Rationalizing
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Rationalize each of the following.

1. ** **

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This is the typical
rationalization problem that you will see in an algebra class. In these kinds of problems you want to
eliminate the square roots from the denominator. To do this we will use

.

So, to rationalize the denominator
(in this case, as opposed to the next problem) we will multiply the numerator
and denominator by . Remember, that to rationalize we simply
multiply numerator and denominator by the term containing the roots with the
sign between them changed. So, in this
case, we had and so we needed to change the “+” to a “-”.

Now, back to the problem. Here’s the multiplication.

Note that the results will often
be “messier” than the original expression.
However, as you will see in your calculus class there are certain
problems that can only be easily worked if the problem has first been
rationalized.

Unfortunately, sometimes you have
to make the problem more complicated in order to work with it.

2. ** **

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In this problem we’re going to
rationalize the numerator. Do NOT get
too locked into always rationalizing the denominator. You will need to be able to rationalize the
numerator occasionally in a calculus class. It works in pretty much the same
way however.

Notice that, in this case there
was some simplification we could do after the rationalization. This will happen occasionally.