Question

How do you rationalize the denominator for `5/(sqrt6+sqrt5)`?

Answer 1

Multiply the top and bottom by `\sqrt{6}-\sqrt{5}` to get `5/(\sqrt{6}+\sqrt{5})=\frac{5(\sqrt{6}-\sqrt{5})}{6-5}=5\sqrt{6}-5\sqrt{5}`.

In general, `\frac{a}{\sqrt{b}+\sqrt{c}}=\frac{a(\sqrt{b}-\sqrt{c})}{b-c}` when `b` is not equal to `c`. In the case where `b=c>0`, then `\frac{a}{\sqrt{b}+\sqrt{c}}=\frac{a}{2\sqrt{b}}=\frac{a\sqrt{b}}{2b}`.