Square Roots

Table of Contents 

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Introduction to Square Root

We have learnt about exponents.

Squares and square roots are special exponents.

Consider the number \(9\)

Which number when multiplied by itself gives \(9\) as the product?

Square and square root - Exponent form is explained with an example, 3 square equals 9

When the exponent is \(2\), it is called a square.

When the exponent is \(\frac{1}{2}\), it is called a square root.

How to Find Square Root

We can use the subtraction method, prime factorisation method, approximation method and long division method to find the square root of a given number.

Repeated Subtraction Method

This is a very simple method.

We will subtract the consecutive odd numbers from the number for which we are finding the square root, till we reach \(0\)

The number of times we subtract is the square root of the given number.

This method works only for perfect square numbers.

Let us find the square root of \(16\) using this method.

16 -1  &= 15 \\
15 - 3 &=12 \\
12 - 5 &= 7 \\
7 - 7 &= 0

You can observe that we have subtracted \(4\) times.

Thus, \(\sqrt{16} = 4\)

Prime Factorisation

To find the square root of a given number through the prime factorization method:

  • Divide the given number into its prime factors.
  • Form pairs of similar factors such that both factors in each pair are equal.
  • Take one factor from the pair.
  • Find the product of the factors obtained by taking one factor from each pair.
  • That product is the square root of the given number.

Let us find the square root of 144 by this method.

Prime factorisation of 144

Square root of 144 by prime factorisation 

This method works when the given number is a perfect square number.

Estimation and Approximation

This method helps in estimating and approximating the square root of a given number.

Let us use this method to find the \(\sqrt {15}\)

Find the nearest perfect square numbers to \(15\)

\(9\) and \(16\) are the perfect square numbers nearest to \(15\)

We know that \(\sqrt{16}\) = \(4\) and \(\sqrt{9}\) = \(3\)

This implies that \(\sqrt {15}\) lies between \(3\) and \(4\)

Now, we need to see if \(\sqrt {15}\) is closer to \(3\) or \(4\)

Let us consider \(3.5\) and \(4\)

\(3.5^2 = 12.25\) and \(4^2 = 16\)

Thus, \(\sqrt {15}\) lies between \(3.5\) and \(4\) and is closer to \(4\)

Let us find the squares of \(3.8\) and \(3.9\) 

\(3.8^2 = 12.25\) and \(3.9^2 = 15.21\)

This implies that \(sqrt {15}\) lies between \(3.8\) and \(3.9\)

We can repeat the process and check between \(3.85\) and \(3.9\)

We can observe that \(\sqrt{15}\) = \(3.872\)

This is a very long process and time-consuming.

Long Division Method

We can find the exact square root of any given number using this method.

Let us understand this process with an example.

Let's find the square root of \(180\)

Step 1: Place a bar over every pair of digits of the number starting from the unit’s place (right-most side).

We will have two pairs, i.e. \(1\) and \(80\)   

Step 2: We divide the left-most number by the largest number whose square is less than or equal to the number in the left-most pair.

Step 1 - Long division method to find square root of 180

Step 3: Bring down the number under the next bar to the right of the remainder.

Add the last digit of the quotient to the divisor.

To the right of the obtained sum, find a suitable number which together with the result of the sum forms a new divisor for the new dividend that is carried down.

Step 2 - long division method to find square root of 180

Step 4: The new number in the quotient will have the same number as selected in the divisor.

The condition is the same as being either less or equal to that of the dividend.

Step 3 - Long division method to find square root of 180

Step 5: Now, we will continue this process further using a decimal point and adding zeros in pairs to the remainder.

Step 4 - long division method to find square root of 180

Step 5- Finding square root of 180 by long division

Step 6: The quotient thus obtained will be the square root of the number.

Square Root Table

Here are the square root list of perfect square numbers and some non-perfect square numbers from \(1\) to \(10\)

Number Square root
 \(1\)  \(1\)
\(2\) \(1.414\)
\(3\) \(1.732\)
\(4\) \(2\)
\(5\) \(2.236\)
\(6\) \(2.449\)
\(7\) \(2.646\)
\(8\) \(2.828\)
\(9\) \(3\)
\(10\) \(3.162\)

Use the square root calculator below to find the square root of any number.

The square roots of numbers that are not a perfect square are part of irrational numbers.

For quick calculations, it is good to memorise the square root list shown above.

Square Root Calculator

Enter the number for which you need to calculate the square root.

Square Root Symbol and Formula

Square root is nothing but the exponent \(\frac{1}{2}\)

Square Root Symbol

It is represented by the radical symbol pronounced as square root.

Square root symbol

Example: \(\sqrt{9}\) = \(3\) as (\(3 \times 3 = 9\))

3 square is 9 and square root of 9 is 3

Square Root Formula

We know the exponent formula

\( \sqrt[\text{n}]{x } = x ^\frac{1}{n}\)

When \(n\) = \(2\), we call it square root.

