To rationalize a quantity means, literally, to make it rational.  A rational number is one that can be expressed as the ratio of two integers, like 2/3, for example, or 4, since 4 can be expressed as 4/1.  The quantity 2.5 is also rational, since it represents 2 and 1/2, or 5/2.  In fact, any number with a terminating decimal part is rational.  Any number whose decimal part begins to repeat is also rational, such as .33333333...., since this can be expressed as 1/3.  Numbers that are not rational are called irrational.  Examples of irrational numbers are the square root of 2, pi, and e.

So, to rationalize the denominator of a fraction, we need to re-write the fraction so that our new fraction has the same value as the original, and has a rational denominator.  The standard method of changing a fraction into an equivalent fraction with a specified denominator is to multiply it by some number over itself, since any non-zero number over itself is 1, and multiplication by 1 doesn't change value.

Type of Problem	Example	Solution
Denominator is one-term; square root
Denominator is one-term, some other root
Denominator has two terms (is a binomial)

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