To find the square root of five or six digit numbers we will expand on the method we have already seen to find the square root of three or four digit numbers. Actually we follow the same method as before and it is only when we do the remainder check we will have some additional work.

## Square Root of a Six Digit Example

We will find the square root of 172227

**Step 1**

The first step is, starting from the right hand side, put a slash after every second digit as you move to the left.

As we have two slashes breaking the number into three groups this means the answer will bave three digitd. Both five-digit and six-digit numbers will always have a three digit square root.

**Step 2**

The first group of digits on the left is 17 so we need to pick the largest single digit, which when squared, the square is less than or equal to 17.

16 is less than 17 but 25 is more than 17 so the first digit of the answer is 4.

**Step 3**

We square take the square of first digit of the answer and subtract it from the first group in the number.

In this case we square 4 to get 16 and subtract this from the 17.

**Step 4**

In this step we need to take the result of the subtraction in the previous step, the 1, halve it and add a zero.

**Half of 1**,

This point was not specifically mentioned in the book, actually it was all but ignored.

When we halve 1 we get a fraction, which is 0.5

The adding a zero is replaced with moving the decimal one place to the right so 0.5 becomes 5.

For 1 I have found that instead of just 0.5 as the half, being odd we still have three choices. The "lower half" is 0.4 and the "higher half" is o.6 which gives us 4 and 6 once the decimal place is moved.

I don't know why it was not simply described as "multiply by ten" rather than "add a zero".

Halving the 1 and moving the decimal one place to the right we get a choice of either 4, 5 or 6.

In this case it does not matter if we use the 4, 5 or 6 as dividing any of these by 4, the first digit of our answer, gives us 1 as the next digit of the answer. We will use 5 here.

**Step 5**

To do our partial calculation of the square of our answer we do the following two steps:

1) - square the units digit of the answer, in this case the 1.

The 1 goes into the third column of our little table.

2) - Multiply the tens digit and the units digit then double the result.

The 8 goes into the second column and the zero goes into the first column of our table.

**Step 6**

We subtract the number in the first column of our table, the 0, from the result of our subtraction in step 3, which was 1.

We will put the 1 up under the 2 which is the third digit of our radicand.

We can then cross out the 0 in the table so we know we have used it. For readability I will just highlight it light blue to indicate we have used it.

**Step 7**

Here we vary from what we did when finding the square root of three and four digit numbers as we still have one more digit to find before we do the check for the remainder.

We bring down the 2 next to the 1. We then subtract the number in the second column of our table, the 8, from the 12 which gives us 4.

We can then cross out the 8 in our table. Again I will just highlight it light blue to indicate we have used it.

**Step 8**

Similar to step 4 we take the result of the subtraction in the previous step, the 4, halve it then add a zero.

Halving the 4 gives us 2 and adding a zero we end up with 20.

We divide the 20 by the first digit of our answer, 4 which gives us 5.

**Step 9**

Now we will add a bit more to the table we have under our answer on the right hand side.

We take the first and last digit of the answer and multiply them then double the answer.

We write the 40 under our answer so that the first digit is on the last column that we crossed out the digit, i.e. on the second column, and the zero is in the third column.

**Step 10**

We take the second and third digits of the answer and do the following:

1) - square the units digit of the answer, in this case the 5.

The 5 is written into the fifth column and we carry the 2.

2) - Multiply the tens digit and the units digit then double the result.

We add the carry from 1) to the 10 giving us 12. We write the 2 in column 4 and the 1 in column 3 of our table.

**Step 11**

We will subtract the 4 of the figure we just calculated in the previous step from the 4 we got in step 7.4 minus 4 is 0 so we put the zero up under the next 2 and we can cross out the 4 that we just used on the right.

**Last Step**

We bring down the remaining digits of the 172227, the 227 next to the 0.

Adding up all the unused numbers in our table column wise we get 225we then subtract this total from the 0227.So our answer to the square root of 172227 is 415 with 2 remainder.

That is how you can find the square root of five or six digit numbers.

Gaurav Bhakuni says

Can you explain this method for the root of 190096 please ??

Tony says

Hi Gaurav, The answer is a bit long to put here in the comments but you click this link to download a pdf with the detailed answer.