Discussion
Simplify:
\[\begin{aligned}\frac{0.251^{3} + 1.749^{3}}{0.251^{2} - 0.251 \times 1.749 + 1.749^{2}}\\\end{aligned}\]
Answer: B
The value is 2. Let a = 0.251 and b = 1.749. The numerator is \(a^3 + b^3\) and the denominator is \(a^2 - ab + b^2\). By the identity \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\] the fraction reduces to a + b = 0.251 + 1.749 = 2. The other options, 1.498, 0.5 and 2.244, do not result from this simplification. Put a = 0.251 and b = 1.749 so that the numerator is a cubed plus b cubed and the denominator is a squared minus ab plus b squared. Since a + b = 2, the identity gives exactly 2, and no calculator is needed. The answers 1.498 and 2.244 would arise only from careless addition or multiplication.
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