime factorisation method
2
900
Factorise the given number
into primes.
2
450
5
225
A natural number, which is greater
than one and is divisible by 1 and
itself only, is called a prime number.
5
45
3
9
3
Step 2
Write the number as the
product of primes.
900 = 2x2x5x5x3x3
RADICE
18
Step 3
Group the factors using
brackets such that one group
contains a pair of same prime.
Step 4
Take one factor from each pair
and find their product which is
the square root of the given
number.
900 = (2x2)x(5x5)x(3x3)
900 = 2 x 5 x 3 = 30
Try yourself
Find the square root of 784 using prime factorisation.
Step 1
Factorise the given number
into primes.
2
784
....
.......
....
196
2
.......
....
49
.......
Step 2
Write the number as the
product of primes.
Step 3
Group the factors using
brackets such that one group
contains a pair of same prime.
Step 4
RADICE
Take one factor from each pair
and find their product which is
the square root of the given
number.
784 = 2x......x......x2x......x......
784 = (2x.....) x (......x......)
x (......x......)
784 = ..... x ...... x ......
19
USING IDENTITIES
Illustration to find 625 using Identities
Step 1
Find two perfect squares such
that one is less than and the
other is greater than the given
number.
Step 2
Use the square root sign in the
inequation in step 1.
400 <
Step 3
Find the numbers between
which the square root of the
given number lies.
20 <
Step 4
Write an equation connecting
the square root and one of the
number involved in the
inequation in step 3 using a
variable.
Step 5
Step 6
400 < 625 < 900
Represent the equation in
step 4 in the form of identity
(a+b)2.
Substitute the values 1,
2, 3, ... for x in the
equation in step 5.
625 <
900
625 < 30
20 + x =
625
or
30 - x =
625
(20 + x)2 = 625
(20+1)2 = 212 = 21 x 21 = 441 ≠ 625
(20+2)2 = 222 = 22 x 22 = 484 ≠ 625
(20+3)2 = 232 = 23 x 23 = 529 ≠ 625
(20+4)2 = 242 = 24 x 24 = 576 ≠ 625
(20+5)2 = 252 = 25 x 25 = 625
The value x = 5 satisfies the equation in step 5.
∴ 625 = 25
RADICE
20
Try yourself
Find the square root of 529 using identities.
Step 1
Find two perfect squares such
that one is less than and the
other is greater than the given
number.
Step 2
Use the square root sign in the
inequation in step 1.
...... <
529 <
Step 3
Find the numbers between
which the square root of the
given number lies.
...... <
529 < ......
Step 4
Write an equation connecting
the square root and one of the
number involved in the
inequation in step 3 using a
variable.
Step 5
Represent the equation in
step 4 in the form of identity
(a+b)2.
Step 6
Substitute the values 1,
2, 3, ... for x in the
equation in step 5.
...... < 529 < ......
.... + x =
529
(.... + x)2 = 529
(... +1)2 = .... ≠ 529
(...+2)2 = ....
≠ 529
(...+3)2 = ....
= 529
The value x = 3 satisfies the equation in step 5.
∴ 529 = .......
RADICE
......
21
DESCRIPTIVE PROBLEMS
1. Find the square root of 4489
by division method.
Ans.
2. Find 15625 by prime
factorisation method.
Ans.
67
5 15625
6 44, 89
36
127 8 8 9
889
0
5
3125
5
625
5
125
5
25
5
∴
4489 = 67
15625 = (5 x 5) x (5 x 5) x (5 x 5)
∴
3
15625 = 5 x 5 x 5 = 125
Find the square roots of the following numbers
(a) 150.0625
(b)
7.7284
(c)
Ans: a)
12.25
b)
2. 78
0.9801
c)
0. 99
1 1, 50. 06, 25
1
22 0 50
44
242
606
484
12225
2445
12225
0
2 7. 72, 84
4
47 372
329
548 4384
4384
0
9 0. 98, 01
81
189
1701
1701
0
∴
∴
∴
150.0625
= 12.25
7.7284
= 2.78
0.9801
= 0.99
RADICE
22
4. A play ground is in the shape of a square. It’s area is 3906.25 m2. It
is to be fenced. Then what would be the length of the fence?
