Chapter 7 - Squares and Square Roots
A number when multiplied by itself gives the square of the number.
For example square of 6 is 6 x 6 i.e. 36 and is represented as
Similarly square of 8 is 8 x 8 i.e.64 and is represented as
.
.
Square root of a number is one of the two equal factors which when multiplied together gives
the number. Square root of 64 is represented as
and is equal to 8 since 8 x 8 = 64.
Note: Questions on squares and square roots are repeatedly being found in competitive tests
and since the conventional methods of finding squares and square roots are time consuming,
we will be attending to entirely different methods in this portion. Due to its importance, the
portion is explained in detail with a lot of practice examples. Though the examples and the
methods adopted may appear to be very easy/simple for the candidates, for instant memorisation
in the examination hall, constant practise is required. So try to achieve mastery in this portion.
Properties of square numbers:
1. A perfect square number always ends with 1, 4, 5, 6, 9 or an even number of zeros.
(It does not mean that a number with last digit 1 is always a square number).
2. A perfect square number never ends with 2, 3, 7, 8 or an odd number of zeros.
3.
A perfect square number having ‘n’or ‘(n – 1)’ digits will have
4.
A perfect square number with an even number as the last digit is always divisible by 4
i.e. the last two digits of the number will be divisible by 4.
A perfect square number with an odd number as the last digit when reduced by 1
is exactly divisible by 8.
5.
digits in its square root.
Example 1 : Which among the following numbers is a perfect square?
(a) 6688
(b) 4422
(c) 4477
(d) 4356
Soln: A perfect square number can never end with 8, 2 and 7 and obviously 4356 is the answer.
Example 2: Which among the following numbers is a perfect square?
(a) 144000
(b) 122500
(c) 12960
(d) 136810
Soln: A number ending with an odd number of zeros cannot be a perfect square and hence
122500 is the answer.
Example 3: Which among the following numbers is a perfect square?
(a) 399426
(b) 412166
(c) 315844
(d) 338754
Soln: There are no possible exclusions based on the last digit. But we know that all options have
even numbers as the last digit. A perfect square number with an even number as the last digit
is always divisible by 4. i.e. the last two digits of the given numbers must be divisible by 4.
Since only 44 satisfies the said condition, the required answer is 315844.
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Excel Speed Maths - Chapter 7 - Squares and Square Roots
Example 4:
Which among the following numbers is a perfect square?
(a) 710669
(b) 36491
(c) 148225
(d) 565507.
Soln: There are no possible exclusions based on the last digit. But we know that all options have
odd numbers as the last digit. A perfect square number with an odd number as the last digit
when reduced by 1 is exactly divisible by 8. i.e. the last three digits of the number will be
divisible by 8. Now, 669 – 1 = 668, is not exactly divisible by 8. 490 is not exactly divisible by 8
but 224 is exactly divisible by 8 and hence the required answer is 148225.
Now, we will see how to find the square root of a number.
Example 1 : Find the square root of 230400.
Factorisation method : We can express 230400 as the product of prime factors or as product
of easily identifiable square numbers. If it is expressed in the prime factor form, one out of
every pair of prime factors is to be selected and if it is a square number, the square root of the
same is to be selected. The product of these selected numbers will be the square root of the
number. So 230400 can be factorised as follows ;
100
230400
We know that 144 is a perfect square and we are not factorizing it,
4
2301
any further.
4
576
So the required answer =
= 10 x 4 x 12 = 480.
144
Conventional Method : 23,04,00. Since the number ends with two zeros, we will find the
square root of 2304 and thereafter will add one zero to the square root.
4 23,04
16
Step 1 : Mark off the digits as pairs from your right hand side.
Step 2 : In respect of the first pair from your left, see which is the 88 x 8 = 704 704
biggest perfect square and write down the square root of the same to
704
your left and square of the same at the bottom of the first pair.
704
Step 3 : Now find the difference between the first pair and the square
0
as you got previously.
Step 4 : Bring down the next pair and you get the trial dividend as 704.
Step 5 : Take double the square root as you got previously and couple it with a suitable number
so that the product of the number so got and the coupled number yields a result less than or
equal to the trial dividend 704. So by trial and error we can understand that the number to be
coupled is 8. The coupled number will be the next digit of the square root. So we get the square
root of 2304 as 48 and the required answer = 480.
Example 2 : Find the square root of 38025.
5
38025
5
7605
3
1521
3
507
So we get
= √5 x 5 x 3 x 3 x 169 = 5 x 3 x 13 = 195.
169
Excel Speed Maths - Chapter 7 - Squares and Square Roots
66
Conventional Method :
3,80,25
1
So finding square root by this method is tedious
280
and cumbersome and is not the method
29 x 9 = 261
261
appropriate for a test where a minimum of 2
1925
or 3 questions involving square roots will be
19 x 2 = 38. 385 x 5 =1925 1925
present. So how to find the square root of these
0
numbers without these elaborate steps?
So before working on the short cut method,first we will study the following table which gives us
an idea of the last digit of the square/squareroot in terms of the smaller root and the bigger root.
Last digit of the
Square number
1
4
5
6
9
1
Smaller Root
Bigger Root
1
2
5
4
3
9
8
5
6
7
We know, if the square number ends with 1, the possible last digits of the square root can be 1
or 9. Here 1 is called the smaller root and 9 is called the bigger root. Similarly with regard to
the other numbers also. Memorise this table for facilitating fast computation of square roots.
Example 3 : Find the square root of 2116.
In order to find the square root, first we will group this number as two blocks. The right block
will have two digits only and the left block constitutes the remaining part of the number. i.e.
21,16. Now in the left block i.e. in 21, which is the biggest perfect square? The biggest perfect
square is 16 and its square root is 4. So our first digit to the answer is 4. As we know that a four
digit number will have two digits in its square root, our next task is to find out the second digit
of our answer. Since 6 is the last digit of the square number, the possible second digit can be the
smaller and bigger roots of 6 i.e. 4 and 6 respectively.
Now, how to decide whether it is the smaller root or the bigger root? If in the given options
either 4 or 6 is only available, there is no requirement of any further exercise in this
regard. We can rightly select that particular number as the second digit of our answer. But if
both the smaller root and the bigger root are available in the given options we have to select the
correct number and how to do it?
Here find the product of the first digit and the next consecutive number. i.e. we have to find the
product of 4 and the next consecutive number i.e. 5. Here we get the product as 4 x 5 = 20.
Now see whether the product you got is smaller or greater than the left block. So here the
product we got, 20 is smaller than 21 and hence we take the bigger root.
The bigger root of 6 is 6. So the required answer is 46.
Note: If the product we get is bigger than the left block, we take the smaller root. (Taking the
smaller and the bigger root must be made very clear).Though the narration has been made
lengthy for the sake of understanding, the actual steps involved are very short.
Example 4 : Find the square root of 4624.
46,24 => Here the biggest square number in 46 is 36 and its square root is 6.
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Excel Speed Maths - Chapter 7 - Squares and Square Roots