The value of $$ \frac{5.71 \times 5.71 \times 5.71 - 2.79 \times 2.79 \times 2.79}{5.71 \times 5.71 + 5.71 \times 2.79 + 2.79 \times 2.79} $$ is

### Concept & Formula This problem is built entirely around the algebraic expansion for the difference of two cubes. The required formula is: $$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$ ### Step-by-Step Solution Assign variables to the repeating numbers: Let $a = 5.71$ and $b = 2.79$. Observe the structure of the given expression: * The numerator represents $a^3 - b^3$. * The denominator represents $a^2 + ab + b^2$. Convert the numerical fraction into its algebraic form: $$ \frac{a^3 - b^3}{a^2 + ab + b^2} $$ Substitute the expanded form of $a^3 - b^3$ into the numerator: $$ \frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2} $$ The term $(a^2 + ab + b^2)$ is common to both the numerator and the denominator, so it cancels out completely. The simplified result is simply $(a - b)$. Finally, subtract the actual decimal values: $$ 5.71 - 2.79 = 2.92 $$ ### Exam Strategy & Shortcut When you see a difference of cubes $a^3 - b^3$ divided by its corresponding trinomial $a^2 + ab + b^2$, simply subtract the second number from the first: $a - b$. By looking at the last digits ($1 - 9$), you know the final digit must end in a $2$. This helps eliminate options quickly before even doing the full subtraction. ### Common Pitfall Students sometimes confuse the signs in the formula, mistakenly adding the numbers instead of subtracting them. Option (C) $8.50$ is exactly what you would get if you mistakenly computed $a + b$. Always match the sign of the operation between the cubed terms! ### Final Answer **Therefore, the correct answer is 2.92.**