$\displaystyle \begin{array}{{>{\displaystyle}l}} 4A.\\ a) \ \sqrt{49-12\sqrt{5}} -\sqrt{49+12\sqrt{5}}\\ =\sqrt{45-2.3\sqrt{5} .2+4} -\sqrt{45+2.3\sqrt{5} .2+4}\\ =\sqrt{\left( 3\sqrt{5} -2\right)^{2}} -\sqrt{\left( 3\sqrt{5} +2\right)^{2}}\\ =|3\sqrt{5} -2|-|3\sqrt{5} +2|\\ =3\sqrt{5} -2-3\sqrt{5} -2\\ =-4\\ b) \ \sqrt{29+12\sqrt{5}} -\sqrt{29-12\sqrt{5}}\\ =\sqrt{20+2.2\sqrt{5} .3+9} -\sqrt{20-2.2\sqrt{5} .3+9}\\ =\sqrt{\left( 2\sqrt{5} +3\right)^{2}} -\sqrt{\left( 2\sqrt{5} -3\right)^{2}}\\ =2\sqrt{5} +3-2\sqrt{5} +3\\ =6\ \\ 4B\ \\ a) \ \sqrt{7+4\sqrt{3}} -\sqrt{7-4\sqrt{3}}\\ =\sqrt{4+2.2\sqrt{3} +3} -\sqrt{4-2.2\sqrt{3} +3}\\ =\sqrt{\left( 2+\sqrt{3}\right)^{2}} -\sqrt{\left( 2-\sqrt{3}\right)^{2}}\\ =2+\sqrt{3} -2+\sqrt{3}\\ =2\sqrt{3} \ \\ b) \ \sqrt{41-12\sqrt{5}} -\sqrt{41+12\sqrt{5}}\\ =\sqrt{36-2.6\sqrt{5} +5} -\sqrt{36-2.6\sqrt{5} +5}\\ =\sqrt{\left( 6-\sqrt{5}\right)^{2}} -\sqrt{\left( 6+\sqrt{5}\right)^{2}}\\ =6-\sqrt{5} -6-\sqrt{5}\\ =-2\sqrt{5}\\ \end{array}$
