Question

Please Solve this question correctly and step by step​

Answer 1

Answer:

$(a) \frac{1}{ \sqrt{3} + \sqrt{2} - 1 } \\ rationalising \: the \: denominator, \\ = \frac{1}{\sqrt{3} + ( \sqrt{2} - 1)} \times \frac{\sqrt{3} - ( \sqrt{2} - 1)}{\sqrt{3} - ( \sqrt{2} - 1)} \\ = \frac{\sqrt{3} - ( \sqrt{2} - 1)}{ { (\sqrt{3}) }^{2} - {( \sqrt{2} - 1) }^{2} } \\ = \frac{ \sqrt{3} - \sqrt{2} + 1 }{3 - 2 - 1 + 2 \sqrt{2} } \\ = \frac{ \sqrt{3} - \sqrt{2} + 1 }{ 2 \sqrt{2} } \\ = \frac{ \sqrt{3} - \sqrt{2} + 1 }{ 2 \sqrt{2} } \times \frac{ \sqrt{2} }{ \sqrt{2} } \\ = \frac{ \sqrt{2} (\sqrt{3} - \sqrt{2} + 1 )}{ 4}$

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$(b) \frac{1}{3 -2 \sqrt{2} + \sqrt{5} } \\ = \frac{1}{ (\sqrt{5 } + 3) - 2 \sqrt{2} } \times \frac{(\sqrt{5 } + 3) + 2 \sqrt{2} }{(\sqrt{5 } + 3) + 2 \sqrt{2} } \\ = \frac{\sqrt{5 } + 3 + 2 \sqrt{2} }{ { (\sqrt{5} + 3) }^{2} - ({2 \sqrt{2} )}^{2} } \\ = \frac{\sqrt{5 } + 3 + 2 \sqrt{2}}{5 + 9 + 6 \sqrt{5} - 8 } \\ = \frac{\sqrt{5 } + 3 + 2 \sqrt{2}}{6 - 6 \sqrt{5} } \\ = \frac{\sqrt{5 } + 3 + 2 \sqrt{2}}{6(1 - \sqrt{5}) } \times \frac{1 + \sqrt{5} }{1 + \sqrt{5} } \\ = \frac{(\sqrt{5 } + 3 + 2 \sqrt{2})(1 + \sqrt{5} )}{6(1 - 5)} \\ = \frac{ \sqrt{5} + 3 + 2 \sqrt{2} + 5 + 3\sqrt{5} + 2 \sqrt{10} }{ - 24} \\ = \frac{2 \sqrt{2} + 2 \sqrt{10} + 4 \sqrt{5} + 8 }{ - 24}$

Answer 2

Answer:

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