Evaluate the product 199992 × 200008 using an appropriate algebraic identity.

Introduction / Context:
This question tests quick calculation using algebraic identities instead of direct multiplication. Numbers that lie symmetrically around a round figure like 100000 or 200000 can be handled very efficiently using the identity for the product of a sum and a difference.

Given Data / Assumptions:

• We need to compute 199992 × 200008.
• Both numbers are close to 200000.
• No calculator is assumed in an exam setting, so algebraic simplification is expected.

Concept / Approach:
When two numbers are equidistant from a central value a, say a - b and a + b, their product can be written as (a - b)(a + b) = a^2 - b^2. Here that central value is 200000, and the distance b is 8. This reduces large multiplications to simple squaring and subtraction.

Step-by-Step Solution:
Write 199992 as 200000 - 8 and 200008 as 200000 + 8. So 199992 × 200008 = (200000 - 8)(200000 + 8). Apply the identity (a - b)(a + b) = a^2 - b^2. Here a = 200000 and b = 8, so the product is 200000^2 - 8^2. Compute 200000^2 = 200000 * 200000 = 40,000,000,000. Compute 8^2 = 64. Subtract: 40,000,000,000 - 64 = 39,999,999,936. Thus the required value is 39,999,999,936, which matches option 39999999936.

Verification / Alternative check:
We can check the order of magnitude. Both numbers are about 2 × 10^5, so their product should be close to 4 × 10^10. Our result 3.9999999936 × 10^10 is extremely close to 4 × 10^10, so it is numerically reasonable.

Why Other Options Are Wrong:
39999799936 is smaller by 200000 from the correct result, which would correspond to an incorrect adjustment term. 39999999836 and 39999999926 differ from the correct value by 100 or 10, reflecting likely arithmetic slips in subtracting 64. 39999999736 is also off by 200, indicating an incorrect square of 8 or a mis-subtraction.

Common Pitfalls:
A common mistake is to try direct multiplication, which is lengthy and error prone. Another frequent error is squaring 200000 incorrectly or miscomputing 8^2 or the final subtraction step. Remembering and confidently applying the identity (a - b)(a + b) = a^2 - b^2 is the fastest and safest approach here.

Final Answer:
The value of 199992 × 200008 is 39999999936.