Discussion
Simplify: \( \dfrac{0.05 \times 0.05 \times 0.05 + 0.04 \times 0.04 \times 0.04}{0.05 \times 0.05 - 0.05 \times 0.04 + 0.04 \times 0.04} \)
Answer: A
Correct answer: 0.09
This is the identity for the sum of two cubes: \[ a^{3} + b^{3} = (a+b)(a^{2} - ab + b^{2}). \] The numerator is \( a^{3} + b^{3} \) and the denominator is exactly the second factor, \( a^{2} - ab + b^{2} \), so the whole expression reduces to \[ \frac{(a+b)(a^{2}-ab+b^{2})}{a^{2}-ab+b^{2}} = a + b. \] With \( a = 0.05 \) and \( b = 0.04 \), the answer is \( 0.05 + 0.04 = 0.09 \). The whole point of the question is recognising the pattern rather than computing anything: multiplying the six decimal products out is slow and invites arithmetic slips. The companion form appears just as often: \[ \frac{a^{3} - b^{3}}{a^{2} + ab + b^{2}} = a - b. \] Note the sign in the middle term flips between the two, which is the detail to get right.
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