Đáp án:
$A=0\\ B=8\\ C=9-a\\ D=0$
Giải thích các bước giải:
$A=\sqrt{\sqrt{2}-1}+\sqrt{\sqrt{2}+1}-\sqrt{2\sqrt{2}+2}\\ =\sqrt{\sqrt{2}-1}+\sqrt{\sqrt{2}+1}-\sqrt{2}\sqrt{\sqrt{2}+1}\\ =\sqrt{\sqrt{2}-1}+(1-\sqrt{2})\sqrt{\sqrt{2}+1}\\ =\sqrt{\sqrt{2}-1}-(\sqrt{2}-1)\sqrt{\sqrt{2}+1}\\ =\sqrt{\sqrt{2}-1}-\sqrt{\sqrt{2}-1}^2\sqrt{\sqrt{2}+1}\\ =\sqrt{\sqrt{2}-1}-\sqrt{\sqrt{2}-1}\sqrt{\sqrt{2}-1}\sqrt{\sqrt{2}+1}\\ =\sqrt{\sqrt{2}-1}-\sqrt{\sqrt{2}-1}\sqrt{(\sqrt{2}-1)(\sqrt{2}+1)}\\ =\sqrt{\sqrt{2}-1}-\sqrt{\sqrt{2}-1}\sqrt{2-1}\\ =\sqrt{\sqrt{2}-1}-\sqrt{\sqrt{2}-1}\\ =0\\ B=(5+\sqrt{21})(\sqrt{14}-\sqrt{6})\sqrt{5-\sqrt{21}}\\ =(5+\sqrt{21})(\sqrt{7}-\sqrt{3})\sqrt{10-2\sqrt{21}}\\ =(5+\sqrt{21})(\sqrt{7}-\sqrt{3})\sqrt{7-2.\sqrt{7}.\sqrt{3}+3}\\ =(5+\sqrt{21})(\sqrt{7}-\sqrt{3})\sqrt{(\sqrt{7}-\sqrt{3})^2}\\ =(5+\sqrt{21})(\sqrt{7}-\sqrt{3})(\sqrt{7}-\sqrt{3})\\ =(5+\sqrt{21})(\sqrt{7}-\sqrt{3})^2\\ =(5+\sqrt{21})(7-2\sqrt{21}+3)\\ =(5+\sqrt{21})(10-2\sqrt{21})\\ =2(5+\sqrt{21})(5-\sqrt{21})\\ =2(25-21)\\ =2.4\\ =8\\ C=\left(3+\dfrac{a-4\sqrt{a}}{4-\sqrt{a}}\right)\left(3+\dfrac{a-5\sqrt{a}}{\sqrt{a}-5}\right) (a \ne 16, a \ne 25, a \ge 0)\\ =\left(3+\dfrac{\sqrt{a}(\sqrt{a}-4)}{4-\sqrt{a}}\right)\left(3+\dfrac{\sqrt{a}(\sqrt{a}-5)}{\sqrt{a}-5}\right) \\ =\left(3-\sqrt{a}\right)\left(3+\sqrt{a}\right) \\ =9-a\\ D=\dfrac{2\sqrt{24}-\sqrt{120}}{4\sqrt{5}-8}+\dfrac{3\sqrt{2}+\sqrt{30}}{\sqrt{20}+\sqrt{12}}\\ =\dfrac{2.2\sqrt{6}-2\sqrt{30}}{4\sqrt{5}-8}+\dfrac{3\sqrt{2}+\sqrt{30}}{2\sqrt{5}+2\sqrt{3}}\\ =\dfrac{2\sqrt{6}-\sqrt{30}}{2\sqrt{5}-4}+\dfrac{3\sqrt{2}+\sqrt{30}}{2\sqrt{5}+2\sqrt{3}}\\ =\dfrac{\sqrt{6}(2-\sqrt{5})}{2(\sqrt{5}-2)}+\dfrac{\sqrt{6}(\sqrt{3}+\sqrt{5})}{2(\sqrt{5}+\sqrt{3})}\\ =-\dfrac{\sqrt{6}}{2}+\dfrac{\sqrt{6}}{2}\\ =0$
