Math 135
Rationalizing the Denominator
Examples
Remember these important identities! For all real numbers a, b:
Difference of Squares: a2 − b2 = (a + b)(a − b)
Difference of Cubes:
a3 − b3 = (a − b)(a2 + ab + b2 )
Sum of Cubes:
a3 + b3 = (a + b)(a2 − ab + b2 )
(a + b)2 = (a2 + 2ab + b2 )
(a + b)3 = (a3 + 3a2 b + 3ab2 + b3 )
WARNING! Avoid the ”Freshman’s Dream”
Remember to multiply all the factors in expressions such as (4) and (5) above. In general:
(a + b)n 6= an + bn
We will be using the difference of squares extensively. For example, how would you
simplify the following expression so that the denominator no longer contains a square
root?
5
√
7+ x
√
√
The key idea is to realize that (7 + x)(7 − x) =
49 − x, i.e. we have a difference of
√
7− x
√
squares. Then if we multiply our equation by 1 = 7− x we are not changing the equation,
but the denominator will no longer have a radical.
√
√
√
5
7− x
5(7 − x)
35 − 5 x
√ ·
√ =
=
49 − x
49 − x
7+ x 7− x
Remember that a and b in the difference of squares expression are parameters. This means
that we can replace a and b with pretty much any expression we wish. Consider the
following example. We are still using the difference of squares.
√
−(x − 2)2
−(x − 2)2
2 x−1+x
√
√
= √
·
(9)
2 x−1−x
2 x−1−x
2 x−1+x
√
−(x − 2)2 · (2 x − 1 + x)
=
(10)
4(x − 1) − x2
√
(x − 2)2 · (2 x − 1 + x)
=
(11)
x2 − 4x
+
4
√
(x − 2)2 · (2 x − 1 + x)
(12)
=
(x − 2)2
√
(13)
= 2 x−1+x
√
√
√
In√the above examples both the pairs 7 − x and 7 + x and the pairs 2 x − 1 − x and
2 x − 1 + x are called conjugates. In the difference of squares formula (a + b) is the conjugate of (a − b). So conjugation amounts to switching the sign in the given expression.
The process of multiplying an expression whose denominator contains a radical by 1 in
the form of a fraction with the numerator and denominator both being conjugates of the
expression’s denominator is called rationalizing the denominator.
University of Hawai‘i at Mānoa
47
R Spring - 2014
Math 135
Rationalizing the Denominator
Examples
Example 1. Rationalize the Denominator.
√
√
5
5
√ =√ ·
3
3
√ ! √ √
√
3
5· 3
15
√
=√ √ =
3
3
3· 3
Example 2. Rationalize the denominator.
!4 √ √
√ √
√ √
√ √
√
5
5
5
5
5
4
2 · 44
2 · 28
2 · 2 · 23
2 · 23
√
=
=
=
=
5
4
4
4
2
4
p
Example 3. Find the conjugate of x2 − x2 − 2 + x.
√
√
2
2
√
= √
·
5
5
4
4
r
x
x2 − − 2 − x.
2
Example 4. Rationalize the denominator.
p
2 x2 −
3x + 4
3x + 4
p
p
= p 2 x
·
2 x2 − x2 − 2 + x
2 x − 2 −2+x
2 x2 −
p
(3x + 4)(2 x2 − x2 − 2 − x)
=
4(x2 − x2 + 2) − x2
p
(3x + 4)(2 x2 − x2 − 2 − x)
=
3x2p− 2x + 8
(3x + 4)(2 x2 − x2 − 2 − x)
=
(3x + 4)(x − 2)
p
2 x2 − x2 − 2 − x
=
x−2
x
2
x
2
+2−x
+2−x
!
(14)
(15)
(16)
(17)
(18)
(19)
University of Hawai‘i at Mānoa
48
R Spring - 2014
Math 135
Rationalizing the Denominator
Worksheet
Rationalize the denominator:
4.
√
√2
3
√
2
√
1− 2
√
1+√3
1− 3
√
a+√b
a− b
5.
√1
5
6.
4
√
3
16
7.
3
√
4
3
8.
3
√
4
27a5 b11
√ √
x+ 2
√ √
x− 2
√
√
x−2 y
√
√
x+2 y
1.
2.
3.
9.
10.
√
+ 4 45
University of Hawai‘i at Mānoa
49
R Spring - 2014