Đáp án:

$2)5+3\sqrt{2}\\ 3)\sqrt{3}+1\\ 4)3\\ 5)2\\ 6)\sqrt{3}+1\\ 13)\sqrt{10}-1$

Giải thích các bước giải:

$2)\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{(2\sqrt{2}+1)^2}}}\\ =\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}\\ 3)\sqrt{6+2\sqrt{5-\sqrt{13+\sqrt{48}}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{13+2\sqrt{12}}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{12+2\sqrt{12}+1}}}\\ =\sqrt{6+2\sqrt{5-\sqrt{(\sqrt{12}+1)^2}}}\\ =\sqrt{6+2\sqrt{5-(\sqrt{12}+1)}}\\ =\sqrt{6+2\sqrt{4-\sqrt{12}}}\\ =\sqrt{6+2\sqrt{4-2\sqrt{3}}}\\ =\sqrt{6+2\sqrt{3-2\sqrt{3}+1}}\\ =\sqrt{6+2\sqrt{(\sqrt{3}-1)^2}}\\ =\sqrt{6+2(\sqrt{3}-1)}\\ =\sqrt{4+2\sqrt{3}}\\ =\sqrt{3+2\sqrt{3}+1}\\ =\sqrt{(\sqrt{3}+1)^2}\\ =\sqrt{3}+1\\ 4)\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+4\sqrt{3}}}}}\\ =\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{4+2.2\sqrt{3}+3}}}}\\ =\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{(2+\sqrt{3})^2}}}}\\ =\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{48-10(2+\sqrt{3})}}}\\ =\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{28-10\sqrt{3}}}}\\ =\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{25-2.5.\sqrt{3}+3}}}\\ =\sqrt{4+\sqrt{5\sqrt{3}+5\sqrt{(5-\sqrt{3})^2}}}\\ =\sqrt{4+\sqrt{5\sqrt{3}+5(5-\sqrt{3})}}\\ =\sqrt{4+\sqrt{25}}\\ =\sqrt{4+5}\\ =\sqrt{9}\\ =3\\ 5)\sqrt{(\sqrt{5}-1)\sqrt{13-\sqrt{69-28\sqrt{5}}}}\\ =\sqrt{(\sqrt{5}-1)\sqrt{13-\sqrt{49-2.7.2\sqrt{5}+20}}}\\ =\sqrt{(\sqrt{5}-1)\sqrt{13-\sqrt{(7-2\sqrt{5})^2}}}\\ =\sqrt{(\sqrt{5}-1)\sqrt{13-(7-2\sqrt{5})}}\\ =\sqrt{(\sqrt{5}-1)\sqrt{6+2\sqrt{5}}}\\ =\sqrt{(\sqrt{5}-1)\sqrt{5+2\sqrt{5}+1}}\\ =\sqrt{(\sqrt{5}-1)\sqrt{(\sqrt{5}+1)^2}}\\ =\sqrt{(\sqrt{5}-1)(\sqrt{5}+1)}\\ =\sqrt{5-1}\\ =\sqrt{4}\\ =2\\ 6)\sqrt{6+2\sqrt{6-2\sqrt{4+2\sqrt{3}}}}\\ =\sqrt{6+2\sqrt{6-2\sqrt{3+2\sqrt{3}+1}}}\\ =\sqrt{6+2\sqrt{6-2\sqrt{(\sqrt{3}+1)^2}}}\\ =\sqrt{6+2\sqrt{6-2(\sqrt{3}+1)}}\\ =\sqrt{6+2\sqrt{4-2\sqrt{3}}}\\ =\sqrt{6+2\sqrt{3-2\sqrt{3}+1}}\\ =\sqrt{6+2\sqrt{(\sqrt{3}-1)^2}}\\ =\sqrt{6+2(\sqrt{3}-1)}\\ =\sqrt{4+2\sqrt{3}}\\ =\sqrt{3+2\sqrt{3}+1}\\ =\sqrt{(\sqrt{3}+1)^2}\\ =\sqrt{3}+1\\ 13)\sqrt{3-2\sqrt{2}}+\sqrt{5-2\sqrt{6}}+\cdots + \sqrt{19-2\sqrt{90}}\\ =\sqrt{2-2\sqrt{2}+1}+\sqrt{3-2.\sqrt{3}.\sqrt{2}+2}+\cdots + \sqrt{10-2.\sqrt{10}.\sqrt{9}+9}\\ =\sqrt{(\sqrt{2}-1)^2}+\sqrt{(\sqrt{3}-\sqrt{2})^2}+\cdots + \sqrt{(\sqrt{10}-\sqrt{9})^2}\\ =\sqrt{2}-1+\sqrt{3}-\sqrt{2}+\cdots + \sqrt{10}-\sqrt{9}\\ =\sqrt{10}-1$