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CHAPTER 4
Square Root & Cube Root
A Civil Servant should be well-versed in concepts of Square Root & Cube Root. In the Civil Services Aptitude
Test Paper 2, in Basic Numeracy, certainly there will be asked some questions from square roots and cube roots of
number to test fast calculation skills of a candidate.
Square Root
The square root of a number is that number the product of which itself gives the given number, ie, the square
root of 400 is 20, the square root of 625 is 25.
The process of finding the square root is called evaluation. The square root of a number is denoted by the
symbol
called the radical sign. The expression “ 9 ” is read as “root time”, “radical nine” or “the square root of
nine”.
How to Find the Square Root of an Integer?
(i) By the method of Prime Factors: When a given number is a perfect square, we resolve it into prime
factors and take the product of prime factors, choosing one out of every two.
Example 1: Find the square root of 4356.
Solution.
2 4356
2 2178
3 1089
3 363
11 121
11
4356 = 2 × 2 × 3 × 3 × 11 × 11 = 22 × 32 × 112
4356 = 2 × 3 × 11 = 66
Thus from the above example it is clear that in order to find the complete square root of a given number every prime
factor of that number should be repeated twice. Thus, we can make a number which is not a perfect square, a perfect
square by multiplying or dividing the number by those factors of it which are not contained in pairs.
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Example 2: Find the least number by which 1800 be multiplied or divided to make it a perfect square.
Solution. 1800 = 2 × 2 × 2 × 3 × 3 × 5 × 5
The least number by which the given number be multiplied or divided is 2.
(ii) By the method of Long Division: This method can be used when the number is large and the factors
cannot be determined easily. This method can also be used when we want to add a least number or to subtract a
least number from a given number so that the resulting number may give a perfect square of some number.
Example 3: Find the square root of 156816.
Solution.
3 156816
396
9
69
668
621
786
4716
4716
×
Explanations: Firstly, mark off the digits in pairs starting from the unit’s digit. Each pair is called a period.
Now, 32 = 9 and 42 = 16. So we take 32 = 9 and on subtracting 9 from 15 we get 6 as remainder.
Now, bring down the next period ie,
Now, double the root figure already found which is 3 and write it to the left.
Now, from trial and error we find 69 × 9 = 621 which is closet and least to 668. So, place 9 to right of 6
changing it to 69. We also put another 9 to the right of the quotient 3 making it 39. Now, we subtract 621 from
668. We get a remainder of 47.
Now, repeat the whole process till there is no period left over to be brought down.
So,
156816 = 396
To Find the Square Root of a Decimal
Example 4: Find the square root of 1.8225
Method 1:
1 1.8225
1.35
1
23
82
69
26 1325
5
1325
×
∴
Method 2:
1.8225 = 1.35
18225 135
18225
=
= 1.35
1.8225 = 10000 =
10000 100
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To Find the Square Root of a Fraction
Example 5: Find the square root of 1
Solution:
1
13
.
36
49 7
1
13
49
=
=
= =1
6
36
36
36 6
• The square of a number other than unity is either a multiple of 4 or exceeds a multiple of 4 by 1.
• A perfect square can never end with (a) an odd number or zeroes. (b) 2, 3, 7 and 8.
• The square root of an integer is not always an integer ie,
•
ab =
•
a
=
b
•
a + b ≠ a +b
•
a − b ≠ a −b
3, 5, 11 are not integers.
a× b
a
b
Example 6: Find the square root of 0.0016.
Solution.
16
4
16
=
= 0.04
0.0016 = 10000 =
10000 100
Example 7: Find the value of x if
Solution. Given
25.6
= 8.
x
25.6
25.6
=8⇒
= 64
x
x
⇒ 64x = 25.6 ⇒ x = 0.4
Example 8: Find the value of 5 upto three places of decimal.
Solution.
2 5.000000 2.236
4
42
100
84
443
1600
1329
4466
27100
26796
×
∴
5 = 2.236
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Example 9: If
Solution.
5 = 2.236, find the value of
245 −
1
80 − 20 =
2
245 −
1
80 − 20 correct to three places of decimal.
2
49 × 5 −
= 7 5−
16 × 5
− 4×5
2
4 5
−2 5 = 7 5 −2 5 −2 5
2
= 3 5 = 3 × 2.236 = 6.708
Example 10: Find the smallest number that must be added to 2400 to make it a perfect square.
Solution.
4 2400
48
16
88 800
704
96
∴
Number to be added = (49)2 – 2400 = 2401 – 2400 = 1
Cube Root
The cube root of a number is that number the cube of which itself gives the given number ie, the cube root of
64 is 4. The cube root of a number is denoted by the symbol
“cube root of eight”.
3
. The expression 3 8 is read as “cube eight”, or the
To Find the Cube Root of an Integer
(i) By the method of prime factors: When a given number is a perfect cube, we resolve it into prime factors
and take the product of prime factors, choosing one out of every three.
Example 11: Find the cube root of 74088.
Solution.
2 74088
2 37044
2 18522
7 9261
7 1323
7 189
3 27
3 9
3 3
1
74088 = 23 × 73 × 33
3
74088 = 2 × 7 × 3 = 42
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To Find the Cube Root of a Decimal
Example 12: Find the cube root of 19.683.
3
Solution. 19.683
3
=3
39 33 27
19683 3 19683
= 2.7
= 3
=
=
=
1000
1000 3 103 10 10
3 19683
3 6561
3 2187
3 729
3 243
3 81
3 27
3 9
3 3
1
⇒
19683 = 39
To Find the Cube Root of Fraction
Example 13: Find the cube root of 1
61
.
64
3
Solution.
31
53 5
1
61 3 125 3 125
= 3
=
= 3 3 = =1
4
4
64
64
64
4
(1)
3
ab =
(3)
3
a + 3b ≠ 3 a +b
3
3
a× b
3
a
3
b
(2)
3
a
=
b
(4)
3
a − 3 b ≠ 3 a −b
Example 14: Find the smallest number by which 2400 be divided to make it a perfect cube.
Solution. 2400 = 25 × 52 × 3
To make it a perfect cube it must be divided by 22 × 52 × 3 = 300
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