Simplify each of the following expressions: <br/> <br/> (i) $(3+\sqrt{3})(2+\sqrt{2})$ <br/> <br/>(ii) $(3+\sqrt{3})(3-\sqrt{3})$ <br/> <br/>(iii) $(\sqrt{5}+\sqrt{2})^{2}$ <br/> <br/>(iv) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$

Solution:

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

$=6+3 \sqrt{2}+2 \sqrt{3}+\sqrt{6}$

(ii) $(3+\sqrt{3})(3-\sqrt{3})=(3)^{2}-(\sqrt{3})^{2}$

$=9-3=6$

(iii) $(\sqrt{5}+\sqrt{2})^{2}=(\sqrt{5})^{2}+(\sqrt{2})^{2}+2(\sqrt{5})(\sqrt{2})$

$=5+2+2 \sqrt{10}=7+2 \sqrt{10}$

(iv) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})=(\sqrt{5})^{2}-(\sqrt{2})^{2}$

$=5-2=3$