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+ 3a2b+ 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)n6=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
squares. Then if we multiply our equation by 1 = 7−√x
7−√xwe are not changing the equation,
but the denominator will no longer have a radical.
5
7 + √x·7−√x
7−√x=5(7 −√x)
49 −x=35 −5√x
49 −x
Remember that aand bin the difference of squares expression are parameters. This means
that we can replace aand bwith pretty much any expression we wish. Consider the
following example. We are still using the difference of squares.
−(x−2)2
2√x−1−x=−(x−2)2
2√x−1−x·2√x−1 + x
2√x−1 + x(9)
=−(x−2)2·(2√x−1 + x)
4(x−1) −x2(10)
=(x−2)2·(2√x−1 + x)
x2−4x+ 4 (11)
=(x−2)2·(2√x−1 + x)
(x−2)2(12)
= 2√x−1 + x(13)
In the above examples both the pairs 7−√xand 7 + √xand the pairs 2√x−1−xand
2√x−1 + xare called conjugates. In the difference of squares formula (a+b)is the con-
jugate 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 M¯
anoa 47 R
Spring - 2014
