12 câu hỏi trắc nghiệm thuộc Dạng 4: Rút gọn tổng hợp (sử dụng trục căn thức, hằng đẳng thức, phân tích thành nhân tử; …)

Câu 1:

Rút gọn:

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

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

Câu 2:

Rút gọn

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

Xem đáp án

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

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

Câu 3:

Rút gọn

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

Xem đáp án

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

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

Câu 4:

Rút gọn

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

Xem đáp án

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

Câu 5:

Rút gọn

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

Xem đáp án

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

Câu 6:

Rút gọn

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

Xem đáp án

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:

Rút gọn

$A = (\sqrt{3} + 4) \sqrt{19 - 8 \sqrt{3}}$

Xem đáp án

$A = (\sqrt{3} + 4) \sqrt{19 - 8 \sqrt{3}} = (\sqrt{3} + 4) \sqrt{{(4 - \sqrt{3})}^{2}} = (\sqrt{3} + 4) (4 - \sqrt{3}) = 16 - 3 = 13$

Câu 8:

Rút gọn

$B = (\sqrt{5} - 2) (\sqrt{5} + 2) - \frac{\sqrt{7 - 4 \sqrt{3}}}{\sqrt{3} - 2}$

Xem đáp án

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

Câu 9:

Rút gọn

$C = \frac{\sqrt{7 + \sqrt{5}} + \sqrt{7 - \sqrt{5}}}{\sqrt{7 + 2 \sqrt{11}}} - \sqrt{3 - 2 \sqrt{2}}$

Xem đáp án

$C = \frac{\sqrt{7 + \sqrt{5}} + \sqrt{7 - \sqrt{5}}}{\sqrt{7 + 2 \sqrt{11}}} - \sqrt{3 - 2 \sqrt{2}}$

Ta có: ${(\frac{\sqrt{7 + \sqrt{5}} + \sqrt{7 - \sqrt{5}}}{\sqrt{7 + 2 \sqrt{11}}})}^{2} = \frac{14 + 2 \sqrt{44}}{7 + 2 \sqrt{11}} = 2 \Rightarrow \frac{\sqrt{7 + \sqrt{5}} + \sqrt{7 - \sqrt{5}}}{\sqrt{7 + 2 \sqrt{11}}} = \sqrt{2}$ $\Rightarrow C = \sqrt{2} - \sqrt{3 - 2 \sqrt{2}} = \sqrt{2} - \sqrt{{(\sqrt{2} - 1)}^{2}} = \sqrt{2} - \sqrt{2} + 1 = 1$

Câu 10:

Rút gọn

$D = \sqrt{\frac{4}{{(2 - \sqrt{5})}^{2}}} - \sqrt{\frac{4}{{(2 + \sqrt{5})}^{2}}}$

Xem đáp án

$D = \sqrt{\frac{4}{{(2 - \sqrt{5})}^{2}}} - \sqrt{\frac{4}{{(2 + \sqrt{5})}^{2}}} = \frac{\sqrt{2^{2}}}{\sqrt{{(2 - \sqrt{5})}^{2}}} - \frac{\sqrt{2^{2}}}{\sqrt{{(2 + \sqrt{5})}^{2}}} = \frac{2}{|2 - \sqrt{5}|} - \frac{2}{|2 + \sqrt{5}|} = \frac{2}{\sqrt{5} - 2} - \frac{2}{\sqrt{5} + 2} = \frac{2 (\sqrt{5} + 2) - 2 (\sqrt{5} - 2)}{(\sqrt{5} + 2) (\sqrt{5} - 2)} = \frac{2 \sqrt{5} + 4 - 2 \sqrt{5} + 4}{5 - 4} = 8$

Câu 11:

Rút gọn

$E = \frac{\sqrt{8 - \sqrt{15}}}{\sqrt{30} - \sqrt{2}}$

Xem đáp án

$E = \frac{\sqrt{8 - \sqrt{15}}}{\sqrt{30} - \sqrt{2}} = \frac{1}{\sqrt{15} - 1} . \sqrt{\frac{16 - 2 \sqrt{15}}{4}} = \frac{1}{\sqrt{15} - 1} . \frac{\sqrt{15} - 1}{2} = \frac{1}{2}$

Câu 12:

Rút gọn

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

Xem đáp án

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