Using a calculator is the quickest method.
Memorizing or just knowing what the first few are.
Then there's prime factorization method. Prime factor and then find 3 groups.
Example:
Square root of 1,728
864 * 2
432 * 2 * 2
216 * 2 * 2 * 2
108 * 2 * 2 * 2 * 2
54 * 2 * 2 * 2 * 2 * 2
27 * 2 * 2 * 2 * 2 * 2 * 2
9 * 3 * 2 * 2 * 2 * 2 * 2 * 2
3 * 3 * 3 * 2 * 2 * 2 * 2 * 2 * 2
Make 3 groups of the same of primes...
(3 * 2 * 2) * (3 * 2 * 2) * (3 * 2 * 2)
12 * 12 * 12 = 1,728
12³ = 1,728