The value of $(1.25)^3 - 2.25 (1.25)^2 + 3.75 (0.75)^2 - (0.75)^3$ is

### Concept & Formula This problem is a direct application of the algebraic identity for the cube of a difference. Recognizing this pattern saves you from doing complex decimal calculations. $$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$ ### Step-by-Step Solution * Let $a = 1.25$ and $b = 0.75$. * Analyze the middle terms of the given expression: * $3a^2b = 3 \times (1.25)^2 \times 0.75$. Notice that $3 \times 0.75 = 2.25$, so the second term is indeed $2.25(1.25)^2$. * $3ab^2 = 3 \times 1.25 \times (0.75)^2$. Notice that $3 \times 1.25 = 3.75$, so the third term is indeed $3.75(0.75)^2$. * The entire expression perfectly matches the expanded form of $(a - b)^3$. * Substitute the values of $a$ and $b$ back into the simplified formula: $$(1.25 - 0.75)^3$$ * Solve the bracket: $1.25 - 0.75 = 0.50$. * Cube the result: $(0.5)^3 = 0.125 = \frac{1}{8}$. ### Exam Strategy & Shortcut Whenever you see a sequence of terms starting with a cube, alternating signs (or specific coefficients like 3), and ending with a cube, immediately test for the $(a-b)^3$ identity. Once you identify $a = 1.25$ and $b = 0.75$, skip the middle verification and jump straight to $(1.25 - 0.75)^3 = 1/8$. ### Common Pitfall The most common mistake is failing to recognize the algebraic identity and attempting to manually calculate the cubes and squares of these decimal numbers, which consumes excessive time and inevitably leads to arithmetic errors. ### Final Answer Therefore, the correct answer is 1/8.