The value of $$ \left( \frac{0.943 \times 0.943 - 0.943 \times 0.057 + 0.057 \times 0.057}{0.943 \times 0.943 \times 0.943 + 0.057 \times 0.057 \times 0.057} \right) $$ is

### Concept & Formula This is an inversion of the standard sum of cubes identity. Here, the quadratic term is in the numerator, and the sum of cubes is in the denominator. The core formula needed is: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ ### Step-by-Step Solution Let $a = 0.943$ and $b = 0.057$. Identify the structure of the fraction: * Numerator is the quadratic form: $a^2 - ab + b^2$ * Denominator is the sum of cubes: $a^3 + b^3$ Express the fraction algebraically: $$ \frac{a^2 - ab + b^2}{a^3 + b^3} $$ Expand the denominator using the sum of cubes formula: $$ \frac{a^2 - ab + b^2}{(a + b)(a^2 - ab + b^2)} $$ The term $(a^2 - ab + b^2)$ cancels out from the numerator and denominator. We are left with the inverse structure: $$ \frac{1}{a + b} $$ Substitute the original values back into this simplified expression: $$ \frac{1}{0.943 + 0.057} $$ $$ \frac{1}{1.000} = 1 $$ Looking at the options, $1$ is not listed among (a), (b), or (c). ### Exam Strategy & Shortcut A sharp eye will catch that the heavy $a^3+b^3$ block is on the bottom this time. This means your shortcut is $1 / (a+b)$, not just $(a+b)$. Because $0.943 + 0.057$ perfectly equals $1$, the answer is instantly $1/1 = 1$. When a test setter makes the denominator sum to exactly $1$ or $10$, it's a clue that the math is meant to be done mentally. ### Common Pitfall The most dangerous pitfall is acting on autopilot. Students often see the $a, b$ patterns and blindly calculate $(a+b)$, forgetting to check whether the cubes are in the numerator or denominator. In this specific case, $a+b=1$, so the error doesn't change the final numeric result, but on other questions, writing $(a+b)$ instead of $1/(a+b)$ will result in a completely wrong answer. Another pitfall is second-guessing yourself when the right answer isn't (a), (b), or (c) and avoiding the "None of these" option. ### Final Answer **Therefore, the correct answer is None of these.**