What follows is an edited and expanded version of comments, and a list of examples, that I posted 12 June 2001 (and later in 26 September 2007, in a more abbreviated form) in the Math Forum discussion group AP-calculus.
I believe rationalizing the denominator was originally positioned so early in the curriculum --- algebra 1 and geometry for division by $\sqrt{n},$ and algebra 2 for division by something like $m + \sqrt{n}$ --- was partly for reasons having to do with numerical calculation, and partly for reasons having to do with algebraic combination and simplification of exact numerical values. Incidentally, if you look at textbooks written 50 to 150 years ago, you don't really see much of an expectation that radicals were numerically approximated (this view being based on worked examples in the text and answers to exercises), except for trigonometry texts. However, the numerical aspect becomes much more important in applications that occurred outside of mathematics (mainly in science courses), so I suspect what happened is that the training in appropriately rewriting radical expressions so that square root tables and such could be easily used was left to the math courses.
I personally think there has been too much emphasis on rationalizing the denominator in the past 40 years (perhaps in the past 20 years the emphasis has been more appropriate), especially in classes below the precalculus level, but I also think it's easy to forget just how often the technique of rationalization shows up in math, even if we restrict ourselves to the lower undergraduate level. As for me, when departmental and/or course supervisor constraints allowed me to do so, I DID NOT REQUIRE answers to be in denominator-rationalized form in high school or college algebra classes, or in precalculus classes. However, I felt it was an important skill for anyone getting at least as far as calculus. Thus, in calculus courses, I tried to make up for this inattention to rationalization (both by me and by other teachers) by working the topic in at a number of places. I did this mainly by working examples in class and by assigning problems (with an appropriate hint) like #1-6 below.
MISCELLANEOUS LIST OF EXAMPLES FOR RATIONALIZING
1. These limits can be evaluated without taking derivatives if you first apply a binomial rationalization step:
$$\lim_{x \rightarrow 1}\frac{x^2 - 1}{\sqrt{2x+2\,} \; - \; 2} \;\;\; \text{and} \;\;\; \lim_{x \rightarrow 0}\frac{1 - \cos x}{x} $$
2. To rewrite
$$ \ln \left( \frac{x + \sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}} \right) \;\;\; \text{as} \;\;\; 2\ln\left (x + \sqrt{x^2 - 1}\right),$$
it helps if you first rationalize the numerator. Note: Putting $x = \sec \theta$ gives an identity that is sometimes useful.
3. To differentiate $x^{\frac{1}{2}},$ $x^{\frac{1}{3}},$ etc. using the limit definition of the derivative, you'll want to rationalize numerators.
4. The derivative of
$$ \frac{\sqrt{a-x} \; + \; \sqrt{a+x}}{\sqrt{a-x} \; - \; \sqrt{a+x}} $$
is much easier to put into the more useful form
$$ \frac{a^2 \; + \; a\sqrt{a^2 - x^2}}{x^2\sqrt{a^2 - x^2}}$$
if you rationalize the denominator BEFORE differentiating.
5. Let $a \neq 0,$ $b,$ and $c$ be real number constants. To verify that
$$ \lim_{n \rightarrow \infty} \left( \sqrt{an^2 + bn} \; - \; \sqrt{an^2 + cn} \right) \;\;\; = \;\;\; \frac{1}{2\sqrt{a}}(b-c),$$
it helps to rationalize the numerator first.
