What is the value of ( 1004^2-996^2 )?

Correct answer: (16000)

The expression has the form \(a^2-b^2\), so the difference-of-squares identity is the quickest method: \\(a^2-b^2=(a+b)(a-b)\\). Take \(a=1004\) and \(b=996\). Their sum is 2000, and their difference is 8. Therefore the required value is \\(2000\times8=16000\\).

Hence option A is correct. Calculating both large squares separately is unnecessary and creates more opportunity for arithmetic mistakes. The value 8000 would result from losing a factor of 2, while 4000 and 12000 do not match the product of the sum and difference. The identity works because the middle terms cancel when the two squares are subtracted.