This document provides instruction on multiplying and dividing radical expressions. It begins with an introduction and examples of multiplying square roots using the product property. It then covers multiplying sums and differences of radicals using FOIL. Finally, it discusses rationalizing denominators by multiplying quotients by forms of 1 to obtain a perfect square radicand in the denominator. Examples are provided throughout to demonstrate each concept along with practice problems at the end.

1. 11-4 Multiplying and Dividing Radical Expressions
Warm Up
Lesson Presentation
California Standards
Preview

2. 11-4 Multiplying and Dividing Radical Expressions
Warm Up
Simplify each expression.
1.
2.
3.
4.

3. 11-4 Multiplying and Dividing Radical Expressions
Extension of 2.0
Students understand and use such operations
as taking the opposite, finding the reciprocal,
taking a root, and raising to a fractional power.
They understand and use the rules of
exponents.
California
Standards

4. 11-4 Multiplying and Dividing Radical Expressions
You can use the Product and Quotient Properties
of square roots you have already learned to
multiply and divide expressions containing square
roots.

5. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 1A: Multiplying Square Roots
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Product Property of Square Roots
Multiply the factors in the radicand.
Factor 16 using a perfect-square factor.
Product Property of Square Roots
Simplify.

6. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 1B: Multiplying Square Roots
Expand the expression.
Commutative Property of Multiplication
Product Property of Square Roots.
Simplify the radicand.
Simplify the square root.
Multiply.
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

7. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 1C: Multiplying Square Roots
Simplify the radicand.
Factor 12 using a perfect-square factor.
Product Property of Square Roots
Simplify.
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

8. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 1a
Product Property of Square Roots
Multiply the factors in the radicand.
Factor 50 using a perfect-square factor.
Product Property of Square Roots
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

9. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 1b
Expand the expression.
Commutative Property of Multiplication
Product Property of Square Roots
Simplify the radicand.
Simplify the square root.
Multiply.
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

10. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 1c
Product Property of Square Roots
Factor 14m.
Product Property of Square Roots
Simplify.
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

11. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 2A: Using the Distributive Property
Product Property of Square Roots.
Multiply the factors in the second
radicand.
Factor 24 using a perfect-square
factor.
Product Property of Square Roots
Simplify.
Distribute
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

12. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 2B: Using the Distributive Property
Product Property of Square Roots
Distribute
Simplify the radicands.
Simplify.
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

13. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 2a
Product Property of Square Roots
Multiply the factors in the first
radicand.
Factor 48 using a perfect-square
factor.
Product Property of Square Roots
Simplify.
Distribute
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

14. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 2b
Product Property of Square Roots
Factor 50 using a perfect-square
factor.
Simplify.
Distribute
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.

15. 11-4 Multiplying and Dividing Radical Expressions
In Chapter 7, you multiplied binomials by using
the FOIL method. The same method can be used
to multiply square-root expressions that contain
two terms.

16. 11-4 Multiplying and Dividing Radical Expressions
First terms
Outer terms
Inner terms
Last terms
See Lesson 7-8.
Remember!

17. 11-4 Multiplying and Dividing Radical Expressions
= 20 + 3

18. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 3A: Multiplying Sums and
Differences of Radicals
Multiply. Write the product in simplest form.
Use the FOIL method.
Simplify the radicand.
Simplify by combining like terms.
Simplify.

19. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 3B: Multiplying Sums and
Differences of Radicals
Multiply. Write the product in simplest form.
Expand the expression.
Use the FOIL method.
Simplify by combining like terms.

20. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 3a
Multiply. Write the product in simplest form.
Expand the expression.
Use the FOIL method.
Simplify by combining like terms.

21. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 3b
Multiply. Write the product in simplest form.
Use the FOIL method.
Simplify by combining like terms.

22. 11-4 Multiplying and Dividing Radical Expressions
A quotient with a square root in the denominator
is not simplified. To simplify these expressions,
multiply by a form of 1 to get a perfect-square
radicand in the denominator. This is called
rationalizing the denominator.

23. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 4A: Rationalizing the
Denominator
Simplify the quotient. All variables represent
nonnegative numbers.
Multiply by a form of 1 to get a perfect-
square radicand in the denominator.
Product Property of Square Roots
Simplify the denominator.

24. 11-4 Multiplying and Dividing Radical Expressions
Additional Example 4B: Rationalizing the
Denominator
Simplify the square root in denominator.
Multiply by a form of 1 to get a perfect-
square radicand in the denominator.
Simplify the quotient. All variables represent
nonnegative numbers.

25. 11-4 Multiplying and Dividing Radical Expressions
Use the square root in the denominator to find
the appropriate form of 1 for multiplication.
Helpful Hint

26. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 4a
Simplify the quotient.
Simplify the square root in denominator.
Multiply by a form of 1 to get a perfect-
square radicand in the denominator.

27. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 4b
Simplify the quotient.
Simplify the square root in denominator.
Multiply by a form of 1 to get a perfect-
square radicand in the denominator.

28. 11-4 Multiplying and Dividing Radical Expressions
Check It Out! Example 4c
Simplify the quotient.
Simplify the square root in denominator.
Multiply by a form of 1 to get a perfect-
square radicand in the denominator.
Factor and simplify the square root in
the numerator.

29. 11-4 Multiplying and Dividing Radical Expressions
Lesson Quiz
1.
3.
5.
2.
4.
6.
7.
Multiply. Write each product in simplest form.
All variables represent nonnegative numbers.
8. 9.
Simplify each quotient. All variables represent
nonnegative numbers.