Rationalizing the denominator is a technique used when demonstrating a quotient, or fraction, and the denominator is irrational. Basically, what you do (in most cases) to rationalize the denominator is multiply by 1/1 in form of the quantity of the denominator divided by the denominator. This causes the denominator of the expression to be rational, and thus, legal.
An Example (using an irrational square root):
This is an expression with an irrational denominator (the square root of 2). In order to do this, we multiply by the denominator divided by the denominator (effectively multiplying by 1, and thus the result is equal to the original value):
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Now, since the square root of two multiplied by the square root of two will give you two (the denominator), we get:
And now the denominator is rationalized. In the case of another radical fractional exponent, the number you use to rationalize is the the number 1 (one) minus the fractional exponent. That is, if the denominator is 3√4, you would multiply by (3√4)2, or 42/3 divided by itself. In the situation of a complex, or imaginary denominator, the approach is basically the same, except the complex conjugate of the denominator is used instead.
Stupot asks: Why do we want to do this again?
Well, according to some math professors (not many engineering professors agree, mind you) you should never have an irrational denominator. Basically, if you have an expression with an irrational denominator, you can simplify it a bit more by rationalizing the denominator.
In theory, this is true. It may look a bit nicer. And it's not that tough to do. But if you are still doing operations on this number or expression, then it's easier to just leave it irrational. This is only my opinion, and should be treated as such. A real math major could tell you exactly why it should be done.
jrn has a whole bucketfull of commentary:
re why: one reason is that certain simple kinds of numbers (e.g. quadratic surds) gain unique representations. It also is good for reminding me how similar the field extension over e.g. √2 is to the ring of polynomials in x.
with surds I think part of the reason it doesn't matter so much is that you can do this stuff in your head. But manipulating ratios of complex numbers is a lot easier if the denominator is rational. But imagine if we always put things in standard forms: then instead of talking about e^x we'd say sum_i=0^infty (1/i!) x^i, and so forth. Sometimes that's useful; usually it's not.