To find the square root seven or eight digit numbers we will continue to expand on the method we have already seen to find the square root of three or four digit numbers and five or six digit numbers.

## A Seven Digit Example

We will find the square root of 4231249

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 three slashes breaking the number into four groups this means the answer will have four digits. Both seven-digit and eight-digit numbers will always have a four digit square root.

### Step 2

The first group on the left has only one digit, 4, so we need to pick the largest single digit, which when squared, the square is less than or equal to 4. This of course is 2.

### Step 3

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

In this case we square 2 to get 4 and subtract this from the 4 in the first group.

### Step 4

In this step we need to take the result of the subtraction in the previous step, the 0, halve it and add a zero. As the result of the subtraction is zero we know that we will still have zero as a result of this step. We will write the zero in parentheses below the zero from the subtraction.

We now divide the zero we just got by the first digit of the answer, zero divided by 2 is zero so the second digit of our answer is zero.

### 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 0.

0 x 0 = 0

The 0 goes into the third column of our little table.

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

2 x 0 = 0

0 x 2 = 00

The a zero goes into the second column and the other 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 0.

0 - 0 = 0

We will put the 0 up under the 2 which is the second 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

We bring down the 2 next to the 0.

We then subtract the number in the second column of our table, the 0, from the 02 which gives us 2.

We can then cross out the 0 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 2, halve it then add a zero.

Halving the 2 gives us 1 and adding a zero we end up with 10.

We divide the 10 by the first digit of our answer, 2 which gives us 5. So 5 is the third digit of our answer.

### Step 9

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

2 x 5 = 10

10 x 2 = 20

We write the 20 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.

5 x 5 = 25

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.

0 x 5 = 0

0 x 2 = 00

We add the carry from 1) to the 0 giving us 02. We write the 2 in column 4 and the 0 in column 3 of our table.

### Step 11

We will subtract the 2 of the figure we just calculated in part 1) of the previous step from the 2 we got in step 7.

2 minus 2 is 0 so we put the zero up under the next 3.

We can cross out the 2 that we just used on the right.

### Step 12

We bring down the 3 next to the 0.

We then subtract the sum of the numbers in the third column of our table, which add up to zero, from the 03 which gives us 3.

03 - 0 = 3

We can then cross out the 0's in column three of our table. Again I will just highlight them light blue to indicate we have used them.

### Step 13

Looking back at the result of the subtraction in the previous step, the 3, we now need to halve it but as it is an odd number we have the choice of the lower half, 1 or the upper half, which is 2.

Adding a zero we have 10 and 20.

As the chances are good that rather than using 10 or 20 it would be best to try the average of the two first, we but we will choose 15 to continue on with.

Dividing the 15 by the first digit of our answer we get 7 as the fourth digit of our answer, ignoring any decimals.

### Step 14

We now do a series of calculations multiplying the last digit of the answer by each of the digits in the answer and, in all but multiplying the last digit by itself, we then double the result.

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

2 x 7 = 14

14 x 2 = 28

We put the 28 in our table so that the first digit, 2, is on the last column that we crossed out the digits, i.e. on the third column, and the 8 is in the fourth column.

2 - We take the second and last digits of the answer and multiply them then double the result.

0 x 7 = 00

0 x 2 = 00

The first 0 is put into the fifth column, under the unit digit of part 1, and the second 0 is put in the sixth column.

3 - We take the third and last digits of the answer and multiply them then double the result.

5 x 7 = 35

35 x 2 = 70

The 7 is put into the sixth column, under the unit digit of part 2, and the 0 is put in the seventh column.

4 - We square the last digit.

7 x 7 = 49

The 4 is put into the seventh column, under the unit digit of part 3, and the 9 is put in the eighth column.

### Step 15

We now subtract the left most digit, not yet used, the 2, in our table and subtract it from the 3 we got in step 12 from the last subtraction we did under the dividend.

3 - 2 = 1

We put the 1 up under the next digit in the dividend.

We can now cross out the 2 in the table to indicate we have used it. As always I will just highlight it light blue.

### Step 16

We now have all four digits we expected for our answer so now we go to the remainder steps to confirm the last digit of the answer and to see if there is a remainder.

We bring down all of the remaining digits of the dividend next to the 1 we just put up.

We now add up the remaining digits, column-wise from right to left, in our table. For any column where the total is two digits we put the unit digit down and carry the tens digit to the next column.

The total from the remaining digits in the table we now subtract that from the working number we now have under the dividend. As both numbers are 11249 the result of the subtraction is zero.

So the square root of 4231249 is 2057 with no remainder.

## An Eight Digit Example

We will find the square root of 23097636

### 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 in our seven digit example we have three slashes breaking the number into four groups this means the answer will have four digits, the difference now is that the first group on the left has two digits where a seven digit number only had one.

### Step 2

The first group on the left has two digits, 23, so we need to pick the largest single digit, which when squared, the square is less than or equal to 23.

