Rationalize the denominator: $\frac{1}{\sqrt{8}+\sqrt{50}+\sqrt{18}}$.
This problem involves rationalizing the denominator, which means getting rid of any square roots in the bottom of the fraction. A common technique for doing this is to multiply the fraction by a clever form of 1, that is, something that has the same numerator and denominator. The tricky part is figuring out what that form of 1 should be. One thing I notice is that the denominator has three square roots, and each of them has a perfect square factor. For example, $\sqrt{8}=\sqrt{4\cdot 2}=2\sqrt{2}$. Similarly, $\sqrt{50}=\sqrt{25\cdot 2}=5\sqrt{2}$ and $\sqrt{18}=\sqrt{9\cdot 2}=3\sqrt{2}$. This suggests that $\sqrt{2}$ is a common factor in the denominator, and I might want to factor it out. If I do that, I get $\frac{1}{\sqrt{8}+\sqrt{50}+\sqrt{18}}=\frac{1}{\sqrt{2}(2+5+3)}=\frac{1}{\sqrt{2}(10)}$. Now I have only one square root in the denominator, which is simpler. To rationalize it, I can multiply the fraction by $\frac{\sqrt{2}}{\sqrt{2}}$, which is 1. This gives me $\frac{1}{\sqrt{2}(10)}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2(10)}=\frac{\sqrt{2}}{20}$. This is the rationalized form of the original fraction, where the denominator is a whole number. 