Rút gọn biểu thức
----- Nội dung ảnh -----
VD2: N.
\( \frac{4 + \sqrt{3}}{4 - \sqrt{3}} \) = \( \frac{\sqrt{27}-10\sqrt{2}}{\sqrt{4+\sqrt{3}}} \)
70°: \( \gamma = \sqrt{2} \left( \sqrt{4+\sqrt{3}} - \sqrt{4-\sqrt{3}} \right) \)
= \( 8 + 2\sqrt{13} \)
= \( -\sqrt{2} \left( \sqrt{4+\sqrt{3}} + \sqrt{4-\sqrt{3}} \right) \)
= \( \sqrt{(4+\sqrt{3}) + 2\sqrt{4+\sqrt{3}}\cdot\sqrt{4-\sqrt{3}} + (4+\sqrt{3})} \)
= \( -\frac{\sqrt{2} \left( \sqrt{4+\sqrt{3}} + \sqrt{4-\sqrt{3}} \right)}{\sqrt{(4 + \sqrt{3}) \cdot (4 - \sqrt{3})}} \)
= \( \sqrt{2 \left( \sqrt{4+\sqrt{3}} + \sqrt{4-\sqrt{3}} \right)} + \sqrt{15 - \sqrt{2}}^2 \)
= \( \sqrt{4+\sqrt{3}} + \sqrt{4-\sqrt{3}} \)
= \( \sqrt{2 + 5 - \sqrt{2}} = 5 \)
VD2: N.
\( \frac{4 + \sqrt{3}}{4 - \sqrt{3}} \) = \( \frac{\sqrt{27}-10\sqrt{2}}{\sqrt{4+\sqrt{3}}} \)
70°: \( \gamma = \sqrt{2} \left( \sqrt{4+\sqrt{3}} - \sqrt{4-\sqrt{3}} \right) \)
= \( 8 + 2\sqrt{13} \)
= \( -\sqrt{2} \left( \sqrt{4+\sqrt{3}} + \sqrt{4-\sqrt{3}} \right) \)
= \( \sqrt{(4+\sqrt{3}) + 2\sqrt{4+\sqrt{3}}\cdot\sqrt{4-\sqrt{3}} + (4+\sqrt{3})} \)
= \( -\frac{\sqrt{2} \left( \sqrt{4+\sqrt{3}} + \sqrt{4-\sqrt{3}} \right)}{\sqrt{(4 + \sqrt{3}) \cdot (4 - \sqrt{3})}} \)
= \( \sqrt{2 \left( \sqrt{4+\sqrt{3}} + \sqrt{4-\sqrt{3}} \right)} + \sqrt{15 - \sqrt{2}}^2 \)
= \( \sqrt{4+\sqrt{3}} + \sqrt{4-\sqrt{3}} \)
= \( \sqrt{2 + 5 - \sqrt{2}} = 5 \)