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) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})=(\sqrt{5})^{2}-(\sqrt{2})^{2}$
$=5-2=3$
(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) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})=(\sqrt{5})^{2}-(\sqrt{2})^{2}$
$=5-2=3$
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