Ans. The area of a square of side having length ‘a’ = a2
Given that the area of the square shaped play ground = 3906.25 m2
i.e., a2 = 3906.25 m2
∴ Length of one side of the play ground = a = 3906.25 m
62.5
6 39,06. 25
36
122
306
244
1245
6225
6225
0
∴
3906.25 = 62.5
Length of one side of the play ground = 62.5m
Length of the fence
= Perimeter of the square
= 4 x 62.5
= 250.0 m
5. A basket contains 125 flowers. A man goes for worship and puts as many
flowers in each temple as there are temples in the city. If the man needs 20
baskets of flower, find the number of flowers that he puts in each temple.
Ans : Let the number of flowers that he puts in each temple be ‘x’
∴ The number of temples in the city
=
x
Thus the total number of flowers used =
the number of temples in the city x
the number of flowers that he puts in each temple
RADICE
}
=
xxx
=
x2
23
The number of flowers in a basket = 125
The number of baskets
= 20
Total number of flowers in 20 baskets = 125 x 20
= 2500
⇒
⇒
= 50
x2
= 2500
x
=
2500
So the number of flowers that he puts
in each temple
= 50
TRAINING FOR COMPETITIVE EXAMINATION
Very challenging problems are marked with
1.
81
a) 9
2.
b)10
b) 0
Find the square root of
a) ±121
4.
The value of
1 x
c) -1
d) none of these
c) 121
d)
c) ± 1
d) none of these
121
± 11
-1 =
b) +1
Which of the following is a perfect square?
a) -121
6.
±
b) 11
a) -1
5.
d) 7
If a real number has one and only one square root, then it is
a) 1
3.
c) 8
0.0196
0.2
a) 0.7
b) 121
c) -64
d) All of them
b) 7
c) 0.07
d) 0.007
=
RADICE
24
196 x
14
7.
8.
9.
17 x
289
78 =
169
a) 2
b) 4
c) 6
1 11 =
25
a) 3
5
b) 5
3
c)
19
5
d) 8.33
d) 1 1
5
The area of a square is 2304, then the length of its side is
a) 48
10.
12.
d) 18
b) 64
c) 16
d) 18
The square root of a perfect square containing ‘n’ digits has —— digits
a) n + 1
b) n
c) Either n + 1 or n
d) none of these
2
2
2
2
x
16
=
49
49
a) 4
13.
c) 28
The least perfect square which is divisible by 2, 4 and 6 is
a) 36
11.
b) 38
then x =
b) 7
c) 16
d) 28
A man plants his 5625 orchid trees and arranges them so that there are as
many rows as there are trees in a row. How many rows are there?
a) 85
14.
b) 5
c) 65
d) 75
Which of the following is not a perfect square?
a) 100
RADICE
b) 2025
c) 324
d) 112
25
15.
16.
If
k = 2 5 then k =
5
a) 50
b) 10
b) 125
20.
b) 9
? then ? =
2.56
=
?
0.1
a) 0.4
b) 0.060
d) 5x625
c) 8
d) 7
c) 0.06
d) 0.09
b) 363 m
c) 336 m
d) 330 m
25
b) 144
5
c) 12
d) None of these
1
1
+
=
16 9
21.
a)
7
12
0.081
0.484
Square root of 0.0064 x 6.25
a) 0.45
23.
c) 6252
The area of a square field is 8190.25m2. The perimeter of square field is
a) 362m
22.
d) 155
A number added to its square gives 56. The number is
a) 12
19.
c) 255
25 times of square of 125 is
a) 125 x 252 b) 5 x 1252
18.
d) none of these
The square root of (11)(12)(13)(14)+1 is
a) 225
17.
c) 25
b) 0.75
c) 0.95
x 2.5
12.1
d) 0.99
Which of the following is a Pythagorean triplet
a) (6,8,10)
b) (3, 4, 7)
c) (5, 12, 18)
d) none of these
RADICE
26
24.
