Extracting the square root is something every mathematics student will have to do time and again during his mathematics career.The majority of the modern students feel that since calculators can find the square roots, they do not need to know the steps for deriving these values using pen and paper. Following this trend, it is not a big deal that your mathematics book is devoid of the lesson to find square root without a calculator. However, the students must appreciate and remember the steps for extracting square root which is essential to understand the concept of this process.

**Method 1 has no dividing but it is a bit slower than other methods**

**>** Suppose you want to extract the square root of 75

**>** As you know [latex]{ 8 }^{ 2 }[/latex] = 64 and [latex]{ 9 }^{ 2 }[/latex] = 81, so you can guess that the answer is somewhere between the two

**>** Try calculating [latex]{ 8.5 }^{ 2 }[/latex] and you will get 72.25; which is too small

**>** Try calculating [latex]{ 8.6 }^{ 2 }[/latex] and you will get 73.96 which is too small but you are getting near 75

**>** Try calculating [latex]{ 8.7 }^{ 2 }[/latex] and you will get 75.69; which is too big

**>** So now you know that the answer is between 8.6 and 8.7

**>** Now try to calculate [latex]{ 8.65 }^{ 2 }[/latex] and you get 74.8225

**>** Try calculating [latex]{ 8.66 }^{ 2 }[/latex] and so on, until you get the accurate answer.

This method can be extremely helpful if you have access to a simple calculator, which does not have a square root button.

*Source: LearnNext Maths Tricks*

**Method 2 has many divisions but it is faster**

**>** This time too, we will try for the square root of 75

**>** As you know [latex]{ 8 }^{ 2 }[/latex] = 64 and [latex]{ 9 }^{ 2 }[/latex] = 81, so you can guess that the answer is somewhere between the two

**>** Guess its 8.5 and then calculate 75/8.5. The answer is 8.8235

**>** Now try to calculate the average of 8.5 and 8.8235, which is 8.66175 (8.5 + 8.8235)/2. 8.66175 is a very close answer.

**>** To get closer, calculate 75/8.66175 = 8.6588

**>** Try to calculate the average of 8.66175 and 8.6588. The answer is 8.660275, which is close enough.

**Method 3 focuses on using a simple formula**

The formula is [latex]\sqrt { x } =\sqrt { s } +(x-s)/\sqrt [ 2 ]{ s } [/latex]

Where; X = the number you want the square root of, and S = the closest square number you know to X

**>** This time too, we will try for the square root of 75

**>** X = 75 and the nearest square number is 81; so, S = 81.

**>** Placing the numbers in the formula gives us:

[latex]\sqrt { 75 } =\sqrt { 9 } +(75-81)/\sqrt [ 2 ]{ 9 } [/latex]

or

[latex]\sqrt { 75 } =9+-\frac { 6 }{ 18 } =9-0.333=8.667 [/latex]

This formula is accurate up to two decimal points.

**Method 4 is very simple and fun to use**

**> ** Say you want to extract the square root of 36

**>** Subtract the odd numbers (1, 3, 5, 7, 11, etc.) until you reach zero.

**>** Then count the number of subtractions and get you answer

For example:

To get the square root of 36,

36-1=35

35-3=32

32-5=27

27-7=20

20-9=11

11-11=0

As the total numbers of subtractions are 6, therefore, the square root of 36 is 6

All of these methods point out one thing-calculating square root of a whole number is quite easy. However, trying to find out the square root of a digit that is not a whole number demands to follow a systemic and logical process. Using a calculator is not required but you do need to have basic knowledge about addition, multiplication and division.