Why not $1$/irrational? An older math teacher told me that I shouldn't leave a fraction with an irrational in the denominator. But lately, I keep hearing this from every math teacher that I have ever had.  
Thus if I have this fraction  
$$\frac1{2^{1/2}}$$
I should always convert it to one that has no irrationals in the denominator
$$\frac{2^{1/2}}{2}$$
But why? My thought are that, just like the fact that we can't write out irrationals, we cant write fractions with irrational as denominators as decimals, but we can approximate them, so I don't see the problem. If we approximate $1/\sqrt{2}$, it would be around 0.70710678118, so I dont understand why I shouldn't include irrationals in denominators, at least if I don't want some points deducted from a test.

If I want to compute $1/\sqrt{2}$, I have to divide by $1.414$, at first glance.  Ick! 4-digit long division!  But if I convert it to $\sqrt{2}/2$, I just have to divide by $2$.  Heck, I can do that in my head.
