A binary operation A is defined on the set of real numbers, R, by (a Delta b = a^{3} – b^{3}). Without using calculator, find the value of ((sqrt{3} + sqrt{2}) Delta (sqrt{3} – sqrt{2})) leaving the answer in surd form.
Explanation
(a Delta b = a^{3} – b^{3})
((sqrt{3} + sqrt{2}) Delta (sqrt{3} – sqrt{2}) = (sqrt{3} + sqrt{2})^{3} – (sqrt{3} – sqrt{2})^{3})
((sqrt{3} + sqrt{2})^{3} = (sqrt{3})^{3} + 3(sqrt{3})^{2}(sqrt{2}) + 3(sqrt{3})(sqrt{2})^{2} + (sqrt{2})^{3})
= (3sqrt{3} + 9sqrt{2} + 6sqrt{3} + 2sqrt{2})
= (9sqrt{3} + 11sqrt{2})
((sqrt{3} – sqrt{2})^{3} = (sqrt{3})^{3} + 3(sqrt{3})^{2}(-sqrt{2}) + 3(sqrt{3})(-sqrt{2})^{2} + (-sqrt{2})^{3})
= (3sqrt{3} – 9sqrt{2} + 6sqrt{3} – 2sqrt{2})
= (9sqrt{3} – 11sqrt{2})
((sqrt{3} + sqrt{2}) Delta (sqrt{3} – sqrt{2}) = 9sqrt{3} + 11sqrt{2} – 9sqrt{3} + 11sqrt{3})
= (22sqrt{2})