104. Освободитесь от иррациональности в знаменателе дроби: 1) 4/√15; 2) 8/√2; 3) 42/5√7; 4) m⁴/n√m; 5) a+6/√a +6; 6) 1/√11-1; 7) 14/√17+ √3; 8) 15/√43-√13.

1) $$ \frac{4}{\sqrt{15}} = \frac{4 \sqrt{15}}{15} $$

2) $$ \frac{8}{\sqrt{2}} = \frac{8 \sqrt{2}}{2} = 4 \sqrt{2} $$

3) $$ \frac{42}{5\sqrt{7}} = \frac{42 \sqrt{7}}{5 \cdot 7} = \frac{6 \sqrt{7}}{5} $$

4) $$ \frac{m^4}{n\sqrt{m}} = \frac{m^4 \sqrt{m}}{nm} = \frac{m^3 \sqrt{m}}{n} $$

5) $$ \frac{a+6}{\sqrt{a+6}} = \frac{(a+6)\sqrt{a+6}}{a+6} = \sqrt{a+6} $$

6) $$ \frac{1}{\sqrt{11}-1} = \frac{\sqrt{11}+1}{(\sqrt{11}-1)(\sqrt{11}+1)} = \frac{\sqrt{11}+1}{11-1} = \frac{\sqrt{11}+1}{10} $$

7) $$ \frac{14}{\sqrt{17}+\sqrt{3}} = \frac{14(\sqrt{17}-\sqrt{3})}{(\sqrt{17}+\sqrt{3})(\sqrt{17}-\sqrt{3})} = \frac{14(\sqrt{17}-\sqrt{3})}{17-3} = \frac{14(\sqrt{17}-\sqrt{3})}{14} = \sqrt{17}-\sqrt{3} $$

8) $$ \frac{15}{\sqrt{43}-\sqrt{13}} = \frac{15(\sqrt{43}+\sqrt{13})}{(\sqrt{43}-\sqrt{13})(\sqrt{43}+\sqrt{13})} = \frac{15(\sqrt{43}+\sqrt{13})}{43-13} = \frac{15(\sqrt{43}+\sqrt{13})}{30} = \frac{\sqrt{43}+\sqrt{13}}{2} $$