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.
$121=11\times{11}$, this is the factors of $121$.
now take square root of both sides,
$\sqrt{121}=\sqrt{11\times{11}}=\sqrt{11^2}=11$
Square cancelled square root according to the law of indices. 
Hence, $11$ is the square root of $121$.
Check: $11\times{11}=121$.

Example
  1. Evaluate square root $3600$,
 the factors of $3600=2\times{2}\times{2}\times{2}$
$\times{3}\times{3}\times{5}\times{5}$
Group the factors into $2$ and square each group inside a square root.
$\sqrt{3600}=\sqrt{2^2}\times\sqrt{2^2}\times\sqrt{3^2}\times\sqrt{5^2}$
Square will eliminate the square root according to the law of indices and we will be left with
$=2\times{2}\times{3}\times{5}=60$.
Hence, $60$ is square root of $3600$.
Check: $60\times{60}=3600$

Example 3
  1.  Evaluate the square root of $41209$. 
The factors of $41209$ after division by possible prime numbers is:
$7\times{7}\times{29}\times{29}$
Square the factors inside a square root. 
$=\sqrt{7^2}\times\sqrt{29^2}$
The squares will eliminate the square root according to the laws of indices 
$=7\times{29}=203$.
Hence, $203$ is the square root of $41209$.
Check: $203\times{203}=41209$. 

Example 4

  1.  Evaluate the square root of $1\frac{19}{81}$
First convert the mixed fractions into improper fraction 
$=\frac{1\times{81}+19}{81}=\frac{100}{81}$
Now the square root of $\frac{100}{81}=\sqrt{\frac{100}{81}}$
First, find the factors of $100$ and then the factors of $81$. 
$100=2\times{2}\times{5}\times{5}$
Take the square root of both sides 
$\sqrt{100}=\sqrt{2^2}\times\sqrt{5^2}$
$\sqrt{100}=2\times{5}=10$
$\therefore$ square root of $100$ is $10$.
Factors of $81$ are:
$81=3\times{3}\times{3}\times{3}$
Take the square root of both sides 
$\sqrt{81}=\sqrt{3^2}\times{3^2}$
$\sqrt{81}=3\times{3}=9$
$\therefore$ square root of $81$ is $9$.
Hence, $\sqrt{\frac{100}{81}}=\frac{10}{9}$

Example 5

  1. Evaluate the square root of $\frac{98}{18}$. 
Remember, you can only evaluate square root of perfect squares. 
So we will need to break this down until we get two perfect squares. 
$\frac{98}{18}=\frac{49}{9}$
So we find the factors of $49$ and $9$.
$49=7\times{7}$
Take their roots 
$\sqrt{49}=\sqrt{7\times{7}}=\sqrt{7^2}=7$
Hence, $7$ is the square root of $49$. 
Factors $9=3\times{3}$
$\sqrt{9}=\sqrt{3\times{3}}=\sqrt{3^2}=3$
Hence, square root of $9$ is $3$.
Therefore, $\frac{49}{9}=\frac{7}{3}$.