Đáp án:

$4)\\ a)1\\ b)5+3\sqrt{2}\\ c)1\\ d)2\sqrt{3}\\ e)\sqrt{6}$

Giải thích các bước giải:

$4)\\ a)\sqrt{\sqrt{5}-\sqrt{3-\sqrt{29-12\sqrt{5}}}}\\ =\sqrt{\sqrt{5}-\sqrt{3-\sqrt{20-2.2\sqrt{5}.3+9}}}\\ =\sqrt{\sqrt{5}-\sqrt{3-\sqrt{(2\sqrt{5}-3)^2}}}\\ =\sqrt{\sqrt{5}-\sqrt{3-(2\sqrt{5}-3)}}\\ =\sqrt{\sqrt{5}-\sqrt{6-2\sqrt{5}}}\\ =\sqrt{\sqrt{5}-\sqrt{5-2\sqrt{5}+1}}\\ =\sqrt{\sqrt{5}-\sqrt{(\sqrt{5}-1)^2}}\\ =\sqrt{\sqrt{5}-(\sqrt{5}-1)}\\ =\sqrt{\sqrt{5}-\sqrt{5}+1}\\ =1\\ b)\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{8+2.2\sqrt{2}.1+1}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{8+2.2\sqrt{2}.1+1}}}\\ =\sqrt{13+30\sqrt{2+\sqrt{(2\sqrt{2}+1)^2}}}\\ =\sqrt{13+30\sqrt{2+(2\sqrt{2}+1)}}\\ =\sqrt{13+30\sqrt{2+2\sqrt{2}+1}}\\ =\sqrt{13+30\sqrt{(\sqrt{2}+1)^2}}\\ =\sqrt{13+30(\sqrt{2}+1)}\\ =\sqrt{43+30\sqrt{2}}\\ =\sqrt{25+2.5.3\sqrt{2}+18}\\ =\sqrt{(5+3\sqrt{2})^2}\\ =5+3\sqrt{2}\\ c)(\sqrt{3}-\sqrt{2})\sqrt{5+2\sqrt{6}}\\ =(\sqrt{3}-\sqrt{2})\sqrt{2+2.\sqrt{2}.\sqrt{3}+3}\\ =(\sqrt{3}-\sqrt{2})\sqrt{(\sqrt{2}+\sqrt{3})^2}\\ =(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})\\ =(=\sqrt{3}^2-\sqrt{2}^2\\ =3-2\\ =1\\ d)\sqrt{5-\sqrt{13+4\sqrt{3}}}+\sqrt{3+\sqrt{13+4\sqrt{3}}}\\ =\sqrt{5-\sqrt{12+2.2\sqrt{3}.1+1}}+\sqrt{3+\sqrt{12+2.2\sqrt{3}.1+1}}\\ =\sqrt{5-\sqrt{(2\sqrt{3}+1)^2}}+\sqrt{3+\sqrt{(2\sqrt{3}+1)^2}}\\ =\sqrt{5-(2\sqrt{3}+1)}+\sqrt{3+(2\sqrt{3}+1)}\\ =\sqrt{4-2\sqrt{3}}+\sqrt{4+2\sqrt{3}}\\ =\sqrt{3-2\sqrt{3}+1}+\sqrt{3+2\sqrt{3}+1}\\ =\sqrt{(\sqrt{3}-1)^2}+\sqrt{(\sqrt{3}+1)^2}\\ =\sqrt{3}-1\sqrt{3}+1\\ =2\sqrt{3}\\ e)\sqrt{1+\sqrt{3+\sqrt{13+4\sqrt{3}}}}+\sqrt{1-\sqrt{3-\sqrt{13-4\sqrt{3}}}}\\ =\sqrt{1+\sqrt{3+\sqrt{12+2.2\sqrt{3}+1}}}+\sqrt{1-\sqrt{3-\sqrt{12-2.2\sqrt{3}+1}}}\\ =\sqrt{1+\sqrt{3+\sqrt{(2\sqrt{3}+1)^2}}}+\sqrt{1-\sqrt{3-\sqrt{(2\sqrt{3}-1)^2}}}\\ =\sqrt{1+\sqrt{3+(2\sqrt{3}+1)}}+\sqrt{1-\sqrt{3-(2\sqrt{3}-1)}}\\ =\sqrt{1+\sqrt{3+2\sqrt{3}+1}}+\sqrt{1-\sqrt{3-2\sqrt{3}+1}}\\ =\sqrt{1+\sqrt{(\sqrt{3}+1)^2}}+\sqrt{1-\sqrt{(\sqrt{3}-1)^2}}\\ =\sqrt{1+(\sqrt{3}+1)}+\sqrt{1-(\sqrt{3}-1)}\\ =\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\\ =\dfrac{\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}}{\sqrt{2}}\\ =\dfrac{\sqrt{3+2\sqrt{3}+1}+\sqrt{3-2\sqrt{3}+1}}{\sqrt{2}}\\ =\dfrac{\sqrt{(\sqrt{3}+1)^2}+\sqrt{(\sqrt{3}-1)^2}}{\sqrt{2}}\\ =\dfrac{\sqrt{3}+1+\sqrt{3}-1}{\sqrt{2}}\\ =\dfrac{2\sqrt{3}}{\sqrt{2}}\\ =\sqrt{6}$