5+2
21 x 0.169
25
1.6
a) 0.91
25.
m+9
If x*y =
a) -7
29.
m+1
c)
m2 + 2m
d) none of these
b) 9
c) 6
d) 3
b) 2.449
c) 2.645
d) 2.828
x2 + y2 , the Value of (1*2 2) (1*-2 2) is
b) 0
c) 2
d) 9
c)5
d) 6.4
c) 7
16
d)
41- 21+ 19 - 9
a) 3
30.
b) m + 2
What is the square root of 6?
a) 2.236
28.
d) 0.95
31+ 21+ 13 + 8 + 1
a) 74
27.
c) 0.94
If m is a square number, then the next immediate square number is
a)
26.
b) 0.92
b) 6
25 x 13 x
2
0.25
18
32
a) 48
35
RADICE
b) 16
7
35
48
27
ANSWERS
1. (a)
2. (b) Except zero, all the other real numbers have 2 square roots.
3. (d)
4. (d)
±
121 = ± 11 since
121 =
± 11
1x -1 = 1x-1 = -1 since negative numbers have no real
square roots the answer is not in the given list.
5. (b) Negative numbers are not perfect squares so the only perfect
square in the list is 121, 121 = 11
6. (a)
0. 0196
0.14
=
= 0.7
0.2
0.2
7. (c)
196
17
x
14
289
8. (d)
1
9. (a)
11
=
25
x
78
169
25+11
=
25
=
36
25
14
17
x
14
17
=
6
5
=1
x
78
13
= 6
1
5
Area of a square = side × side = 2304
side2 = 2304
side = 2304 = 48
10. (a) In the given list 36, 64 and 16 are perfect squares. But 16 is not
divisible by 6 and 64 is not divisible by 6. 36 is divisible by 2, 4 and
6 which is also a perfect square
11. (c) Consider some squares and their square roots,
Let the number of digits in the square = n
If n is even then the square root of the number have
n
digits
2
If n is odd then the square root of the number have n+ 1 digits.
2
RADICE
28
12. (d) 16
x
=
49 49
13. (d)
x
4
=
7
49
x = 49 x 4
7
Let the number of rows
= x
Then the number of trees in a row
= x
∴ total number of trees
= x. x = x2
Given that x2
28
= 5625
x
14. (d)
=
=
5625
= 75
No perfect square ends in 2, 3, 7, 8
∴
112 is not a perfect square or
on the other hand you can find out the square roots
100 = 10,
2025 = 45
324 = 18
but 112 is not a square of an integer.
15. (b) k
5
=2 5 ⇒ k =2 5x 5
= 2 ( 5)2
=2x5
= 10
16. (d)
(11) (12) (13) (14) + 1
= 24024 + 1
= 24025
Square root of (11) (12) (13) (14) + 1 =
24025
= 155 (using division method)
17. (c)
RADICE
Square of 125 is 1252
25 times of square of 125 is 25 x 1252 = 25 (25 x 5)2 since 125 = 25x5
= 25 x 252 x 52
29
=
=
=
18. (d)
Let x be the number
Then the square of x is x2
∴by given condition x2 + x
x2 +x – 56
that is (x + 8) (x – 7)
25 x 252 x 25
(25 x 25) 252
625 x 625 = (625)2
= 56
= 0
= 0 factorised using the
identity (x+a) (x+b) =
x2+(a+b)x + ab
i.e. x + 8 = 0, x - 7 = 0
i.e.,
x = -8, x = 7
-8 is not in the given list so the number is 7.
19. (a) 2.56 = ? ⇒ 1.6
?
0.1
?
Let ? =
= ?
0.1
x
1.6
x
=
x
0.1
ie, 1.6 (0.1) = x2
then 0.16
x
20 (a)
= x2
= 0.16 = 0.4
Area of square
= side x side
i.e, side2
= 8190.25
side
=
perimeter
= 4 x side
8190.25
= 4 x 90.5
ie, ? = 0.4
= 8190.25
(using division
= 90.5 method)
= 362m.
RADICE
30
21. (c)
1
1
+
=
16 9
=
22. (a)
9+16
16x9
25
16x9
25
16 x 9
=
0.081 x 0.484 x 2.5 =
0.0064
6.25
12.1
=
5
4x3
=
5
12
81
484 25
x
x
1000 1000 10
64 x 625 x 121
1000 100
10
=
81 x 484 x 25
64 x 625 x121
=
9 x 22 x 5
8 x 25 x 11
=
9
20
= 0.45
23. (a) A Pythogorean triplet is a set of 3 numbers where the sum
of the squares of the first two is equal to the square of the third number.