6. The linearization of
$$\frac{1+x}{1-x} \;\; \text{at} \;\; x=0$$
is easy if you begin by multiplying both the numerator and denominator by $1+x.$ After doing this, you get to ignore the $x^2$ terms that appear additively with constants or with multiples of $x.$ The result will be $1 + 2x.$ More generally, rationalization ideas can be used to obtain the quotient rule for derivatives by multiplying/dividing by an appropriate conjugate and ignoring all but first order terms, and similar methods can be used to approximate $f(x+h,\,y+k)$ for rational functions $f(x,y)$ when $h$ and $k$ are close to $0.$
In the same way, one can show by rationalization methods that for $\delta$ and $\epsilon$ near $0,$ we have
$$\frac{1}{1 + \delta} \; \approx \; 1 - \delta \;\;\; \text{and} \;\;\; \frac{1}{1-\delta} \; \approx \; 1 + \delta \;\;\; \text{and} \;\;\; \frac{1+\epsilon}{1+\delta} \; \approx \; 1 + \epsilon - \delta $$
These and other approximations are discussed in Philip L. Alger's 1957 text Mathematics for Science and Engineering (see pp. 145-155 of Chapter 6: Numerical Calculations) and in William Charles Brenke's 1917 text Advanced Algebra (see Chapter IX, Section 146: Useful Approximations, pp. 126-127). These approximations are often more important for giving approximations that are valid over a range of variable values than for giving individual and isolated numerical approximations. This is especially useful when an exact algebraic form is difficult to work with, such as in a differential equation (recall the pendulum equation). For instance,
$$\tanh M \;\; = \;\; \frac{e^{M} \; - \; e^{-M}}{e^{M} \; + \; e^{-M}} \;\; = \;\; \frac{1 \; - \; e^{-2M}}{1 \; + \; e^{-2M}} \;\; \approx \;\; 1 - 2e^{-2M}$$
is an approximation that is correct to $16$ decimal places when $M = 10.$ This particular approximation for $\tanh M$ is obtained in the same manner I've just shown, and then used to find the lowest eigenvalue in the high barrier limit for a quantum mechanical particle confined to a double potential well, in Charles S. Johnson and Lee G. Pedersen's 1974 Problems and Solutions in Quantum Chemistry and Physics (see Problem 4.8(b) on pp. 105-106).
Another example can be found in Jerry B. Marion's 1970 text Classical Dynamics of Particles and Systems (see p. 270). Marion uses the approximation
$$\theta \;\; = \;\; \frac{2\pi}{1 - \frac{\delta}{\alpha}} \;\; \approx \;\; 2\pi\left(1 + \frac{\delta}{\alpha}\right) \;\; = \;\; 2\pi + \frac{2\pi\delta}{\alpha}$$
near the end of a derivation of the precession of Mercury's orbit as predicted by Einstein's Theory of Relativity. The term $\frac{2\pi\delta}{\alpha}$ represents the approximate precession per orbit, which in Mercury's case works out to approximately $43$ seconds (angle measure) per century.
7. To express the quotient of two complex numbers in rectangular form, when each of the complex numbers is given in rectangular form, you'll want to use a "rationalization of the denominator" technique. Related to this is finding the real and imaginary parts of a rational function of a complex variable (e.g. verifying the Cauchy-Riemann equations, finding a harmonic conjugate of a rational function, investigating certain orthogonal families of curves, etc.).
8. For numerical purposes (e.g. reducing round-off errors during a computer computation), the quadratic formula
$$ x \;\; = \;\; \frac{-b \; \pm \; \sqrt{b^2 - 4ac}}{2a}$$
is in some cases more usefully expressed as
$$ x \;\; = \;\; \frac{2c}{-b \; \pm \; \sqrt{b^2 - 4ac}}$$
9. To show that ${\mathbb Q}[\sqrt{2}]$ (i.e. real numbers of the form $r + s\sqrt{2}$ where $r$ and $s$ are rational numbers) is a field, a "rationalization of the denominator" technique is useful when verifying the multiplicative inverse part of the definition of a field.
10. Rationalizing techniques are useful to obtain non-radical forms for the general equation of a hyperbola and an ellipse directly from their geometric definitions. Related to this is the general idea of rationalizing an algebraic equation (say, for an algebraic curve or an algebraic surface -- see Cayley's 1868 paper On Polyzomal Curves, otherwise the Curves $\sqrt{U} + \sqrt{V} +$ &c. $=0,$ which begins on p. 470 here, for some eye-opening stuff) and of solving radical equations.
11. It is easy to find a simple expression for the following sum if each denominator is rationalized:
$$\frac{1}{\sqrt{1} + \sqrt{2}} \; + \; \frac{1}{\sqrt{2} + \sqrt{3}} \; + \;\frac{1}{\sqrt{3} + \sqrt{4}} \; + \; \cdots \; + \; \frac{n}{\sqrt{n} + \sqrt{n+1}}$$