4^{2} = 16

5^{2} = 25

25 is more than 23 so the first digit of our 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 23 in the first group.

### Step 4

In this step we need to take the result of the subtraction in the previous step, the 7, halve it and add a zero. Since 7 is an odd number we have the choice of the lower half, 3, or the upper half, 4. Adding a zero to these we get 30 or 40. Additionally we can take the average of these two which is 35.

Having a quick look at our choices, knowing we have to divide the number we choose by the 4.

40 ÷ 4 = 10

30 ÷ 4 = 7 (ignoring any remainder or decimal)

35 ÷ 4 = 8 (ignoring any remainder or decimal)

We know we can't use 10 but we could reduce it to 9 and try that.

We have no more hints as which to use, 7, 8 or 9 so we will just have to pick one. More often than not it would be correct to go with the middle choice so that is what we will do.

### 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 8.

8 x 8 = 64

The 4 goes into the third column of our little table and we carry the 6.

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

4 x 8 = 32

32 x 2 = 64

Adding the carry from 1) we get:

64 + 6 = 70

The zero goes into the second column and the 7 goes into the first column of our table.

### Step 6

We subtract the number in the first column of our table, the 7, from the result of our subtraction in step 3, which was 7.

7 - 7 = 0

We will put the 0 up under the 0 which is the third digit of our radicand.

We can then cross out the 7 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

We bring down the 0 next to the 0.

We then subtract the number in the second column of our table, the 0, from the 00 which gives us 0.

We can then cross out the zero in the table we just used, I will just highlight it in blue.

### Step 8

Similar to step 4 we take the result of the subtraction in the previous step, the 0, halve it then add a zero.

Halving the 0 gives us 0 and adding a zero we end up with 0.

We divide the 0 by the first digit of our answer, 4 which gives us 0. So 0 is the third digit of our answer.

### Step 9

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

4 x 0 = 0

0 x 2 = 00

We write the 00 under our answer so that the first digit is on the last column that we crossed out a digit, i.e. on the second column, and the other 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 0.

0 x 0 = 0

The 0 is written into the fifth column.

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

0 x 5 = 0

0 x 2 = 00

We add the carry from 1) to the 0 giving us 2. We write the units 0 in column 4 and the tens 0 in column 3 of our table.

### Step 11

We will subtract the 0 from column 2 of the table from the 0 we got in step 7.

0 - 0 = 0

We put the zero up under the next digit, the 9.

We can cross out the 0 that we just used in the table.

### Step 12

We bring down the 9 next to the 0.

We then subtract the sum of the numbers in the third column of our table, which add up to 4, from the 09 which gives us 5.

We can then cross out the numbers in column three of our table. Again I will just highlight them light blue to indicate we have used them.

### Step 13

Looking back at the result of the subtraction in the previous step, the 5, we now need to halve it but as it is an odd number we have the choice of the lower half, 2 or the upper half, which is 3.

Adding a zero we have 20 and 30.

As we have done before we will choose the third alternative which is the average of the two, 25 to continue on with.

Dividing the 25 by the first digit of our answer we get 6 as the fourth digit of our answer, ignoring any decimals.

### Step 14

We now do a series of calculations multiplying each digit of the answer with the last digit in the answer and, in all but multiplying the last digit by itself, we then double the result.

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

4 x 6 = 24

24 x 2 = 48

We put the 48 in our table so that the first digit, 4, is on the last column that we crossed out the digits, i.e. on the third column, and the 8 is in the fourth column.

2 - We take the second and last digits of the answer and multiply them then double the result.

8 x 6 = 48

48 x 2 = 96

The 9 is put into the fifth column, under the unit digit of part 1, and the 6 is put in the sixth column.

3 - We take the third and last digits of the answer and multiply them then double the result.

0 x 6 = 00

00 x 2 = 00

The first 0 is put into the sixth column, under the unit digit of part 2, and the second 0 is put in the seventh column.

4 - We square the last digit.

6 x 6 = 36

The 3 is put into the seventh column, under the unit digit of part 3, and the 6 is put in the eighth column.

### Step 15

We now subtract the left most digit, not yet used, the 4, in our table and subtract it from the 5 we got in step 12 from the last subtraction we did under the dividend.

5 - 4 = 1

We put the 1 up under the next digit in the dividend.

We can now cross out the 4 in the table to indicate we have used it. As always I will just highlight it light blue.

### Step 16

We now have all four digits we expected for our answer so now we go to the remainder steps to confirm the last digit of the answer and to see if there is a remainder.

We bring down all of the remaining digits of the dividend next to the 1 we just put up.

We now add up the remaining digits, column-wise from right to left, in our table. For any column where the total is two digits we put the unit digit down and carry the tens digit to the next column.

The total from the remaining digits in the table we now subtract that from the working number we now have under the dividend. As both numbers are 17363 the result of the subtraction is zero.

So the square root of 23097636 is 4086 with no remainder.

Rahul Bharadwaj says

very nice trick