To simplify the expression \( 1008 \cdot 1007 \) using the Product Rule of Exponents, we first need to express both numbers in terms of powers of 10:
- \( 1008 = 10^2 \cdot 10^6 \)
- \( 1007 = 10^2 \cdot 10^5 \)
However, let's look at it directly. We can also express it as:
- \( 1008 = 10^3 \cdot 10^{5} \) and
- \( 1007 = 10^3 \cdot 10^{4} \)
Now the general rule for multiplying powers with the same base is:
\[ a^m \cdot a^n = a^{m+n} \]
So applying this to our expression:
\[ 1008 \cdot 1007 = 10^8 \cdot 10^7 = 10^{8+7} = 10^{15} \]
Now looking at the possible responses:
- \( 10015 \)
- \( 100^{15} \)
- \( 10056 \)
- \( 100^{56} \)
- \( 20015 \)
- \( 200^{15} \)
- \( 10,000^{15} \)
- \( 10,000^{15} \)
Out of these, \( 10^{15} \) would be a correct simplification since \( 10^{15} \) is equal to \( 100^7.5 \) as well but does not appear as an option. Therefore, none of the provided options accurately reflect the simplification.
However, the most appropriate transformation of \(1008 \cdot 1007\) as per the product rule results in the exponentiations involving "10" and the sum of the exponents that adds up to 15, which is closest to \(100^{15}\) assuming a base representation, if the base of \(100 \text{ (which is } 10^2\)) was a consideration in the options.
Ultimately, assuming we can't realize better choices from the options listed, the correct application of exponent product rule points out that \(100^{15}\) would be accepted as a representation of \(10^{15}\), thus of the listed options, the best approximate answer would be
\(100^{15}\).