Answer:
See answers below.
Step-by-step explanation:
All of these use the same idea: multiply numerator and denominator by the conjugate radical (an identical expression but with the sign between terms changed).
[tex]\frac{1}{13-\sqrt{15}}\cdot\frac{13+\sqrt{15}}{13+\sqrt{15}}=\frac{13+\sqrt{15}}{169-15}}\\\\=\frac{13+\sqrt{15}}{154}[/tex]
[tex]\frac{1}{\sqrt{13}-\sqrt{11}} \cdot \frac{\sqrt{13}+\sqrt{11}}{\sqrt{13}+\sqrt{11}}=\frac{\sqrt{13}+\sqrt{11}}{13-11}\\\\=\frac{\sqrt{13}+\sqrt{11}}{2}[/tex]
[tex]\frac{1}{\sqrt{21}+\sqrt{29}}\cdot\frac{\sqrt{21}+\sqrt{29}}{\sqrt{21}+\sqrt{29}}=\frac{\sqrt{21}+\sqrt{29}}{21-29}\\\\=\frac{\sqrt{21}+\sqrt{29}}{-8}[/tex]
Note: Here's an example of what happens when you multiply conjugate radicals.
[tex](\sqrt{13}-\sqrt{11}})(\sqrt{13}+\sqrt{11})=13-\sqrt{11}\sqrt{13}+\sqrt{13}\sqrt{11}-11=13-11[/tex]
When you FOIL those binomials, the square roots go away!
