The value of $$ \left( \frac{0.051 \times 0.051 \times 0.051 + 0.041 \times 0.041 \times 0.041}{0.051 \times 0.051 - 0.051 \times 0.041 + 0.041 \times 0.041} \right) $$ is

### Concept & Formula This problem relies on expanding the sum of two cubes. When you see numbers repeated three times and added together, it is a massive hint to use the cube identity. The foundational formula is: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ ### Step-by-Step Solution Let $a = 0.051$ and $b = 0.041$. Notice how the expression maps perfectly to variables: * Numerator: $a^3 + b^3$ * Denominator: $a^2 - ab + b^2$ Rewrite the entire fraction algebraically: $$ \frac{a^3 + b^3}{a^2 - ab + b^2} $$ Expand the numerator using the sum of cubes formula: $$ \frac{(a + b)(a^2 - ab + b^2)}{a^2 - ab + b^2} $$ The entire quadratic expression $(a^2 - ab + b^2)$ cancels out from the top and bottom. We are left with just $(a + b)$. Now, substitute the original values back in: $$ 0.051 + 0.041 = 0.092 $$ ### Exam Strategy & Shortcut Look at the structure. Sum of cubes on top, quadratic on the bottom? The answer is just $a + b$. Add the two distinct numbers together in your head: $51 + 41 = 92$, then place the decimal correctly ($0.092$). You can solve this in under 5 seconds. ### Common Pitfall A major mistake is misaligning the decimals when adding $(0.051 + 0.041)$. Exam setters exploit this by offering options like $0.0092$ and $0.92$. Always double-check your decimal places during the final addition step. ### Final Answer **Therefore, the correct answer is 0.092.**