The value of $(.98)^3 + (.02)^3 + 3 \times .98 \times .02 - 1$ is

### Concept & Formula This expression is a cleverly disguised application of the algebraic identity for the cube of a sum. $$ (a + b)^3 = a^3 + b^3 + 3ab(a + b) $$ ### Step-by-Step Solution * Identify the components of the expression: $(0.98)^3 + (0.02)^3 + 3 \times 0.98 \times 0.02 - 1$. * Let $a = 0.98$ and $b = 0.02$. * Notice that the sum of these variables is exactly 1: $$ a + b = 0.98 + 0.02 = 1 $$ * Look at the $3ab$ term in the problem: $3 \times 0.98 \times 0.02$. Since multiplying by $1$ changes nothing, we can invisibly multiply this term by $(a + b)$ because $a + b = 1$. * The expression $(0.98)^3 + (0.02)^3 + 3(0.98)(0.02)(0.98 + 0.02)$ perfectly matches the expansion of $(a + b)^3$. * Collapse the expansion back into the cubed binomial: $$ (0.98 + 0.02)^3 $$ * Solve the cube: $$ (1)^3 = 1 $$ * The original problem asks for this entire expression minus 1: $$ 1 - 1 = 0 $$ ### Exam Strategy & Shortcut In competitive exams, anytime you see two cubed decimal numbers that add up to exactly $1.0$, accompanied by a $3 \times \text{number} \times \text{number}$ term, immediately recognize the $(a+b)^3$ identity. Since $a+b=1$, the entire first segment of the equation simply evaluates to $1^3$, which is $1$. The final step $1 - 1 = 0$ is immediate. ### Common Pitfall Failing to realize that the $(a+b)$ term is "missing" from the equation because it equals $1$. Students often try to manually cube $0.98$, which is a brutal calculation that guarantees a massive loss of time. ### Final Answer Therefore, the correct answer is 0.