The only step by step way of finding the square root of a number which doesn't radicalise is to use an arithmetic technique as follows:
We know that the answer is close to 6 because 6 squared is 36, so there will be decimals. We write digits in pairs thus:
| 34 |•00 | 00 | 00 | 00
The answer is in single digits for each pair, so we will be writing the answer over the top. The decimal point is inserted in front of the digit that will go over the first pair of zeroes. I've put in 4 pairs of zeroes so we'll be working out 4 decimal places only.
We know the first digit in the answer is 5, because the answer is a little less than 6, so we write 5 over 34.
We write the square of 5, i.e., 25, under 34 and subtract it:
..........5...• 8.....3.....0.....9
...... | 34 |•00 | 00 | 00 | 00
.........25
108 |...900
...........864
1163 |...3600
.............3489
................11100
116609 |..1110000
................1049481
....................60519
I'm using dots to space out and align the working so that you can see what I'm doing.
Now we double 5 to make 10 and we need to guess what digit should follow 10 so that when we multiply by the digit we get a number which is smaller than or equal to the remainder so far, 900. The digit we need to add is 8, so we write 108 and multiply by 8=864, which is a little less than 900. We subtract 864 from 900 leaving 36 and we pull down another pair of zeroes. We write 8 next to the decimal point, so now we have 5.8.
The next step is to double the answer we have so far ignoring the decimal point, so we have 116, and again we need to put a digit on the end of this and multiply by the same digit so that the product is as close to 3600 as we can get without exceeding it. Let's guess 3, so we have 1163*3=3489 (note that 4 would be too big). The remainder is 111 and we bring down the next pair of zeroes to make 11100.
We carry on. Double the answer so far=1166. What digit are we going to add this time? The smallest is 1, but 11661 is bigger than the remainder 11100, so we have to put zero in the answer and bring down the next pair of zeroes. Double the answer 11660. Now we can add a digit, which is going to be high, so we'll pick 9 and multiply it to give 116609*9=1049481.
We could carry on repeating the same process, bringing down pairs of zeroes as necessary.
There are other ways of finding square roots, but I find this one fairly straightforward and it doesn't require a calculator. I think this method is based on Newton's method.
