Square root
8
After studying this lesson, you can get a good understanding of
 the square root of a perfect square number.
 the square root of a positive whole number.
.

8.1 Square of a whole number.
1
1×1
4
9
16
64
3×3
4×4
×2
8×8
Given above are the dot patterns of square numbers. The number of rows and
the number of columns of the dot patterns are equal to the relevant square
number.
2
1st Square number
2nd Square number
3rd Square number
.
.
.
.
.
.
.
.
.
= 1 × 1 = 12 = 1
= 2 × 2 = 22 = 4
= 3 × 3 = 32 = 9
.
.
.
10th Square number = 10 × 10 = 102 = 100
Accordingly, the nth square number is n2.
The nth square number is n2 .
Accordingly, the square of any positive whole number can be written.
50
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Activity 8.1
Number
1
2
3
.
.
.
20
Value
Square
1 × 1 = 12
2 × 2 = 22
..............
1
4
............
...............
............
(ii) Are 49,81,121 square numbers? If they are square numbers what is the position
each gets in the above table?
8.2 The square root of a positive whole number
Observe the dot patterns of the square numbers again. It was emphasized
before that the number of rows and the number of columns of the dot patterns
of all square numbers are equal. The number of rows or the number of columns of
the dot pattern of a square number is called the square root of that number.
According to this dot pattern of the square number 9 the number
of rows is 3 and the number of columns is 3. Hence the square
root of 9 is 3.
9=3
3 = 32
i.e. when
32 = 9 the square root of 9 is 3.
Similarly, since 42 = 16 the square root of 16 is 4.
Accordingly the square root of any square number is the number used for
squaring.
Since 12 = 1 , 1 is the square root of 1. When‘x’and ‘y’are whole numbers
2
Since 22 = 4 , 2 is the square root of 4. and if x = y
Since 52= 25, 5 is the square root of 25. ‘x’is the square root of ‘y’.
i.e.
is read as the square root of 25.
The symbol for square root is “
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”.
51
i.e. 12 = 1 = 1
22 = 4 = 2
32 = 9 = 3
a2 = a
n2 = n
study well the examples below.
=6
Similarly,
=
=2×6
= 12
= 12
Accordingly
When a and b are positive numbers
Accordingly, is
ab = a ×
b
4 + 36 = 4 + 36 true? Discuss with your teacher.
Exercise 8.1
(1) Using the table completed in activity 8.1, complete the following table.
Number
when written
as a square
Finding the square
root
Square root
1
12
22
1
1
2
4
9
.
.
.
400
52
4
9
32
.
.
.
.
.
.
202
400
3
.
.
.
20
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(2) Find the square root of each of the following numbers.
(i)
(ii)
(v)
(vi)
(iii)
(iv)
(3) If the area of a square plot of land is 9m2, f i
of the land.
.
, fi
(4)
2
(5) If the area of a square land is 100m2,
(i) what is the length of one side of the land?
(ii) what is the perimeter of the land?
9 × 36 = 9 × 36, give reasons.
8.2 Finding the square root by using prime factors
Example 1
Let us write 144 as a product of prime factors.
144 = 2 × 2 × 2 × 2 × 3 × 3
This can be written as,
144 = 22 × 22 × 32
144 = (2 × 2 × 3) ( 2 × 2 × 3)
144 = (2 × 2 × 3 )2
Hence the square root of 144 can be obtained as follows.
1
=2 ×2 ×3
= 12
Example 2
write 676 as a product of prime factors.
= 2 × 2 × 13 × 13
= 22 × 132
= (2 × 13)2
,
676
676
676
Hence the square root of 676 can be obtained as give on page 54.
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1
53
(ii)
= 2 × 13
= 26
Exercise 8.2
(1) Find the square root of each of the following.
(i)
(iv)
22 × 32
a 2 × b2
(ii)
2 2 × 52
(iii)
22 × 32 × 52
(v)
32 × b 2 × c 2
(vi)
x2 × y 2 × z2
(2) Find the square root of each of the following numbers using factors.
(i)
256 (ii)
729
(iii)
576
(iv)
784
(v)
900
(3) Find the length of a side of a square piece of paper of which the area is
324 cm2.
(4) Find the length of a side of a square piece of cardboard of which the area is
361 cm2.
(5) If the area of a square land is 2025 m2
(6) The length of a rectangular piece of paper is 20 cm and the breadth is 10 cm. If
this paper has to be separated into square shaped pieces, without any waste,
draw the ways how that separation could be done and write the length of one
side of each square piece.
Find the area of each of the separated square pieces.
2
200 cm.
(i) What is the area of ∆ BCD ?
(ii) What is the area of the square BEFD?
(iii) What is the length of one side of the square BEFD?
A
B
D
C
E
54
F
For Free Distribution only
3
4
hall has to be completely covered with square tiles of area 225 cm2 each.
(i) What is the length of one side of a tile?
(ii) How many tiles are needed to cover one row lengthwise?
(iii) How many tiles are needed to cover one row breadthwise?
?
(iv)
(8) The length of a rectangular hall is 6 m and the width is 3
(9) A pyramid of Giza built in the past which is a world heritage is given in
2
. If a model of it has been
constructed of which the area of the base in 625 cm2
side of the base of the model.
Summary
 A perfect square number can be obtained by squaring any whole
number.
 The square root of a perfect square number is a whole number.
 The square root of a perfect square number can be obtained by
(i) observation.
(ii) expressing it as a product of prime factors.
For Free Distribution only
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