Note: This is not a proof, but an explanation. I misread the question. I hope this helps anyway.
First, I will explain what, AFAIK, is a more common version of prime factorization (which is what the rule in your question is): any positive integer greater than one can be expressed as the product of one or more prime numbers, where there can be multiple of any of the prime numbers. Prime numbers are their own prime factorizations. Composite numbers have multiple numbers in their prime factorizations. For example, the prime factorization of $7$ is $7$, because $7$ is prime. The prime factorization of $4$ is $2*2$. Most composite number have much larger prime factorizations, such as $5280$, whose prime factorization is $2*2*2*2*2*3*5*11$.
Now for the version in your question. Let's go back to the prime factorization of $5280$. See all those $2$s? Those could be expressed as a power, so the new version is $2^5*3*5*11$. Each number there has its degree (in your question, the variables $a$ through $m$). In fact, you can think of $2^5*3*5*11$ as $2^5*3^1*5^1*11^1$. Then $a$ is $5$ and $b$, $c$ and $d$ are 1. Now you can think of $p_k$ as one of those prime numbers (obviously replacing $k$ with the number you are referring to), and $a$ through $m$ as their degrees (powers).