If I have $5^{200}$, can I rewrite it in terms of $2$'s and $3$'s to some powers? If I have $5^{200}$, can I rewrite it in terms of $2$'s and $3$'s to some powers? For example, if I had $4^{250}$ can be written in terms of $2$'s like so: $2^{500}$.

No. This is the Fundamental theorem of algebra: every integer $n\geq 2$ can be written in exactly one way (up to the order of factors) as a product of powers of prime numbers. 
