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19) $\frac{1}{3+ \sqrt{7} } + \frac{1}{3- \sqrt{7} }= \frac{(3- \sqrt{7})+(3+ \sqrt{7})}{(3- \sqrt{7})(3+ \sqrt{7})} = \frac{3- \sqrt{7}+3+ \sqrt{7}}{3^2-7} = \frac{6}{9-7}=3$
20) $\frac{5- \sqrt{15} }{ \sqrt{15} -3} = \frac{(5- \sqrt{15})(\sqrt{15} +3)}{(\sqrt{15} -3)(\sqrt{15} +3)} = \frac{5 \sqrt{15}-15+15-3 \sqrt{15} }{15-9}= \frac{2\sqrt{15}}{6}= \frac{ \sqrt{15} }{3}$
23) $(1+ \frac{1}{4} )(1+ \frac{1}{5} )(1+ \frac{1}{6} )(1+ \frac{1}{7} )(1+ \frac{1}{8} )(1+ \frac{1}{9} )= \frac{5}{4} \frac{6}{5} \frac{7}{6} \frac{8}{7} \frac{9}{8}\frac{10}{9}= \frac{10}{4}=2 \frac{1}{2}$
24) $\frac{1}{9-4 \sqrt{5} } + \frac{1}{9+4 \sqrt{5} }= \frac{(9+4 \sqrt{5})+(9-4 \sqrt{5})}{(9-4 \sqrt{5})(9+4 \sqrt{5})} = \frac{9+4 \sqrt{5}+9-4 \sqrt{5}}{9^2-4^2*5} = \frac{9+9}{81-16*5} =18$