Given a natural number say $n \in \mathbb{N}$ with a prime factorization $p_1^{m_1} \cdot p_2^{m_2} \dots p_k^{m_k}$. If you take product of the prime factors $p_1 \cdot p_2 \dots p_k$ then the following holds:
$$ \exists i,j \in \mathbb{N} ~~ n^i = (p_1 \cdot p_2 \dots p_k) \cdot j $$
I think that is correct. Is it? If have no idea how to proof it.
Best Answer
Let $n=20$. Then $n^i =2^{2i} 5^i$ for all $i$. Suppose that this is equal to $(2\cdot 5)^j = 2^j 5^j$. Then you immediately see that $j=i$ and $j=2i$. Contradiction. So it's incorrect.