Section 13.3
Rationalizing Denominators and
Simplifying Quotients of
Radical Expressions
Copyright © 2011 Pearson Education, Inc.
Rationalizing Denominators of Rational Expressions
Example
4
.
Simplify
5 3x
Solution
Copyright © 2011 Pearson Education, Inc.
Slide 2
Rationalizing Denominators of Rational Expressions
Solution
Example
Continued
• Use a graphing calculator table to verify solution
Copyright © 2011 Pearson Education, Inc.
Slide 3
Rationalizing Denominators of Rational Expressions
Example
Simplify
Solution
Copyright © 2011 Pearson Education, Inc.
Slide 4
Rationalizing Denominators of Rational Expressions
Solution
Example
Continued
• Use a graphing calculator table to verify solution
Copyright © 2011 Pearson Education, Inc.
Slide 5
Rationalizing Denominators of Rational Expressions
Solution
Example
Continued
Copyright © 2011 Pearson Education, Inc.
Slide 6
Rationalizing Denominators of Rational Expressions
Property
To rationalize the denominator of a radical
A
expression of the form n k , we multiply the
x
n k
x
expression by a fraction of the form n k so
x
that the radical in the denominator has a perfect nthpower radicand.
Copyright © 2011 Pearson Education, Inc.
Slide 7
Quotient Property
Property
If
n
n
a and b are defined and b is nonzero, then
n
a
a
n
n
b
b
In words: The nth root of a quotient is the quotient of
the nth roots.
Copyright © 2011 Pearson Education, Inc.
Slide 8
Using the Quotient Property to Simplify Radical Expressions
Example
Simplify.
Solution
Copyright © 2011 Pearson Education, Inc.
Slide 9
Using the Quotient Property to Simplify Radical Expressions
Solution
Example
Continued
Simplify.
Copyright © 2011 Pearson Education, Inc.
Slide 10
Using Conjugates Rationalize Denominators
Examples
Example
5
Simplify
.
3 x
Copyright © 2011 Pearson Education, Inc.
Slide 11
Using Conjugates Rationalize Denominators
Solution
Simplify
Copyright © 2011 Pearson Education, Inc.
Slide 12
Using Conjugates Rationalize Denominators
Warning
Process
To rationalize the denominator of a square root
expression if the denominator is a sum or difference
involving radicals,
1. Determine the radical conjugate of the
denominator.
Copyright © 2011 Pearson Education, Inc.
Slide 13
Using Conjugates Rationalize Denominators
Warning
Process
2. Multiply the original fraction by the fraction
conjugate/conjugate.
3. Find the product of the denominators by using
Example
Simplify
Copyright © 2011 Pearson Education, Inc.
Slide 14
Using Conjugates Rationalize Denominators
Warning
Solution
2. Multiply the original fraction by the fraction
conjugate/conjugate.
3. Find the product of the denominators by using
Simplify
Copyright © 2011 Pearson Education, Inc.
Slide 15
Using Conjugates Rationalize Denominators
Example
Warning
Simplify
.
Solution
Copyright © 2011 Pearson Education, Inc.
Slide 16
Using Conjugates Rationalize Denominators
Solution
Example
Warning
Continued
Simplify
Copyright © 2011 Pearson Education, Inc.
Slide 17