We can use any of the above methods for finding the square root like prime factorisation, long division and so on.

Exponent form and square root form of 9

tips and tricks
Tips and Tricks
  1. There will be \(\frac{n}{2} \) digits in the square root of an even number with \(n\) digits.

    There will be \(\frac{n+1}{2} \) digits in the square root of an odd number with \(n\) digits.
  2. Square root of a number ending with:
      \(1\) will end in \(1\) or \(9\) 
      \(4\) will end in \(2\) or \(8\) 
      \(9\) will end in \(3\) or \(7\) 
      \(6\) will end in \(4\) or \(6\) 

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Solved Examples on Square Root

Example 1



Find the square root of 169 by repeated subtraction method.


\[\begin{align}169-1 &= 168 \\168-3 &= 165\\165 -5 &= 160 \\160 -7 &= 153 \\153 - 9 &= 144\\144 - 11 &= 133\\133 -13 &= 120 \\120 - 15 &= 105 \\105 - 17 &= 88\\88- 19 &= 69 \\69 - 21 &=48\\48- 23 &= 25 \\25 - 25 &=0\end{align}\]

We have done repeated subtraction \(13\) times.

\(\therefore \sqrt{169}  = 13\)
Example 2



Find the square root of \(529\) by prime factorisation method.


Prime factorisation of \(529\)

prime factorisation of 529

We can see that,

 529 &= 23 \times 23  \\
\Rightarrow\sqrt{529}  &= 23

\(\therefore \sqrt{529}  = 23\)
Example 3



Find the square root of \(6\) to \(3\) decimal places by long division method.


square root of 6 by long division

\(\therefore \sqrt{6}  = 2.449\)
Example 4



Estimate the square root of \(7\) to \(2\) decimal places using the approximation method.


The nearest square numbers to \(7\) are \(4\) and \(9\)

Therefore, \(\sqrt{7}\) lies between \(\sqrt{4}\) and \(\sqrt{9}\) which are \(2\) and \(3\)

We know that,

\(2.5^2 = 6.25\) , \( 2.6^2 = 6.76\) and \(2.7^2 = 7.29\)

So \(\sqrt{7}\) lies betwen \(2.6\) and \(2.7\)

Let us find the square of \(2.65\)

\(2.65^2  = 7.0225 \)  

\(\therefore \sqrt{7} = 2.65 \)
Example 5



John was baking a square cake for his friend's birthday party.

He decided to give each guest a square piece of cake of sides \(1\:\text{inch}\).

There are \(121\) guests attending the party.

What should be the length of the sides of his square cake tin so that he can serve all the guests \(1\) piece each?

John has to serve 121 guests square slices of cake of sides 1 inch side each. We need to find the length of the sides of the baking tray.


Number of pieces of size \(1\:\text{inch}\) by \(1\:\text{inch}\)  =


This is the area of the square cake tin.


Length of the cake tin =

\[\begin{align}\sqrt{121} = 11\end{align}\]

Therefore, the cake tin size choosen by John is:

\(11\:\text{inch}\) by \(11\:\text{inch}\)

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Practice Questions

Here are few problems for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.



Challenge your math skills
Challenging Questions
  1. Evaluate \( \sqrt{\frac{169}{25}} - \sqrt{\frac{64}{49}} +\sqrt{\frac{196}{9}}\)
  2. For what value of \(x\) is this statement valid?
    \( \frac{\sqrt{128}}{x} = \frac{x}{\sqrt{162}} \)

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here

Frequently Asked Questions(FAQs)

1. How do I calculate square root?

  • To calculate the square root of perfect square numbers, we can use: 
      Prime factorisation
      Repeated subtraction method
  • To calculate the square root of a non-perfect square number, we can use:
      Approximation method
      Long division method

2. How do you simplify square roots?

To simplify a square root, we need to find the prime factorisation of the given number.

If a factor cannot be grouped, retain them under the square root symbol.

Example: \(\sqrt{12} = \sqrt{2 \times 2\times3} = 2\sqrt{3}\)

3. What is the square root of 80?

Square root of \(80\) in the decimal form is \( {\displaystyle \pm}8.9442 \) rounded to \(4\) decimal places.

Use the calculator above to find the square root of any given number

\(\sqrt{80} \!= \!\sqrt{5 \times2 \times 2 \!\times\! 2\times2}\! =\!4\sqrt{5}\)

4. What is the square root of 4?

The square root of 4 is \({\displaystyle \pm}2\)

( \(2 \times 2 = 4\)) and
(\(-2 \times -2 = 4 \))

5. What are the real square roots of 100?

The real square roots of 100 are \({\displaystyle \pm}10\)
( \(10 \times 10 = 100\)) and
(\(-10 \times -10 = 100 \))

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