Theory question 101
Binary OperationsA binary operation Δ is defined on the set of real numbers, R, by m Δ n = 1/m3 − 1/n3. Evaluate (√3 − √2) Δ (√3 + √2)
Show the model answer
Model answer
Let m = √3 − √2, n = √3 + √2; mn = 3 − 2 = 1, so 1/m = n and 1/n = m.
m Δ n = 1/m³ − 1/n³ = n³ − m³.
n³ = (√3 + √2)³ = 3√3 + 9√2 + 6√3 + 2√2 = 9√3 + 11√2.
m³ = 9√3 − 11√2.
(√3 − √2) Δ (√3 + √2) = (9√3 + 11√2) − (9√3 − 11√2) = 22√2.
Explanation
Notice that the two surds multiply to 1, so each is the reciprocal of the other.