What is the probability of rolling a product of 6 on the first throw of two dice and a sum of 8 on the second throw?

When two fair dice are thrown, each die has 6 possible outcomes. The total number of possible outcomes is $6 \times 6 = 36$.

The combinations that result in a product of six are:
$(1, 6)$, $(2, 3)$, $(3, 2)$, $(6, 1)$
There are 4 favorable outcomes for a product of six.
The probability of getting a product of six, $P(\text{product=6})$, is:

The combinations that result in a sum of eight are:
$(2, 6)$, $(3, 5)$, $(4, 4)$, $(5, 3)$, $(6, 2)$
There are 5 favorable outcomes for a sum of eight.
The probability of getting a sum of eight, $P(\text{sum=8})$, is:

Since the two throws are independent events, the probability of both events happening is the product of their individual probabilities:

The probability of obtaining a product of six in the first throw and a sum of eight in the second throw is $\boxed{\frac{5}{324}}$.