Question

racionalize os denominadores de:
a)15
4-√15

b)11
11-√11

c)5
√7

d)√2
√3-√5

Answer 1

a) 

$\dfrac{15}{4- \sqrt{15} } \\ \\ \\ \dfrac{15.(4- \sqrt{15})}{(4- \sqrt{15}).(4- \sqrt{15})} \\ \\ \\\dfrac{60 -15 \sqrt{15}}{(4- \sqrt{15})^2}} \\ \\ \\\dfrac{60 -15 \sqrt{15}}{31 - 8 \sqrt{15}}} \\ \\ \\ =\ \textgreater \ 60 +15 \sqrt{15}$

===
b) 

$\dfrac{11}{11 - \sqrt{11}} \\ \\ \\ \dfrac{11.(11 - \sqrt{11})}{(11 - \sqrt{11}).(11 - \sqrt{11})} \\ \\ \\ \dfrac{121- 11\sqrt{11}}{(11 - \sqrt{11})^2} \\ \\ \\ \dfrac{121- 11\sqrt{11}}{132 -22 \sqrt{11}}\\ \\ \\ =\ \textgreater \ \dfrac{11 + \sqrt{11}}{10}$

===
c) 

$\dfrac{5}{ \sqrt{7} } \\ \\ \\ \dfrac{5. \sqrt{7}}{( \sqrt{7}).( \sqrt{7}) } \\ \\ \\ \dfrac{5. \sqrt{7}}{( \sqrt{7})^2} \\ \\ \\ =\ \textgreater \ \dfrac{5. \sqrt{7}}{7}$

===
d)

$\dfrac{ \sqrt{2}}{ \sqrt{3} - \sqrt{5}} \\ \\ \\ \dfrac{ \sqrt{2}.( \sqrt{3} - \sqrt{5})}{(\sqrt{3} - \sqrt{5}).( \sqrt{3} - \sqrt{5})} \\ \\ \\ \dfrac{ \sqrt{2}.( \sqrt{3} - \sqrt{5})}{(\sqrt{3} - \sqrt{5})^2} \\ \\ \\ \dfrac{ \sqrt{2}.( \sqrt{3} - \sqrt{5})}{(\sqrt{3} - \sqrt{5})^2} \\ \\ \\ \dfrac{-(- \sqrt{6} + \sqrt{10})}{8-2\sqrt{3}. \sqrt{5}} \\ \\ \\ =\ \textgreater \ -\dfrac{ \sqrt{6} + \sqrt{10}}{2}$