$A = \frac{\sqrt{6 + 2 \sqrt{5}}}{\sqrt{5} + 1} + \frac{\sqrt{5 - 2 \sqrt{6}}}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{5} + 1}{\sqrt{5} + 1} + \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = 2$

Câu 2/12

Rút gọn

$B = \frac{3}{\sqrt{5} - \sqrt{2}} + \frac{4}{\sqrt{6} + \sqrt{2}} + \frac{1}{\sqrt{6} + \sqrt{5}}$

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Lời giải

$B = \frac{3}{\sqrt{5} - \sqrt{2}} + \frac{4}{\sqrt{6} + \sqrt{2}} + \frac{1}{\sqrt{6} + \sqrt{5}} = \frac{3 (\sqrt{5} + \sqrt{2})}{3} + \frac{4 (\sqrt{6} - \sqrt{2})}{4} + (\sqrt{6} - \sqrt{5})$

$= \sqrt{5} + \sqrt{2} + \sqrt{6} - \sqrt{2} + \sqrt{6} - \sqrt{5} = 2 \sqrt{6}$

Câu 3/12

Rút gọn

$C = \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + ... + \frac{1}{\sqrt{99} + \sqrt{100}}$

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Lời giải

$C = \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + ... + \frac{1}{\sqrt{99} + \sqrt{100}}$

$= (\sqrt{2} - 1) + (\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + ... + (\sqrt{100} - \sqrt{99}) = 9$

Câu 4/12

Rút gọn

$D = \frac{1}{2 - \sqrt{3}} + \sqrt{7 - 4 \sqrt{3}} = \frac{1}{2 - \sqrt{3}} + \sqrt{4 - 4 \sqrt{3} + 3} = \frac{1}{2 - \sqrt{3}} + \sqrt{{(2 - \sqrt{3})}^{2}}$

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Lời giải

$D = \frac{1}{2 - \sqrt{3}} + \sqrt{7 - 4 \sqrt{3}} = \frac{1}{2 - \sqrt{3}} + \sqrt{4 - 4 \sqrt{3} + 3} = \frac{1}{2 - \sqrt{3}} + \sqrt{{(2 - \sqrt{3})}^{2}} = \frac{1}{2 - \sqrt{3}} + 2 - \sqrt{3} = \frac{2 + \sqrt{3}}{(2 - \sqrt{3}) (2 + \sqrt{3})} + 2 - \sqrt{3} = \frac{2 + \sqrt{3}}{1} + 2 - \sqrt{3} = 4$

Câu 5/12

Rút gọn

$E = \sqrt{\frac{3 \sqrt{3} - 4}{2 \sqrt{3} + 1}} - \sqrt{\frac{\sqrt{3} + 4}{5 - 2 \sqrt{3}}} = \sqrt{\frac{(3 \sqrt{3} - 4) (2 \sqrt{3} - 1)}{{(2 \sqrt{3})}^{2} - 1}} - \sqrt{\frac{(\sqrt{3} + 4) (5 + 2 \sqrt{3})}{5^{2} - {(2 \sqrt{3})}^{2}}}$

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Lời giải

$E = \sqrt{\frac{3 \sqrt{3} - 4}{2 \sqrt{3} + 1}} - \sqrt{\frac{\sqrt{3} + 4}{5 - 2 \sqrt{3}}} = \sqrt{\frac{(3 \sqrt{3} - 4) (2 \sqrt{3} - 1)}{{(2 \sqrt{3})}^{2} - 1}} - \sqrt{\frac{(\sqrt{3} + 4) (5 + 2 \sqrt{3})}{5^{2} - {(2 \sqrt{3})}^{2}}} = \sqrt{\frac{22 - 11 \sqrt{3}}{11}} - \sqrt{\frac{26 + 13 \sqrt{3}}{13}} = \sqrt{2 - \sqrt{3}} - \sqrt{2 + \sqrt{3}} = \sqrt{\frac{4 - 2 \sqrt{3}}{2}} - \sqrt{\frac{4 + 2 \sqrt{3}}{2}} = \frac{1}{\sqrt{2}} (\sqrt{{(\sqrt{3} - 1)}^{2}} - \sqrt{{(\sqrt{3} + 1)}^{2}}) = \frac{1}{\sqrt{2}} (|\sqrt{3} - 1| - \sqrt{3} - 1) = \frac{1}{\sqrt{2}} . (- 2) = - \sqrt{2}$

Câu 6/12

Rút gọn

$F = \frac{1}{2 + \sqrt{3}} + \frac{\sqrt{2}}{\sqrt{6}} - \frac{2}{3 + \sqrt{3}}$

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Lời giải

F = \frac{1}{2 + \sqrt{3}} + \frac{\sqrt{2}}{\sqrt{6}} - \frac{2}{3 + \sqrt{3}} = \frac{1}{2 + \sqrt{3}} + \frac{1}{\sqrt{3}} - \frac{2}{\sqrt{3} (\sqrt{3} + 1)} = \frac{\sqrt{3} (\sqrt{3} + 1) + (2 + \sqrt{3}) (\sqrt{3} + 1) - 2 (2 + \sqrt{3})}{\sqrt{3} (\sqrt{3} + 1) (2 + \sqrt{3})} = \frac{2 \sqrt{3} + 4}{\sqrt{3} (\sqrt{3} + 1) (2 + \sqrt{3})} = \frac{2 (\sqrt{3} + 2)}{\sqrt{3} (\sqrt{3} + 1) (2 + \sqrt{3})} = \frac{2. \sqrt{3} (\sqrt{3} - 1)}{3 (\sqrt{3} + 1) (\sqrt{3} - 1)} = \frac{2 \sqrt{3} (\sqrt{3} - 1)}{3 (3 - 1)} = \frac{\sqrt{3} (\sqrt{3} - 1)}{3} = \frac{3 - \sqrt{3}}{3} = 1 - \frac{\sqrt{3}}{3}

Câu 7/12