A number raised to an exponent is called a power. Power sometimes is used to refer to the exponent alone.
(8^2)^3 is not an example of product of powers. It is the power of a power, which is a power raised to an exponent.
The powers being multiplied in (2^4)(5^4) are 2^4 and 2^5. The rule of product of powers cannot be applied here because the bases are different. Here you can apply the rule of power of a product.
