Rationalizing the Denominator
Examples with Solutions
Example 1:
Simplify 
solution:
There is no factorization of the expression in the square root
possible here, so the only simplification necessary is to
rationalize the denominator of the fraction. Since the square
root in the denominator is already as simple as possible, we need
to just multiply the numerator and denominator by that square
root to achieve the required result.
Although there is a t in both the numerator and the
denominator, it is not a factor in the denominator, and therefore
we cannot cancel the t’s between the numerator and
denominator of this last form. In fact, there is obviously no
common factors to cancel between the numerator and denominator of
this last form, so this last expression is as simple as possible
and so it must be the required final answer.
NOTE: Sometimes as they focus on simplifying
fractions such as the final result of Example 1 above, people
note that the numerator and denominator contain a power of the
same expression – in this case of (s + t). Without thinking
they do the following:
or
Superficially, it may appear that one or both of these lines
represents a useful simplification, but in fact, both lines
really just amount to reversing the simplification previously
done. The second line obviously just gives back the original
problem, and the first line does as well, though in a somewhat
less recognizable form. So, while it is very useful to break the
simplification of complicated algebraic expressions down into a
series of relatively simple independent steps, it is also
important to review your entire solution to make sure that you
haven’t lost track of what the problem actually requires you
to accomplish, and make sure that that is indeed what your final
answer has accomplished.
Example 2:
Rationalize the denominator in  and simplify the result
as much as possible.
solution:
If we simply follow the steps illustrated in the first and
second examples above, we get
which has a rational denominator, but clearly can be
simplified further. First
8x 3 = 2 2 · 2 · x 2 · x
so
Thus,
as the final simplified result.
Alternatively, we could have noted that the square root in the
denominator of the original fraction is not in simplest form, and
so we could have started by simplifying both the square root and
the fraction itself first:
Now the denominator of this fraction can be rationalized by
multiplying the numerator and denominator by  :
giving the same final simplified result as we obtained
earlier.
Example 3:
Simplify 
solution:
No common factors can be found to cancel between the numerator
and denominator here. Also, the square roots which appear here
obviously cannot be simplified further. So, all that’s left
to do to simplify this expression is to rationalize the
denominator. This is accomplished by multiplying the numerator
and denominator by  , giving:
as the final result (since there are clearly no common factors
that can be cancelled between the numerator and denominator of
this final expression). Notice that we used brackets explicitly
to ensure that the entire numerator was multiplied by in the first step here.
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