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.
\(a \Delta b = a^{3} - b^{3}\)
\((\sqrt{3} + \sqrt{2}) \Delta (\sqrt{3} - \sqrt{2}) = (\sqrt{3} + \sqrt{2})^{3} - (\sqrt{3} - \sqrt{2})^{3}\)
\((\sqrt{3} + \sqrt{2})^{3} = (\sqrt{3})^{3} + 3(\sqrt{3})^{2}(\sqrt{2}) + 3(\sqrt{3})(\sqrt{2})^{2} + (\sqrt{2})^{3}\)
= \(3\sqrt{3} + 9\sqrt{2} + 6\sqrt{3} + 2\sqrt{2}\)
= \(9\sqrt{3} + 11\sqrt{2}\)
\((\sqrt{3} - \sqrt{2})^{3} = (\sqrt{3})^{3} + 3(\sqrt{3})^{2}(-\sqrt{2}) + 3(\sqrt{3})(-\sqrt{2})^{2} + (-\sqrt{2})^{3}\)
= \(3\sqrt{3} - 9\sqrt{2} + 6\sqrt{3} - 2\sqrt{2}\)
= \(9\sqrt{3} - 11\sqrt{2}\)
\((\sqrt{3} + \sqrt{2}) \Delta (\sqrt{3} - \sqrt{2}) = 9\sqrt{3} + 11\sqrt{2} - 9\sqrt{3} + 11\sqrt{3}\)
= \(22\sqrt{2}\)
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