How do I prove that every powerful number can be written as the product of a perfect square and a perfect cube?

Question

anonymous · Answer

Think about it, what do you know about a square?

anonymous · Answer

they are even

anonymous · Answer

there you go thats all it is

kinggeorge · Answer

Just curious, but what class is this for?

asnaseer · Answer

how does that prove it? I didn't know what "powerful numbers" were until I just looked them up.
if I understand it correctly then a powerful number is a positive integer m such that for every prime number p dividing m, p^2 also divides m.

anonymous · Answer

Proofs class

anonymous · Answer

I didn't really understand the question

asnaseer · Answer

:/

kinggeorge · Answer

I was curious because I helped out on the same question yesterday. See http://openstudy.com/study#/updates/50552fcde4b02986d370aedd

anonymous · Answer

The first part makes sense but why are u subtracting 3 form ei when ei is odd?

kinggeorge · Answer

So that I get $$\large p_i^{e_i}=p_i^{f_i+3}=p_i^{f_i}\cdot p_i^3$$Note that since $e_i$ is odd, $e_i-3$ is even, so $\displaystyle p_i^{e_i-3}=p_i^{f_i}$ is a perfect square.

kinggeorge · Answer

Additionally, $p_i^3$ is a perfect cube.

anonymous · Answer

oh that makes perfect sense, thanks alot

kinggeorge · Answer

You're welcome.