Basically what you have to do is find √3969 without using complex /long calculations
We know it's equal to the square root of 27 times the square root of 147.
The square root of 27 is between 5 and 6, closer to 5 than to 6.
The square root of 147 is between 12 and 13, much closer to 12 than to 13.
So it's between 5*12=60 and 6*13=78, but closer to 60 than 78, so really we know it's between 60 and 69.
The last digit of 3969 is 9, so the last digit of the square root has to, when you square it, give a number whose last digit is 9.
The only single digits like that are 3 and 7.
Therefore, if the square root is a whole number, it must be either 63 or 67.
But if it was 67, when you multiply the 60 by the 7 you carry a 4, and 60 * 60 =3600, with a carry of 4 gives 4000, which puts you over the 3969.
So it must be 63 if it's a whole number.
Edit: The line eliminating 67 was wrong, fixed by jeff-zeitlin.
I don't really know what counts as "complex" or "long", but I think the following is simpler than the other solutions posted so far.
Both numbers are multiples of 3, so let's begin by dividing both by 3. (Then we'll need to multiply the square-root-of-product by 3 when we're done.) We get 9 and 49. We recognize those as the squares of 3 and 7. So we have 3x3x7=63. Done.
[EDITED to add:] Actually, this is pretty similar to Michael Seifert's answer. I think my version is infinitesimally easier, though.