What is the value of 9991 × 10009 when expressed as an exact product?

Introduction / Context:
This problem seems to be a straightforward multiplication exercise, but the numbers 9991 and 10009 are chosen to encourage recognition of algebraic patterns. It tests both arithmetic skill and understanding of products involving numbers close to 10,000.

Given Data / Assumptions:
- We must compute 9991 × 10009 exactly.
- Both factors are close to 10,000, which suggests looking for a pattern using (a − b)(a + b) if possible.

Concept / Approach:
We can write 9991 and 10009 in terms of 10,000. Note that 9991 = 10,000 − 9 and 10009 = 10,000 + 9. This fits the algebraic identity (a − b)(a + b) = a^2 − b^2, with a = 10,000 and b = 9. Using this identity simplifies the multiplication significantly.

Step-by-Step Solution:
Express the numbers: 9991 = 10,000 − 9 and 10009 = 10,000 + 9. Use the identity (a − b)(a + b) = a^2 − b^2. Here a = 10,000 and b = 9, so the product is 10,000^2 − 9^2. Compute 10,000^2 = 100,000,000. Compute 9^2 = 81, so the result is 100,000,000 − 81 = 99,999,919.

Verification / Alternative Check:
We can quickly compare this result with the options. The number 99,999,919 matches option 99999919 when the commas are removed, so it is consistent. Performing long multiplication would give the same answer, but using the difference of squares identity is much faster and less error prone, especially in an exam setting.

Why Other Options Are Wrong:
99999099 is smaller than 99,999,919 and does not follow from the identity a^2 − b^2 with a = 10,000 and b = 9.
99999819 and 99999019 are also incorrect, as subtracting 81 from 100,000,000 gives 99,999,919 exactly, not these numbers.
100000000 would be the product if one factor were exactly 10,000 and the other were 10,000, which is not the case here because of the adjustment by 9.

Common Pitfalls:
One common mistake is miscomputing 10,000^2 or 9^2, or subtracting incorrectly. Another is failing to see the pattern and attempting full long multiplication, which increases the chance of arithmetic errors. Remembering key algebraic identities like (a − b)(a + b) = a^2 − b^2 allows for quick and reliable calculation in problems involving numbers symmetric around a central value.

Final Answer:
The exact value of 9991 × 10009 is 99999919.