Simplify $1398 \times 1398$

### Concept & Formula The problem requires squaring a number that is very close to a round base (1400). We can simplify this by using the algebraic identity for the square of a difference: $$ (a - b)^2 = a^2 + b^2 - 2ab $$ ### Step-by-Step Solution * We can express $1398$ as $(1400 - 2)$. * Therefore, $1398 \times 1398 = (1400 - 2)^2$. * Applying the formula where $a = 1400$ and $b = 2$: $$ (1400 - 2)^2 = (1400)^2 + (2)^2 - 2 \times 1400 \times 2 $$ * Calculate the individual terms: $$ (1400)^2 = 1960000 $$ $$ (2)^2 = 4 $$ $$ 2 \times 1400 \times 2 = 5600 $$ * Combine the terms: $$ 1960000 + 4 - 5600 = 1954404 $$ ### Exam Strategy & Shortcut Instead of performing a 4x4 digit multiplication, converting the problem to a base of 1400 reduces the math to simple subtraction. Also, check the unit digit: $8 \times 8 = 64$, so the answer must end in 4. If only one option ends in 4, you can pick it immediately without calculating. ### Common Pitfall A common error is forgetting to subtract the $2ab$ term (5600) and instead adding it, which would yield the incorrect distractor 1965604. Pay close attention to the minus sign in the $(a - b)^2$ identity. ### Final Answer Therefore, the correct answer is **1954404**.