Consider 62 + 82 = 36 + 64 = 100 = 102
so (6, 8, 10) is a Pythagorean triplet.
24. (a)
5+2 21 x 0.169 =
25
1.6
=
=
RADICE
5+ 71 x 0.169 =
25
1.6
125+71 x 0.169
25
1.6
196 x 169 x 10
=
25 1000 x 16
14 x 13
= 0.91
5 x 10 x 4
196 x 169
25 100 x 16
31
25. (b) It is given that m is a square number
i.e., m = x2 for some x
⇒
x= m
Then next immediate square is (x+1)2
(x+1)2 = x2+2x+1 = m+2 m + 1
26. (c)
=
31+ 21+ 13 + 8 + 1
31+ 21+ 13 + 8 + 1
since 1 = 1
27.(b)
=
31+ 21+ 13 + 9
=
31+ 21+ 13+3
=
31+ 21+ 16
=
31+ 21+4
=
31+ 25
=
31+ 5
=
36
since 9 = 3
since 16 = 4
since
25 = 5
= 6
The square root of 6, approximating to 3 digits is 2.449
(using division method)
28. (d) given x*y =
x2 + y2
=
12+(2 2)2 =
1+4 x 2 =
1+8 =
1*- 2 2 =
12+(-2 2)2 =
1+4 x 2 =
9 = 3
1* 2 2
9 = 3
(1*+2 2) (1*- 2 2) = 3 x 3 = 9
RADICE
32
29. (b)
30. (d)
41- 21+ 19 - 9
25 x 2 13 x 0.25 =
32
18
=
41- 21+ 19-3
=
41- 21+
=
=
16
=
41- 21+ 4
41- 25
=
41 - 5
36
=
6
25 x 49 x 25
32 18 100
=
25 x 49 x 25
32 9x2 100
since 18 = 9 x 2
=
25 x 49 x 25 =
9
100
64
5 x 7 x 5 = 35
10 x 3 x 8
48
ADVANCED TOPIC
Properties of squares:•
No perfect square ends in 2, 3, 7 or 8.
•
The number of zeroes at the end of a perfect square is even.
•
The square of odd and even numbers are odd and even respectively.
•
If n2 is a perfect square then 2n2 cannot be a perfect square.
Similarly the product of a prime and a perfect square also is not a
perfect square.
•
The square of every non-zero real number is positive.
RADICE
33
Properties of Square Roots:•
We cannot find the square root of a negative number as a real
number.
•
If there is any number ends in 2, 3, 7, 8, then it has no integer
square root
•
If a number ends in odd number of zeroes, then it is not a perfect
square.
•
The square root of an even number is even and that of an odd
number is odd.
•
•
•
xy = x y
x =
y
n =
n
x
y
2
DESCRIPTIVE PROBLEMS
1. Why the following are not perfect squares? Give reason
a) 23453
b) 22222
c) 125000
d) -9876
Ans: (a) Here the unit digit is 3 which is not possible for a perfect square.
(b) Here also the unit digit is 2 and since no square number ends
in 2, 3, 7, or 8, the given number is not a perfect square.
(c) Here the number of zeroes at the end is 3, which is an odd
number. A perfect square cannot have odd numbers of zeroes
at the end thus the given number is not a perfect square.
(d) Since the square of a non zero real number is always positive,
it is not a perfect square.
RADICE
34
2. Classify the following numbers according to whether their squares are
even or not?
a) 431
b) 9999
c) 13984
d) 9847
e) 46432
f) 23718
Ans: We know the square of odd and even numbers are odd and even
respectively. Thus we can classify them as
Numbers having odd squares
3. Find out
Numbers having even squares
431
13984
9999
46432
9847
23718
4.2025
4.2025 =
42025 .
10000
=
42025
10000
=
205
100
=
2.05
So
since
42025
10000
4.2025 =
x
y
42025 = 205,
=
x
y
10000 = 100
MULTIPLE CHOICE QUESTIONS
Very challenging problems are marked with
1.
If x is an even then x2 is
a) an even number b)an odd number c)either even or odd
d) neither even nor odd
RADICE
35
2.
If x2 is odd then x is
a) an even number b) an odd number c) either even or odd
d) neither even nor odd.
3.
Which of the following is a perfect square?
a) -121
b) 121
c) -64
d) none of these.
4.
