Square 1996 in your head. Give the exact value and the identity that makes it a two-step calculation.

Square 1996 in your head. Give the exact value and the identity that makes it a two-step calculation.

Approach: Anchor on the nearest round base and expand the square, or use the identity that turns a square into a product of two numbers equally spaced around it.

3984016. Write 1996 as 2000 - 4 and expand the binomial square: (a - b)^2 = a^2 - 2ab + b^2 = 4000000 - 16000 + 16 = 3984016. The faster route on a desk uses the difference of squares in reverse, since n^2 = (n + d)(n - d) + d^2. With d = 4 that reads 1996^2 as 1992 * 2000 + 16, which is one doubling of a four-digit number and one addition. Both routes need the same round base of 2000, and the second avoids the doubled cross term entirely.

Follow-up: What is 1996 * 2004 without multiplying the digits, and how does the answer relate to 1996 squared?

Key concepts: binomial square, round base, difference of squares.