Rút gọn:
\(\dfrac{2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}\)
= \(\dfrac{2\sqrt{3+\sqrt{5-\sqrt{13+4\sqrt{3}}}}}{\sqrt{6}+\sqrt{2}}\)
= \(\dfrac{2\sqrt{3+\sqrt{5-\sqrt{\left(1+2\sqrt{3}\right)^2}}}}{\sqrt{6}+\sqrt{2}}\)
= \(\dfrac{2\sqrt{3+\sqrt{5-\left(1+2\sqrt{3}\right)}}}{\sqrt{6}+\sqrt{2}}\)
= \(\dfrac{\sqrt{3+\sqrt{5-\left(1+2\sqrt{3}\right)}}.\left(\sqrt{6}-\sqrt{2}\right)}{2}\)
= \(\dfrac{\sqrt{\left[3+\sqrt{5-\left(1+2\sqrt{3}\right)}\right].6}-\sqrt{\left[3+\sqrt{5-\left(1+2\sqrt{3}\right)}\right].2}}{2}\)
= \(\dfrac{\sqrt{\left(3+\sqrt{5-1-2\sqrt{3}}\right).6}-\sqrt{\left(3+\sqrt{5-1-2\sqrt{3}}\right).2}}{2}\)
= \(\dfrac{\sqrt{\left(3+\sqrt{4-2\sqrt{3}}\right).6}-\sqrt{\left(3+\sqrt{4-2\sqrt{3}}\right).2}}{2}\)
= \(\dfrac{\sqrt{\left[3+\sqrt{\left(1-\sqrt{3}\right)^2}\right].6}-\sqrt{\left[3+\sqrt{\left(1-\sqrt{3}\right)^2}\right].2}}{2}\)
= \(\dfrac{\sqrt{\left(3+\sqrt{3}-1\right).6}-\sqrt{\left(3+\sqrt{3}-1\right).2}}{2}\)
= \(\dfrac{\sqrt{\left(2+\sqrt{3}\right).6}-\sqrt{\left(2+\sqrt{3}\right).2}}{2}\)
= \(\dfrac{\sqrt{12+6\sqrt{3}}-\sqrt{4+2\sqrt{3}}}{2}\)
= \(\dfrac{\sqrt{\left(3+\sqrt{3}\right)^2}-\sqrt{\left(1+\sqrt{3}\right)^2}}{2}\)
= \(\dfrac{3+\sqrt{3}-\left(1+\sqrt{3}\right)}{2}\)
= \(\dfrac{3+\sqrt{3}-1-\sqrt{3}}{2}\)
= \(\dfrac{2}{2}\)
= \(1\)
