How to find the square root of whole numbers and fractions without using calculator

Have you ever found yourself in an awkward situation probably in an exam hall or aptitude test where you need to evaluate the square root of whole numbers or fractions and you are not required to use calculator or you don't have a calculator. This lesson will give you a step by step guide on how to evaluate problems of this nature.


Example 1
  1.  Evaluate the square root of 121
To evaluate this, you will need to find all the factors of 121.
Note: To find the factor of any number, divide the number by the smallest possible prime number until 1 is obtained.
121=11×11, this is the factors of 121.
now take square root of both sides,
121=11×11=112=11
Square cancelled square root according to the law of indices. 
Hence, 11 is the square root of 121.
Check: 11×11=121.

Example
  1. Evaluate square root 3600,
 the factors of 3600=2×2×2×2
×3×3×5×5
Group the factors into 2 and square each group inside a square root.
3600=22×22×32×52
Square will eliminate the square root according to the law of indices and we will be left with
=2×2×3×5=60.
Hence, 60 is square root of 3600.
Check: 60×60=3600

Example 3
  1.  Evaluate the square root of 41209
The factors of 41209 after division by possible prime numbers is:
7×7×29×29
Square the factors inside a square root. 
=72×292
The squares will eliminate the square root according to the laws of indices 
=7×29=203.
Hence, 203 is the square root of 41209.
Check: 203×203=41209

Example 4

  1.  Evaluate the square root of 11981
First convert the mixed fractions into improper fraction 
=1×81+1981=10081
Now the square root of 10081=10081
First, find the factors of 100 and then the factors of 81
100=2×2×5×5
Take the square root of both sides 
100=22×52
100=2×5=10
square root of 100 is 10.
Factors of 81 are:
81=3×3×3×3
Take the square root of both sides 
81=32×32
81=3×3=9
square root of 81 is 9.
Hence, 10081=109

Example 5

  1. Evaluate the square root of 9818
Remember, you can only evaluate square root of perfect squares. 
So we will need to break this down until we get two perfect squares. 
9818=499
So we find the factors of 49 and 9.
49=7×7
Take their roots 
49=7×7=72=7
Hence, 7 is the square root of 49
Factors 9=3×3
9=3×3=32=3
Hence, square root of 9 is 3.
Therefore, 499=73.