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Rationalizing
Denominators
Get that pesky radical OUT of my denominator
John Reece
7/13/2010
What does “Rationalize the Denominator” mean?
It means to change a fraction, without changing its value, so that any irrational
or complex numbers in the denominator of the fraction become rational numbers.
For example,
whereas
1
is a fraction with an irrational number in the denominator,
2
2
is an equivalent number, but with a rational number denominator.
2
Before the invention of
readily accessible calculators,
when people did calculations by
hand, sometimes supported by
logarithm tables or by a slide
rule, it was much easier to
calculate
the
numerical
approximation of a number such
as
2
than
2
it
would
calculate the value of
be
to
…mathematicians have…decreed
that numbers with rational
denominators are simpler than
numbers with irrational
denominators.
1
.
2
Hence, by convention, mathematicians have, in general, decreed that numbers
with rational denominators are simpler than those with irrational denominators.
Notice that the convention is qualified with the phrase in general. That is
because there are situations, albeit infrequent, when you may not want to rationalize
the denominator, or perhaps even rationalize a numerator at the expense of an
irrational denominator. However, in the vast majority of cases, you will be either
asked or required to perform the process of rationalizing denominators.
Rationalizing a monomial denominator.
The simplest situation is that illustrated in the example given above where the
denominator consists of a single radical term.
The process of rationalizing the denominator makes use of the definition of the
radical symbol and the concept of the multiplicative identity.
Definition of the Radical Symbol:
y  x  y2  x
Multiplicative Inverse:
1
1
a  , a  0,    such that a   1
a
a
Multiplicative Identity:
a  ,  I  1 such that a  I  a
So how do we apply these ideas to a number that looks like:
2
3
and alter it so that it retains its current value, yet has a rational number for a
denominator?
The definition of the radical symbol tells us that if we multiply the denominator
3 that we will end up with 3, a rational number, in our denominator. The idea of
1
a multiplicative inverse tells us that if we multiply the value
that we need to get rid
3
by
of the radical in the denominator by
3
we will have a multiplier of
1
3
 1 , and then
3
the idea of the multiplicative identity guarantees that using this multiplier will not
change the value of our original fraction.
2  3


3  3 
is the required multiplication – simply proceed as in multiplying any other fractions,
numerator times numerator and denominator times denominator:
2  3 2 3


3
3  3 
And the task is complete.
Rationalizing a binomial denominator.
This is a bit trickier involving a fractional expression where the denominator
consists of two terms at least one of which is a radical. Example:
1
2 3
If we try the method of the previous section, that is to multiply by a factor of the
denominator divided by itself, we won’t get any closer to our goal of eliminating the
radical in our original denominator. Here’s why:
 2  3  2  3   4  4
33 74 3
We still have a radical in the denominator! Obviously, we need a different
strategy. The strategy to solve this type of problem involves the concept of a conjugate
binomial. The conjugate of any binomial is simply the binomial with the sign between
the two terms reversed. For example, a  b and a  b are conjugates. The important
property of conjugates is that when you multiply them, you get the difference of two
squares.
 a  b  a  b   a 2  b2
Ah ha! This looks promising. If we multiply our two-term denominator by its
conjugate, we can get rid of the original radical without introducing a new one.
 2  3  2  3   4  3  1
Much better. Of course, we need to consider the multiplicative identity and
multiply our original fraction by 1, but this time in terms of the conjugate of the
denominator divided by itself:
1  2 3  2 3
 2 3


1
2  3  2  3 
Whoa! We got rid of the denominator altogether – swwwwweet!
Rationalizing a monomial denominator with index >2.
Ok, we have handled monomial and binomial denominators with square roots,
but what about higher order roots, i.e. cube roots or 4th roots and such?
3
5
4
If we simply multiply by the denominator divided by the denominator, we are
out of luck again.
4
5  4 5  4 52 and we still have a radical. Let’s look at what it
means to take the 4th root of something.
y  4 x  y4  x
But notice that y 4  y  y 3 , so that means if I multiply
4
5  4 53  4 54  5 and bye-
bye you nasty old radical. Which leads us to the solution:
3  4 53

4
5  4 53
 3 4 53


5

And that’s all I have to say about that.
Rationalizing a complex denominator.
A complex denominator is one that is of the form a  bi where a and b are real
numbers and i is the imaginary number defined by i 2  1 . Example:
5
3  2i
Again, we need to return to our concept of the conjugate. Complex conjugates
are formed the same way any other binomial conjugates are formed; just change the
sign on the imaginary part coefficient.
For the example:
5  3  2i 
15  10i
15  10i



3  2i  3  2i  9  (1)(4)
13
The only trick is to remember that i 2  1 .
Conclusion:
This tutorial reference deals with the sorts of situations that are typically
encountered in Algebra I and II, Trigonometry, Pre-Calculus, and most, if not all, of
Calculus, but is not intended to be an exhaustive treatment of the subject.
If you have a situation not covered in this tutorial, by all means ask a question
on the Q&A board.
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