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2. 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)\((\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})\)
Answer :

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, (\(a + b)(a - b) = a^2 - b^2\)
i.e., \(= 9 - 3\)
Therefore, \((3 + \sqrt{3})(3 - \sqrt{3}) = 6\)

iii)\(= (\sqrt{5} + \sqrt{2})^2\)
\(= (\sqrt{5})^2 + 2 × \sqrt{5}\sqrt{2} + (\sqrt{2})^2\)
Because, We know that, \((a + b)^2 = a^2 + 2 × a × b + b^2\)
i.e., \(= 5 + 2\sqrt{10} + 2\)
Therefore, \( (\sqrt{5} + \sqrt{2})(\sqrt{5} + \sqrt{2}) = 7 + 2\sqrt{10}\)

iv)= \((\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})\)
\(= (\sqrt{5})^2 - (\sqrt{2})^2\)
Because, We know that, \((a + b)(a - b) = a^2 - b^2\)
i.e., \(= 5 - 2\)
Therefore, \((\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2}) = 3\)