Simplify $1605 \times 1605$

### Concept & Formula The problem requires finding the square of a number close to a base of 1600. We can simplify the calculation by using the algebraic identity for the square of a sum: $$ (a + b)^2 = a^2 + b^2 + 2ab $$ ### Step-by-Step Solution * We can write $1605$ as $(1600 + 5)$. * Therefore, $1605 \times 1605 = (1600 + 5)^2$. * Applying the formula where $a = 1600$ and $b = 5$: $$ (1600 + 5)^2 = (1600)^2 + (5)^2 + 2 \times 1600 \times 5 $$ * Calculate the individual terms: $$ (1600)^2 = 2560000 $$ $$ (5)^2 = 25 $$ $$ 2 \times 1600 \times 5 = 10 \times 1600 = 16000 $$ * Add them together: $$ 2560000 + 25 + 16000 = 2576025 $$ ### Exam Strategy & Shortcut When squaring a number ending in 5, the last two digits of the answer will always be 25. For numbers near a round base like 1600, breaking it into $(1600 + 5)$ allows you to calculate entirely using mental math and simple additions, bypassing tedious multi-digit multiplication. ### Common Pitfall A frequent mistake is applying the distributive property incorrectly by forgetting the $2ab$ term, calculating only $1600^2 + 5^2$ to get 2560025. Always remember the middle term when expanding binomial squares. ### Final Answer Therefore, the correct answer is **2576025**.