Yes, you can write the number as a product of prime numbers. If it is a perfect square, you will have an even number of each prime. For example
15876 = 2*2*3*3*3*3*7*7
Multiply one of each pair together (2*3*3*7)=126
Check: 126^2=15876
Obviously, you have to know how to prime factor to do this.
For your example, 529, this won't really work as it is already a product of primes. For this one, I'd just guess and check.
First, estimate:
20^2=400 too small
30^2=900 too big
25^2=625 too big
At this point, I know that if it is a perfect square it has to be 23, as that is the only number between 20 and 25 that multiplied by itself will leave a 9 in the ones column.
If it isn't a perfect square, I'd either round it to a more manageable number or guess and check.