Applications of Unique Factorization

Proposition 3.1. Let k ≥ 2 be an integer and let m be a positive integer. Then m is a kth power if and only if all the exponents in the prime factorization of m are multiples of k.

not a power since the exponent 3 is not a multiple of 6. Similarly, 2000 = 2453 is not a cube since the exponent 4 is not a multiple of 3.