How can you calculate square root of any number accurately?

We know how to find square root of a perfect square, but it’s difficult to find square root of an imperfect square number. There are various techniques that can be used to determine square root of a number accurately (without using a calculator or googling it, of course!!), to the closest decimal point. Here are some of the secrets to mentally calculate square root of any number.

Method 1: Approximate method

Step 1:

Consider you are calculating square root of a number“x” and x=981. The first step is to find the last digit of the square root. To do this, take the square of numbers 1-10. i.e., 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The last digit of the squared number must be equal to last digit of “x”. In this case, it is 9²=81 and 1²=1. So the last digit of square root of 981 can be either 9 or 1

Step 2:

Now you need to calculate the rest of the number. Eliminate the last 2 digits of x, so there is ‘9’ left. Once again, take the square of numbers 1-10 until the squared number is just near or equal to the number 9. In this case 1²=2, 2²=4, 3²=9, this is equal to the first digit of x. So the first digit of the square root is 3

Step 3:

Now you need to decide if the last integer is either 1 or 9. In order to do this, take the first integer of the square root we calculated, in this case it is 3. Multiply 3(3+1) =12. However, 12>9. So the last integer is not 9. So it has to be 1. The approximate square root of 981 is 31.

Method 2: An almost exact method

Step 1:

Consider you are calculating square root of an integer “x” and x=40. The first step is to know between which square numbers 40 falls, i.e., 40 is between 6²=36 and 7²=49.

Step 2:

Now we need to find what is the mid-point of 36 and 49 is 42.5. The square root of the number we need is 40. So we can deduce that it is closer to 6²=36. So it has to be either 6.2 or 6.3

This seems to be rather simpler than the approximate method.

Method 3: Traditional Division method

Remember our school days learning division for the first time? Yes, we can use similar method to find square root of any perfect or imperfect number. Let us see how to find square root of 586

Step 1:

Consider the first digit 5 and calculate the nearest perfect square number. In this instance it is 2²=4. Let us now consider 2 as quotient and 2 as divisor and 2 as reminder, in traditional division style.

Step 2:

Now consider the remaining digits 86. Along with the reminder the number is 186. Now add quotient with the divisor, i.e. 2+2=4. Now with the new divisor 4, calculate the perfect square nearing 186. So it is 44*4=176. The new reminder is 10. New quotient is 24. New divisor is 44.

Step 3:

Now add new quotient 24 with new divisor 44, the new divisor is 68 and the remainder is 10, make this 100 by adding zero. Place a decimal near the new quotient 24. So the approximate square root of 586 is 24.2

This might sound a little tricky, with practice you can come close to the actual answer and impress your friends with your Math skills.

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