[tex]a) \ \frac{10}{\sqrt{3}+1} = \frac{10(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)} = \frac{10(\sqrt{3}-1)}{3-1} =\frac{10(\sqrt{3}-1)}{2}=5(\sqrt{3}-1) = 5\sqrt{3}-5\\\\b) \ \frac{1}{\sqrt{2}-1} = \frac{1(\sqrt{2}+1)}{(\sqrt{2}-1)(\sqrt{2}+1)} = \frac{\sqrt{2}+1}{2-1} =\frac{\sqrt{2}+1}{1} =\sqrt{2}+1\\\\c) \ \frac{4}{\sqrt{5}+1}=\frac{4(\sqrt{5}-1)}{(\sqrt{5}+1)(\sqrt{5}-1)}=\frac{4(\sqrt{5}-1)}{5-1} =\frac{4(\sqrt{5}-1)}{4} = \sqrt{5}-1[/tex]
[tex]d) \ \frac{2}{\sqrt{5}-3} = \frac{2(\sqrt{5}+3)}{(\sqrt{5}-3)(\sqrt{5}+3)} =\frac{2(\sqrt{5}+3)}{5-9} =\frac{2(\sqrt{5}+3)}{-4} = \frac{-\sqrt{5}-3}{2}\\\\e) \ \frac{\sqrt{2}}{\sqrt{2}02} = \frac{\sqrt{2}(\sqrt{2}+2)}{(\sqrt{2}-2)(\sqrt{2}+2)} = \frac{2+2\sqrt{2}}{(\sqrt{2}-2)(\sqrt{2}+2)} = \frac{2(1+\sqrt{2})}{2-4} = \frac{-2(-1-\sqrt{2})}{-2} = -1-\sqrt{2}[/tex]