If a perfect square has 8 zeroes at its end how many zeroes will be there
in its square root at its end
a) 8
b) 4
c) 2
d) 6
5.
-4 = ?
a) 2
b)-2
c) ± 2
d) none of these
ANSWERS
1. (a)
Square of an even number is always even
x is even ⇒ x2 is even.
2. (b)
Square root of an odd number is always odd
x2 is odd ⇒ x is odd.
3. (b)
Negative numbers are not perfect squares.
4. (b)
The numbers of zeroes at the end of a square root of a number is half the
number of zeroes at the end of the given number.
square has 8 zeroes at the end
square root has 4 zeroes at the end.
5. (d)
Negative numbers have no real square roots.
RADICE
36
PROJECT
Introduction:This project is to derive an easy method to find square of numbers
having its unit digit 5
Aim :
Find the square of numbers having unit digit 5 and derive an easy relation
between the digits in the number and its square.
Sl. Number The
No. having square
of the
unit
number
digit 5
The new number after
deleting the unit digits
5 from the considerig
number in coloumn 2
The new number
after deleting last
digits 25 from the
square in coloumn 3
Relationship
between the new
numbers
in
coloumns 5 & 4
1
5
25
0
0
0 = 0 x (0+1)
2
15
225
1
2
2 = 1 x (1+1)
3
25
625
2
6
6 = 2 x (2+1)
4
35
1225
3
12
12 = 3 x (3+1)
Conclusion
The easy steps to be followed to find the square of a number having its
unit digit 5
• Find the new number after deleting the unit digit 5
• Multiply this new number with the natural number immediately after
it. Let the product be P.
•
The square of the given number can be written in the form P 25
e.g.: To find square of 105
The new number after deleting unit digit 5 from the given number=10
The natural number immediately after 10=11
RADICE
37
P = 10 x 11
=
110,
Replacing P with 110, the form P 25 =
11025
∴ (105)2
11025
=
PRACTICE TEST
“Practice makes perfect”. Test the level of your accuracy and speed by answering
the given problems. Please note the time you spend for each problem. Write all
the necessary steps. But if you are thorough with the ideas you can skip some
steps. Please send the answers to our office in the business envelope which is
free of cost. We will provide you proper guidance. If you want to get a proper
evaluation and assessment, you have to do it sincerely.
Very challenging problems are marked with
1. The area of a square is 100 square inches. Find the perimeter of the
square.
a) 40 inches
b) 20 inches
c) 100 inches
d) 10 inches
2. Evaluate x y when x = 15 and y = 15
a) ± 225
3.
b) 11
c) 12
d) 13
b) 900
c) 400
d) 225
If 67* = (62+7) hundred + 72 = 4349 then the value of 69* is
a) 3481
6.
d) both b and c
The least perfect square exactly divisible by each of the numbers 6, 9,
15 and 20 is
a) 36000
5.
c) 15
Find out the difference in number of zeroes in square and square root of
ten crores.
a) 10
4.
b) -15
b) 4581
c) 3681
d) 3281
A Gardiner plants trees in a row and finds that each row contains as
many trees as there are rows. If the number of trees be 2704, find the
RADICE
38
number of trees in each row.
a) 52
7.
8.
9.
b) 104
c) 8
d) 5
Digit at unit place in the square root of 4624 is
a) 52
b) 7
4.2025
a) 205
100
2025 is
=
b)
a) 35
b) 45
c) 8
205
100
c)
d) 5
205
10
d) none of these
c) 55
d) 65
c) 11025
d) 44025
10. Square of 205 is
11.
12.
a) 42025
b) 21025
If x = 1.69 -
0.01 then the value of x is
a) 11.99
b) 1.68
d) 12
x + 36 = 25% of 400 then the value of x is
a) 3
13.
c) 1.20
b) 36
c) 8
d) None of these
c) 12
d) 30
35 x 23 x 6 = ?
a) 108
b) 36
14. Some persons contributed Rs. 1089. Each person gave as many rupees
as they are in number. Find their number
a) 33
15.
c) 45
d) 25
c) 8
d) 10
2n = 32, then the value of n is
a) 512
16.
b) 66
-4 = ?
