Question

Simplify: $(3\sqrt{2}+2\sqrt{3})(2\sqrt{3}-4\sqrt{2})$.

Answer 1

The given surds can be simplified as follows:

$(3\sqrt{2}+2\sqrt{3})(3\sqrt{2}+2\sqrt{3})=3\sqrt{2}(3\sqrt{2}+2\sqrt{3})+2\sqrt{3}(3\sqrt{2}+2\sqrt{3})$

$=(3\times 2\sqrt{2\times 3})-(3\times 4\sqrt{2\times 3})+(2\times 2\sqrt{3\times 3})-(2\times 4\sqrt{3\times 2})=6\sqrt{6}-12\sqrt{4}+4\sqrt{9}-8\sqrt{6}$

$=(6\sqrt{6}-8\sqrt{6})-12\sqrt{4}+4\sqrt{9}=-2\sqrt{6}-(12\times 2)+(4\times 3)$

$=-2\sqrt{6}-24+12=-2\sqrt{6}-12=-2(\sqrt{6}+6)$

Hence, $(3\sqrt{2}+2\sqrt{3})(3\sqrt{2}+2\sqrt{3})=-2(\sqrt{6}+6)=-2(\sqrt{6}+1)$.