If $a = 7$, $b = 5$, then the value of $a^3 - b^3 + 3a^2b$ is

### Concept & Logic This problem requires evaluating a cubic polynomial. While it resembles algebraic identities like $(a-b)^3 = a^3 - b^3 - 3a^2b + 3ab^2$, it does *not* match a standard simplified identity perfectly. Therefore, the most direct and reliable approach is straightforward substitution. ### Step-by-Step Solution Given the expression: $$a^3 - b^3 + 3a^2b$$ * **Step 1: Substitute the given values** Plug in $a = 7$ and $b = 5$: $$(7)^3 - (5)^3 + 3(7)^2(5)$$ * **Step 2: Evaluate the exponents** Calculate $7$ cubed: $7 \times 7 \times 7 = 343$ Calculate $5$ cubed: $5 \times 5 \times 5 = 125$ Calculate $7$ squared: $7 \times 7 = 49$ * **Step 3: Evaluate the multiplication term** Substitute the square back into the final term: $$3 \times 49 \times 5$$ Multiply in an easier order: $(3 \times 5) \times 49 = 15 \times 49$ *(Shortcut for $15 \times 49$: $15 \times 50 - 15 = 750 - 15 = 735$)* * **Step 4: Combine all terms** Now assemble the entire expression: $$343 - 125 + 735$$ $$218 + 735 = 953$$ ### Exam Strategy & Shortcut Use the **Unit Digit Method** to save time. Evaluate only the last digits of the expression: * $7^3$ ends in $3$ (since $7 \times 7=49$, $9 \times 7=63$). * $5^3$ ends in $5$. * $3 \times 7^2 \times 5 \Rightarrow 3 \times 9 \times 5 = 135$, which ends in $5$. Now combine the unit digits according to the equation signs: $3 - 5 + 5 = 3$. Looking at the options, only $953$ ends in a $3$. You can select option (d) without completing the heavy math. ### Common Pitfall Students often waste time trying to force the expression into the form of $(a-b)^3$ or $(a+b)^3$, leading to confusion and errors. If an expression is short but doesn't perfectly align with a memorized identity, direct substitution is usually faster than algebraic manipulation. ### Final Answer **Therefore, the correct answer is 953.**