For those of you who are really keen we will take our square roots a step further and look at finding the square root of 9 or 10 digit numbers, which will always be a five digit number. Most of this method, except for the final steps, is the same as for finding the square root of 7 or 8 digit numbers.

As the method is virtually identical for both 9 or 10 digit numbers I will just give an example with a nine digit number.

**Step 1:** Starting from the right-hand side, put a slash after every second digit as you go to the left.

**Step 2:** We have four slashes and the group on the left-hand side is only one digit, 6, so we need to pick the largest single digit, which when squared, the square is less than or equal to 6.

So 2 is the first digit of our answer.

**Step 3:** We square the first digit of the answer and subtract it from the first group in the number. In this case, we square 2 and subtract the square from the 6 in the first group.

**Step 4:** The result of the subtraction in step 3 was 2. We use this to find the second digit of our answer. We halve the 2 then multiply by ten.

Dividing the 10 by the first digit of our answer, 2, gives us the second digit of our answer.

**Step 5:** Under our answer we do the following calculations:

**a)** square the right-hand digit.

We write down the 5 under the right-hand digit of the answer, and carry the 2.

**b)** Multiply the two digits of the answer then double the result.

We add the carry from a)

We now write the 22 in front of the 5 we wrote down in a) giving us 225.

**Step 6: **We subtract the left-hand digit of our calculation under the answer from the result of the subtraction in step 3.

We put the 0 under and to the left of the next digit in the dividend then bring down the digit from the dividend.

As we use the digits in our calculations under the answer we cross them out, here I will just highlight the digit in light blue to indicate it has been used.

**Step 7:** We now subtract the middle digit from our calculation under the answer from the 03 under the dividend.

**Step 8:** We now use the 1 to find the next digit of the answer. We half the 1 then multiply by ten.

We divide the 5 by the first digit of the answer to get the third digit of the answer.

We ignore the decimal and just take the 2 as the third digit of our answer.

**Step 9:** We now do some more calculations under our answer.

**a)** We multiply the first and last digits of the answer then double the result. We write the result as a two digit number, adding a leading zero if necessary.

We then write the 08 under the 225 so the 0 is under the middle digit of the 225.

**b)** We then square the third digit of the answer and write just the unit value down below the digit and carry any tens digit to the next calculation. In this case we have no carry.

**c)** We multiply the second and third digit of the answer then double the result. We would also add any carry from b).

We write the 20 so that the 2 is lined up under the 8 from a).

**Step 10:** We now subtract the tens digit from 9a from the result of the last subtraction under the dividend.

We put the 1 under and to the left of the next digit in the dividend.

**Step 11:** We bring down the next digit from the dividend, the 5, next to the 1 giving 15. We then add up the third column of numbers under the answer and subtract this value from the 15.

**Step 12:** The result of the subtraction is 0 which we use to find the next digit of our answer. The result of dividing zero is zero so 0 should be the next digit of the answer.

We will choose 1 as the next digit of the answer.

**Step 13:** We now do some more calculations under our answer.

**a)** We multiply the first and last digits of the answer then double the result. We write the result as a two digit number, adding a leading zero if necessary.

We then write the 04 under the 204 so the 0 is under the first digit of the 204.

**b)** We multiply the second and fourth digit of the answer then double the result.

We write the 10 so that the 1 is lined up under the unit digit of the result from 13a.

**c)** We multiply the third and fourth digit of the answer then double the result.

We write the result as a two digit result adding a leading zero if necessary. We write the 04 so that the 0 is lined up under the unit digit from the result of 13b.

**d)** We then square the fourth digit of the answer.

We write the result as a two digit result adding a leading zero if necessary. We write the 01 so that the 0 is lined up under the unit digit from the result of 13c.

**Step 14:** We now subtract the unused value in the third column of our calculations under the answer, 0, from the result of the subtraction in step 11.

**Step 15:** We put the zero under and to the left of the next digit in the dividend.

We bring down the next digit from the dividend, the 6, next to the 0 giving 06. We then add up the fourth column of numbers under the answer and subtract this value from the 06.

**Step 16:** We use the 1 to find the last digit of our answer. We half the 1 then multiply by ten.

We divide the 5 by the first digit of the answer.

We ignore the decimal and just take the 2 as the fifth digit of our answer.

**Step 17:** We now do a series of calculations under the answer.

**a)** We multiply the first and last digits of the answer then double the result. We write the result as a two digit number, adding a leading zero if necessary.

We then write the 08 below the previous result but we put the 0 under the last column of numbers we used in step 15, ie the fourth column.

**b)** We multiply the second and last digit of the answer then double the result.

We write the 20 so that the 2 is lined up under the unit digit of the result from 17a.

**c)** We multiply the third and last digit of the answer then double the result.

We write the result as a two digit result adding a leading zero if necessary. We write the 08 so that the 0 is lined up under the unit digit from the result of 17b.

**d)** We multiply the fourth and last digit of the answer then double the result.

We write the result as a two digit result adding a leading zero if necessary. We write the 04 so that the 0 is lined up under the unit digit from the result of 17c.

**e)** We then square the fifth digit of the answer.

We write the result as a two digit result adding a leading zero if necessary. We write the 04 so that the 0 is lined up under the unit digit from the result of 17d.

**Remainder Step:** We already have the five digits of our answer so this step will be used to confirm the last digit and also find the remainder.

We subtract the 0 in the fourth column of our calculations under the answer from the result of the subtraction in step 15.

We put the 1 under and to the left of the next digit in the dividend.

We bring down the next digit from the dividend, the 7, next to the 0 giving 17. We then add up the fifth column of numbers under the answer and subtract this value from the 17.

We now just put the result of the subtraction up under and to the left of the next digit in the dividend.

We bring down all the remaining digits of the dividend.

We add up column-wise, starting on the right-hand side, all the remaining digits in the calculations under the answer.

We put the total from the unused digits under the answer up under the numbers bought down in the dividend and do one large subtraction.

The result of the subtraction is 26131 which means the answer is confirmed as 25212 with a remainder of 26131.

Yes it does look like a

If you have any questions or comments then please leave them below.

Tip:It will quite often, but not always, save time if you choose 1 as the next digit of the answer rather than zero. You have a choice if you pick the wrong number it will become obvious within the next few steps and you will need to backtrack to here then continue on with the alternate choice.