Square Roots and Cube roots.

Square roots of Numbers which are not perfect squares :

To find the square root of 500:
 
Let us guess that the square root is 20.

Divide 500 by 20 to get the quotient 25.

Take the average of the factor 20 and the quotient 25 which is 22.5.

This 22.5 is closer to the actual root of 500 than the initial estimate of 20.

Repeating the above process:

500/22.5 = 22.2222

Average of 22.5 and 22.2222 is 22.3611.

For more accuracy, we can repeat the step once again to get the next estimate as 22.36068.

The actual square root of 500 is 22.36068.

Cube Root of Numbers which are not perfect Cubes :

To find the cube root of 78654

Let the initial guess be 40.

Step 1: 78654 / 40 = 1966.35

Step 2: 1966.35 / 40 = 49.15875

The average of 40, 40 and 49.15875 is 43.05292.

You can repeat the above process with the starting number as 43 (No need to start with 43.05292).
Actual cube root of 78654 is 42.84567.

Even if you start with a very wild initial guess, you will only need a few more iterations to reach the answer.

Finding Square root of Perfect square numbers

For finding the square roots of perfect squares we need to observe the last digit of the square of the number.

1 ^2 = 1

2^2  = 4

3^2 = 9

4^2 = 6 (Forget 1 before 6 , we are only interested in the last digit after squaring a number)

5^2 = 5 (Forget 2 before 5 , we are only interested in the last digit after squaring a number)

6^2 = 6 (Forget 3 before 6 , we are only interested in the last digit after squaring a number)

7^2 = 9 (Forget 4 before 9 , we are only interested in the last digit after squaring a number)

8^2 = 4 (Forget 6 before 4 , we are only interested in the last digit after squaring a number)

9^2 = 1 (Forget 8 before 1 , we are only interested in the last digit after squaring a number)

0^2 = 0

Now the obvious question which must come to your mind is why do we find / observe the last digit ?

The answer is simple. We observe the squares of numbers to estimate the units digit of the square root of the  number.( We are now concerned only with finding the unit's digit of the number ) Estimating the unit's digit will enable us to find the other digits.

 Example : Find the unit's digit of the square root of 1024

Observe that the last digit is 4 ( hence from the observation we see that both 2^2 and 8^2 results in 4) So the units digit of the root so formed can be either 2 or 8.

 Example : Find the unit's digit of the square root of 625

From the observation table we know that the unit's digit will be 5 only when the number 5 is squared.So the units digit of the root is 5


 Example : Find the unit's digit of the square root of 676

From the observation table we know that the unit's digit will be 6 only when the number 6 or 4 is squared. So the units digit of the root is either 4 or 6



Now when we know how to find the unit's digit of any number let's proceed to find the Hundredth's and Tenth's digit.

Lets understand it with the help of an example :


Find the square root of 361
 

Step 1 : Here the first thing we have to do is separate the numbers in pairs from right to left 

( For 3 digit number separate it as 2 numbers from left side , For 4 digit number separate is 2 from the left and 2 from the right ,For 5 digit number cut out 2 digits from the right and leave 3 numbers in the left , for 6 digit numbers cut out evenly)

We separate the digits as follows

3 61

Step 2 : Next from our observation we can say that the units digit of the square root of the number is 1 or 9 ( Concept which we have learned earlier)

61
Sow can get an approximate value of  the units digit as either 1 or 9

Step 3 : Now when the units digit have been estimated we must find the Hundredth's and/or the Tenth's digit.

From the above  observation we have to find the number ( We have to find the number , not the square of the number) whose highest square which can be taken away from 3

Here from the table of squares which we know  the highest square which can be taken away from 3 is 1 and the corresponding number is 1.

Hence the square root of the number is either 11 or 19.

 Now the question is which one is the answer. Simply take any one and square it.

11^2 =121

19^19 = 361 ( You can guess it also , The highest square above 361 is 400  which is nothing but square of 20 , so your answer will be slightly less than 20 )

 Lets take one more example.

Find the square root of 12321

Step 1 : Separate the digits 123 21

Step 2: Estimate the units digit Either 1 or 9

Step3 : From 123 the highest square which can be taken away from it is 121 , which corresponds to the square of 11

Hence the answer is either 113 or 119.

Check out by multiplying either 113 or 119