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5.06 Rationalizing Denominators from Basic Algebra: One Step at a Time © 2002 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
As you can see, it is easier to multiply 10(1.414),
than to divide
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
Solution:
Multiply the numerator and
denominator by
Calculate the decimal value to nearest hundredth:
EXERCISES. Rationalize the denominators. Calculate the values to the nearest hundredth.
1.
5.
9.
13.
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 =
EXERCISES. Simplify the radicals, reduce the fractions, and calculate the values.
17.
________
20.
23.
26.
Sometimes the radical contains the entire fraction,
like
Product Property of Square Roots:
Quotient Property of Square Roots:
EXAMPLE 3.
Simplify the radical and rationalize the denominator
Solution:
EXERCISES. Simplify the radicals using the quotient property of square roots.
29.
32.
EXAMPLE 4.
Simplify the radical and rationalize the denominator
Solution:
STEP 1: Simplify the
radical:
STEP 2:
Rationalize the denominator:
STEP 3:
Reduce the fraction (if possible):
Calculator values:
EXERCISES. Rationalize the denominators and simplify the radical expressions. Calculate the values.
35.
38.
EXAMPLE 5.
Simplify the radical and rationalize the denominator
Solution:
STEP 1: Simplify the
radical:
STEP 2:
Rationalize the denominator:
STEP 3:
Reduce the fraction (if possible):
41.
__________
BINOMIAL DENOMINATORS
When the
denominator of the fraction involves binomial radical expressions, such
as
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:
EXERCISES. In each of the following exercises, rationalize the denominators and reduce each fraction to lowest terms.
49.
___________
___________
51.
53.
55.
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:
EXERCISES. Rationalize the denominators and simplify the radical expressions.
57.
59.
61.
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.
65.
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44.
47.
This is a work in progress. Please check back SOON!! ANSWERS 5.06 p. 436 - 446: 1.
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