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2. Radicals and Rational Exponents
Chapter 10

3. 10 Radicals and Rational Exponents 10.1 Finding Roots
10.3 Simplifying Expressions Containing Square Roots
10.4 Simplifying Expressions Containing Higher Roots
10.5 Adding, Subtracting, and Multiplying Radicals
10.6 Dividing Radicals
Putting it All Together
10.7 Solving Radical Equations
10.8 Complex Numbers

4. 10.6
Dividing Radicals
To eliminate a radical from its denominator we use a process called rationalizing
the denominator.
We will be looking at two types of rationalization problems.
1) Rationalize a denominator containing one term.
2) Rationalize a denominator containing two terms.
We will use the idea that multiplying a numerator and denominator of a fraction by
The same quantity results in an equivalent fraction.

5. Rationalize a Denominator: One Square Root
The goal of rationalizing is to eliminate the radical from the denominator. Recall that,
We will use this property to rationalize denominator.
Example 1
Rationalize the denominator of each expression.
Solution
Rationalize denominator

6. Rationalize the denominator of each expression.
Note
Incorrect!
Example 2
Rationalize the denominator of each expression.
Solution
To rationalize the denominator of begin by simplifying
Simplify square root of 20.
Rationalize denominator

7. Rationalize the denominator of each expression.
Example 3
Rationalize the denominator of each expression.
Solution
Rationalize denominator

8. To Simplify we must rationalize the denominator.
Example 4
Simplify completely.
Solution
To Simplify we must rationalize the denominator.
Rationalize denominator
Rationalize denominator

9. Example 5 Simplify Completely. Solution
Simplify the radicand using the quotient rule for exponents.
After simplifying now we can rationalize the denominator.
Simplify
Rationalize
denominator
Multiply

10. Rationalize a Denominator: One Higher Root
Many student assume that to rationalize denominators we simply multiply the
denominator and numerator of the expression by the denominator as in the
previous section.
Let’s practice eliminating radicals before we move on to rationalizing.
Example 6
Fill in the blank.
Solution

11. Example 7
Fill in the blank.
Solution

12. Rationalize the denominator.
Example 8
Rationalize the denominator.
Solution
a) First identify what we want the denominator to be after multiplying.
Rationalize
denominator
Multiply
Simplify
Rationalize
denominator
Multiply
Simplify

13. Rationalize the denominator.
Example 9
Rationalize the denominator.
Solution
a) First identify what we want the denominator to be after multiplying.
Rationalize
denominator
Multiply
Simplify

14. Rationalize a Denominator: Containing Two Terms

15. Each method, i and ii, both give the same result.
Example 10
Solution
i) Use FOIL to multiply.
F O I L
Each method, i and ii, both give the
same result.
ii) Use

16. Example 11 Rationalize the denominator and simplify completely.
Solution
Multiply by the conjugate.
Use FOIL or
Simplify.
Subtract.

17. Example 12 Rationalize the denominator and simplify completely.
Solution
Multiply by the conjugate.
Use FOIL or
in denominator. Use FOIL in the numerator.
Simplify.

18. Rationalize a Numerator
In higher-level math courses, sometimes it is necessary to rationalize the
numerator of a radical expression so that the numerator does not contain a radical.
Example 13
Rationalize the numerator and simplify completely.
Solution
Rationalize
Numerator.
Multiply
Simplify.

19. Divide Out Common Factors from the Numerator and Denominator
Sometimes it is necessary to simplify a radical expression by dividing out common factors from the numerator and denominator. This is a skill we will need in chapter 11 to solve
Quadratic equations, so we look at an example here.
Example 14
Simplify completely.
Solution
It is tempting to do one of the following:
Each is incorrect because is a term in a sum and 18 is a term in
a sum. Be
Careful
Incorrect!
Incorrect!
The correct way to simplify is to begin by factoring out a 9 in the numerator and then
divide the numerator and denominator by any common factors. We will see this on
the next slide.

20. Factor out 9 from the numerator.
Divide by 9.
Simplify.