Why eliminate radicals in the denominator? [rationalizing the denominator] Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to $\sqrt{3}/3$. 
Is there a mathematical or other reason?
And does the same apply to exponential notation -- are students expected to "simplify" $3^{-1/2}$ to $3^{1/2}/3$ ?

The form with neither denominators in radicals nor radicals in denominators and with only squarefree expressions under square-root signs, etc., is a canonical form, and two expressions are equal precisely if they're the same when put into canonical form.
When are two fractions equal? How do you know that $\dfrac{51}{68}$ is the same as $\dfrac{39}{52}$?  They're both the same when reduced to lowest terms.
How do you know that $\dfrac{1}{3+\sqrt{5}}$ is the same as $\dfrac{3\sqrt{5}-5}{4\sqrt{5}}$?  Again, they're the same when put into canonical form.
How do you know that $\dfrac{13+i}{1+2i}$ is the same as $\dfrac{61}{6+5i}$?  Same thing.
