Find the product of the divisors of $420^4$.

$420^4=2^8 \cdot 3^4 \cdot 5^4 \cdot 7^4$. I tried the following method which I soon realized was flawed: for $2^8$ there are $5^3$ possible divisors, for $2^7$ there are also $5^3$ and so on... for $3^4$ there are $9 \cdot 5^2$ possible divisors and for $3^3$ and so on and so forth.

Though I seem to be over-counting the powers loads here, e.g. I will count any one divisor maybe four times. But I still get the answer as $420^{2250}$ which is correct, how can this be?

And yes I know the formula for working out the product easily.