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# Simplify each of the following expressions: i) $$(3 + \sqrt{3})(2 + \sqrt{2})$$ii) $$(3 + \sqrt{3})(3 - \sqrt{3})$$iii) $$(\sqrt{5} + \sqrt{2})(\sqrt{5} + \sqrt{2})$$iv)$$(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})$$

i) $$(3 + \sqrt{3})(2 + \sqrt{2})$$
$$\therefore$$ $$(3 + \sqrt{3})(2 + \sqrt{2})$$ $$= 6 + 3\sqrt{2} + 2\sqrt{3} + 6$$

ii) $$(3 + \sqrt{3})(3 - \sqrt{3})$$)
$$= 3^2 - (\sqrt{3})^2$$
Because, We know that,
($$a + b)(a - b) = a^2 - b^2$$
$$\therefore (3 + \sqrt{3})(3 - \sqrt{3})$$
$$= 9 -3 = 6$$

iii)$$(\sqrt{5} + \sqrt{2})^2$$
$$= (\sqrt{5})^2 + 2 × \sqrt{5}\sqrt{2} + (\sqrt{2})^2$$
Because, We know that, $$(a + b)^2 = a^2 + 2 × a × b + b^2$$
$$= 5 + 2\sqrt{10} + 2$$
$$\therefore (\sqrt{5} + \sqrt{2})^2 = 7 + 2\sqrt{10}$$

iv) $$(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})$$
$$= (\sqrt{5})^2 - (\sqrt{2})^2$$
Because, We know that, $$(a + b)(a - b) = a^2 - b^2$$
$$\therefore$$ $$(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})$$
$$= 5 -2 = 3$$