RADICE
b) 4
39
a) +2
b) -2
c) ± 2
d) None of these
17. The square of a negative number is.
a) positive
18.
b) 9904
c) 9804
d) 9809
A certain number of people collected Rs. 125. If each person contributed
as many five paise as they are in number, the number of persons were.
a) 25
20.
c) both positive and negative d) Neither positive nor negative
The largest four digit number which is a perfect square
a) 9801
19.
b) negative
b) 50
81 , the value of x is
0.09
a) 3
b) 30
c) 100
d) 125
c) 300
d) 0.3.
x=
RADICE
40
PUZZLES
Which is your favourite subject in school? Is it Maths? Some of you don’t
like Maths. But every one likes the Mathematical magic. Let’s have some
fun with Maths.
Try the following and enjoy yourself
I will tell your date of birth
Suppose your date of birth (only date)
Add 2 to it.
Multiply by 100
Add 5 to it.
Add the number corresponding to month (If Jan, add 1)
Subtract 205 from the result
If the result is 2606,
(2606-205 =2401)
Pair it from right 24 , 01
24 corresponds to date
01 corresponds to month Jan
If you tell me the result I will
tell your date of birth.
RADICE
41
FOR BRAIN EXERCISE
Now try to answer the following. Answer will be provided
in the next volume.
1. Can you name the smallest integer that can be written with 2
digits?
2. Can you make 100 using eight 8’s and using standard
mathematical operations?
3. Combine 3 numbers in each group to get the same result in
each of the 3 groups. You can use addition, subtraction,
multiplication and division
Group 1: 15, 19, 24
2: 11, 30, 36
3: 20, 22, 36
4. How many squares are there in the following figure?
5. Can you make 2 squares and 4 right angled triangles using
only 8 straight lines?
6. Can you name the biggest number that can be written with
four 1’s?
7. Can you make 100 using 6 match sticks?
8. Which is the only number that is the perimeter and area of
the same square?
RADICE
42
MATHEMATICS QUIZ
Did you know this?
You have heard about the Nobel prizes. Nobel prizes are awarded in
Physics, Chemistry, literature, peace Physiology or medicine and
Economics. It is not awarded to Mathematics.
Have you ever heard about field’s medal? It is often described as
the “Nobel prize” of Mathematics. The field’s medal is a prize
awarded to 2,3,or 4 mathematicians not over 40 years at
International Mathematical union, a meeting that takes place every
4 years.
FIRST! FIRST! FIRST!
•
•
It is the greatest
award in
Mathematics .It is
awarded to 2, 3 or 4
Mathematicians not
over than 40 years.
•
•
•
•
RADICE
Which is the first calculating machine?
Abacus.
Which is the first composite number?
4
Which is the first irrational number to be
discovered?
2
Who is the first Indian Mathematician to be
elected as a fellow of Royal society, London?
Ramanujan.
Which is the first Mathematical book in
Malayalam?
Yukthi Bhasha
Which is the first printed book on Mathematics?
Treviso Arithmetic.
43
SRINIVASA RAMANUJAN
(The great Indian Mathematician )
1887 -1920
Srinivasa Ramanujan was a great Mathematician
born in South India. He was from a poor Tamil family. He
was a clerk in Madras. He lived off the charity of friends.
The English mathematician Hardy discovered his
Mathematical talents. With the help of Professor Hardy,
he went to Cambridge University for further studies.
Ramanujan’s years in England were mathematically
productive. Cambridge granted him a Bachelor of Science
degree “by research” in 1916, and he was elected a Fellow
of the Royal Society (the 1st Indian to be so honoured) in
1918.But Ramanujan had always lived in a tropical climate.
In 1917 he was hospitalized. At that time professor Hardy paid him a visit. Prof .Hardy
told Ramanujan that he rode a taxi cab to the hospital, with a very unlucky number. When
Ramanujan enquired what the number was, Prof. Hardy replied that: 1729
Ramanujan’s face lit up with a smile and he said that it was not an unlucky number
at all, but a very interesting number, the smallest number that can be expressed as
the sum of cubes of 2 different numbers in 2 different ways.
1729 = 103+93
1729 = 123+13
The number 1729 is known as Ramanujan’s number.
In spite of his illness, he discovered the mathematical formulas. By late 1918 his
health condition became worst; he returned to India in 1919. But his health failed again
and he died in the next year.
RADICE
44
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45
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46
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