How does the (a ± b)² identity let you square awkward numbers?

Split the number into a round part a and a small part b, then use a² ± 2ab + b².

Choose a as a nearby multiple of 10:

• 41² = (40 + 1)² = 1600 + 2·40·1 + 1 = 1600 + 80 + 1 = 1681.
• 39² = (40 − 1)² = 1600 − 80 + 1 = 1521.
• 104² = (100 + 4)² = 10000 + 800 + 16 = 10816.

Check: 41² = 1681 ✓, 39² = 1521 ✓. The middle term 2ab is where errors hide — double the product of the two parts, carefully.