What is the square root of 1728 using prime factorization?

Answer 1

To use prime factorization to find a square root, after factorizing the number into its primes split each prime into two groups so that the same number (half the number) are in each group - the square root is the product of the primes in one of the groups (both groups are equal). If any prime occurs an odd number of times, then the square root is irrational and to leave the answer in surd form, collect the extra primes into a separate group and multiply the product of one of the equal groups by the square root of the product of the primes in this extra group:

1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3

= (2 x 2 x 2) x (2 x 2 x 2) x (3) x (3) x 3

= (2 x 2 x 2 x 3) x (2 x 2 x 2 x 3) x 3 = 24 × 24 × 3

→ √1728 = 24 × √3