5.06
Rationalizing Denominators
from Basic Algebra: One Step at
a Time © 2002-2011
P. 435-446
Dr. Robert J. Rapalje
Seminole State College of Florida
ANSWERS TO ALL EXERCISES ARE INCLUDED AT THE
END OF THIS PAGE
There is a tradition in mathematics of eliminating
the radicals from the denominators (or numerators) of fractions. The
process is called rationalizing the denominator (or numerator) of the
fraction. This may be done to simplify the radical expression or to make
calculation of the expression easier, especially in days when calculators
were not available. For example, knowing the value of
to
be approximately 1.414, to calculate
without
a calculator would require long division of 20 divided by 1.414. It is
much easier to multiply numerator and denominator by
,
.
As you can see, it is easier to multiply 10•(1.414),
than to divide 20 ÷ 1.414.
MONOMIAL DENOMINATORS
When rationalizing a
monomial square root denominator, you need to multiply numerator and
denominator by "something" that makes the denominator result in a perfect
square. The next examples and exercises in this section illustrate the
process.
EXAMPLE 1.
Rationalize the denominator for
.
Calculate the decimal value.
Solution:
Multiply the numerator and
denominator by 
.
Calculate the decimal value to nearest hundredth:
=
6.32; =
6.32.
435
EXERCISES.
Rationalize the denominators.
Calculate the values to the nearest hundredth.
1. 
2. 
3. 
4.
5.
6.
7.
8.
9.
10.
11. 
12. 
436
In the next examples
and exercises, there are essentially three steps:
EXAMPLE 2.
Simplify the radical and rationalize the denominator
.
Solution:
STEP 1:
Simplify the radical:
STEP 2:
Rationalize the denominator:
STEP 3:
Reduce the fraction:
Check
by calculating the values:
Problem =
=
1.79; Answer =
=
1.79.
EXERCISES.
Simplify the radicals, reduce the
fractions, and calculate the values.
17. 
18.
19. 
=
________
= ________
437
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
Sometimes the radical contains the entire fraction,
like 
.
In these cases, the quotient property for square roots applies, and you
take separate square roots of the numerator and denominator.
Product Property of Square Roots:
,
a>0 and b>0.
Quotient Property of Square Roots:
,
a>0 and b>0.
438
EXAMPLE 3.
Simplify the radical and rationalize the denominator
.
Solution:
which
is 
.
EXERCISES.
Simplify the radicals using the quotient property of square
roots.
29.
30. 
31. 
32.
33. 
34. 
EXAMPLE 4.
Simplify the radical and rationalize the denominator
.
Solution:
STEP 1:
Simplify the
radical: means
STEP 2:
Rationalize the denominator:
STEP 3:
Reduce the fraction (if possible):
Calculator values:
=
1.55; =
1.55.
439
EXERCISES.
Rationalize the denominators and simplify the radical
expressions. Calculate the values.
35.
36. 
37. 
38.
39. 
40. 
EXAMPLE 5.
Simplify the radical and rationalize the denominator
.
Solution:
STEP 1:
Simplify the
radical: means
STEP 2:
Rationalize the denominator:
STEP 3:
Reduce the fraction (if possible):
440
41.
42. 
43. 
__________
441
BINOMIAL DENOMINATORS
When the
denominator of the fraction involves binomial radical expressions, such
as 
,
a special procedure is used. Multiplying the numerator and denominator
by 
will
eliminate the radicals from the denominator. For the fraction
,
multiply numerator and denominator by 
.
In general, whatever the binomial denominator may be, you multiply the
numerator and denominator by the same quantity as the denominator but
with the opposite sign in the middle. This is called the conjugate of
the denominator.
EXAMPLE 6. Rationalize the denominator and simplify
.
Solution: Multiply numerator and denominator by
: 
Leave the numerator factored, multiply denominator:
Notice that the middle term subtracts
out:
Reduce the fraction:
442
EXERCISES.
In each of the following exercises, rationalize the
denominators and reduce each fraction to lowest terms.
49.
50.
___________
___________
51.
52. 
53.
54. 
443
55.
56. 
EXAMPLE 7.
Rationalize the denominator and simplify 
.
Solution: First, it may help to simplify the numerator: 
Multiply numerator and denominator by
:
Multiply numerator and denominator:
Simplify the radicals (the middle term subtracts
out): 
Continue to simplify: 
Factor the common factor which is 9 :
Reduce the fraction:
or
444
EXERCISES. Rationalize the denominators and simplify the radical expressions.
57.
58. 
59.
60. 
61.
62. 
445
EXAMPLE 8. Rationalize the denominator and simplify
.
Solution:
Multiply numerator and denominator by
:
Multiply numerator
and denominator :
Simplify the radical
:
Factor the numerator
:
Reduce the
fraction:
EXERCISES. Rationalize the denominators and simplify the radical expressions.
63.
64.
65.
66.
446
ANSWERS 5.06
p. 436 - 446:
1.
4.24;
2. 8.94;
3. 
13.86;
4. 
12.65;
5.
3.46;
6. 
9.95;
7. 8.25;
8. 12.25;
9. 33.53;
10. 37.04;
11. 
4.95;
12. 
4.04;
13. 
3.10;
14. 6.12;
15. 4.08;
16. 
1.90;
17. 
1.41;
18. 8.94;
19. 4.47;
20. 
2.83;
21. 2.68;
22. 1.79;
23. 1.15;
24. 0.89;
25. 1.34;
26. 
1.22;
27. 2.12;
28. 1.77;
29. 
;
30. 
;
31. 
;
32. ;
33. 
;
34. ;
35. 0.71;
36.0.82;
37. 1.90;
38. 0.61;
39. 0.54;
40. 0.76;
41. 
;
42. 
;
43. 
;
44. 
;
45. ;
46. 
;
47. 
;
48. ;
49.
;
50.
;
51. 
;
52. ;
53. 
;
54. ;
55. 
; 56. 
or
;
57. 
;
58. 
;
59. ;
60. 
; 61.
62.
63.
64. 
65.
66.
.
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