WAEC 2015 · Paper 2 · Q1

(a)
A binary operation $\Delta$ is defined on the set of real numbers $\mathbb R$ by $a\,\Delta\,b = a^3 - b^3$ . Without using a calculator, find the value of $(\sqrt3 + \sqrt2)\,\Delta\,(\sqrt3 - \sqrt2)$ , leaving the answer in surd form.

Worked solution (try it first)

By the definition, the answer is $(\sqrt3 + \sqrt2)^3 - (\sqrt3 - \sqrt2)^3$ .
Expand with $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ : $(\sqrt3)^3 = 3\sqrt3$ , $3(\sqrt3)^2\sqrt2 = 9\sqrt2$ , $3\sqrt3(\sqrt2)^2 = 6\sqrt3$ and $(\sqrt2)^3 = 2\sqrt2$ .
So $(\sqrt3 + \sqrt2)^3 = 3\sqrt3 + 9\sqrt2 + 6\sqrt3 + 2\sqrt2$
$= 9\sqrt3 + 11\sqrt2$ .
With a minus sign, the terms with an odd power of $\sqrt2$ change sign: $(\sqrt3 - \sqrt2)^3 = 9\sqrt3 - 11\sqrt2$ .
Subtract: $(9\sqrt3 + 11\sqrt2) - (9\sqrt3 - 11\sqrt2) = 22\sqrt2$ .