Discussion
Simplify: \( \dfrac{(0.98)^{2} - (0.02)^{2}}{0.96} \)
Answer: A
Correct answer: 1
Apply the difference of two squares to the numerator: \[ (0.98)^{2} - (0.02)^{2} = (0.98 + 0.02)(0.98 - 0.02) = 1 \times 0.96 = 0.96. \] Dividing by 0.96 gives \[ \frac{0.96}{0.96} = 1. \] The numbers are chosen so that the sum is exactly 1 and the difference is exactly the denominator, which is the signal that the identity is intended. Squaring both terms and subtracting - \( 0.9604 - 0.0004 = 0.96 \) - gives the same answer but far more slowly and with more scope for a slip in the decimal places. The general form is worth recognising in any arrangement: \[ \frac{a^{2} - b^{2}}{a - b} = a + b, \qquad \frac{a^{2} - b^{2}}{a + b} = a - b. \] Spotting which factor the denominator matches is the whole of the work.
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