Teaching Notes 7.3: Rationalizing the Denominator
To express radicals in simplest form, no radicals should appear in the denominator of a fraction.
Students often make mistakes in the procedure for rationalizing the denominator.- Explain that a radical expression is in simplest form when there are no perfect square factors
other than 1 in the radicand and no radicals appear in the denominator of a fraction. When a
radical appears in the denominator of a fraction, students must rationalize the denominator
in order to express the radical in simplest terms. - Demonstrate how to rationalize the denominator using the example
2
√
3
. Explain that to
eliminate the radical in the denominator, students must multiply the fraction by√
3
√
3
.Note
that this fraction is equivalent to 1; therefore, they are not changing the value of the original
fraction. Explain that2
√
3
·
√
3
√
3
=
2
√
3
√
9
, which is simplified as2
√
3
3
.
- Review the information and example on the worksheet with your students.
EXTRA HELP:
You can have a radical in the numerator of your answer but you cannot have a radical in the
denominator.ANSWER KEY:
(1)
5
√
7
7
(2)
√
2 (3)
5
√
6
3
(4)
√
10
5
(5)
√
15
5
(6)
√
21
7
(7)
√
10
4
(8)−
√
3 (9)
2
√
6
3
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(10)
√
33
11
(11)
√
2
2
(12)
√
3
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(Challenge)If the radicand in the denominator is a perfect square, it is not necessary to rationalize
the denominator. It is only necessary to find the square root. For example,5
√
9
=
5
3
.
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