11-4
Multiplying and Dividing Radical Expressions
Preview
Warm Up
California Standards
Lesson Presentation
11-4
Multiplying and Dividing Radical Expressions
Warm Up
Simplify each expression.
1.
2.
3.
4.
11-4
Multiplying and Dividing Radical Expressions
California
Standards
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.
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.
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.
11-4
Multiplying and Dividing Radical Expressions
Additional Example 1B: Multiplying Square Roots
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Expand the expression.
Commutative Property of Multiplication
Product Property of Square Roots.
Simplify the radicand.
Simplify the square root.
Multiply.
11-4
Multiplying and Dividing Radical Expressions
Additional Example 1C: Multiplying Square Roots
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Factor 12 using a perfect-square factor.
Simplify the radicand.
Product Property of Square Roots
Simplify.
11-4
Multiplying and Dividing Radical Expressions
Check It Out! Example 1a
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Product Property of Square Roots
Multiply the factors in the radicand.
Factor 50 using a perfect-square factor.
Product Property of Square Roots
11-4
Multiplying and Dividing Radical Expressions
Check It Out! Example 1b
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Expand the expression.
Commutative Property of Multiplication
Product Property of Square Roots
Simplify the radicand.
Simplify the square root.
Multiply.
11-4
Multiplying and Dividing Radical Expressions
Check It Out! Example 1c
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Factor 14m.
Product Property of Square Roots
Product Property of Square Roots
Simplify.
11-4
Multiplying and Dividing Radical Expressions
Additional Example 2A: Using the Distributive Property
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Distribute
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.
11-4
Multiplying and Dividing Radical Expressions
Additional Example 2B: Using the Distributive Property
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Distribute
Product Property of Square Roots
Simplify the radicands.
Simplify.
11-4
Multiplying and Dividing Radical Expressions
Check It Out! Example 2a
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Distribute
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.
11-4
Multiplying and Dividing Radical Expressions
Check It Out! Example 2b
Multiply. Write the product in simplest form.
All variables represent nonnegative numbers.
Distribute
Product Property of Square Roots
Factor 50 using a perfect-square
factor.
Simplify.
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.
11-4
Multiplying and Dividing Radical Expressions
Remember!
First terms
Outer terms
Inner terms
Last terms
See Lesson 7-8.
11-4
Multiplying and Dividing Radical Expressions
= 20
+3
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 by combining like terms.
Simplify the radicand.
Simplify.
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.
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.
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.
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.
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 perfectsquare radicand in the denominator.
Product Property of Square Roots
Simplify the denominator.
11-4
Multiplying and Dividing Radical Expressions
Additional Example 4B: Rationalizing the
Denominator
Simplify the quotient. All variables represent
nonnegative numbers.
Multiply by a form of 1 to get a perfectsquare radicand in the denominator.
Simplify the square root in denominator.
11-4
Multiplying and Dividing Radical Expressions
Helpful Hint
Use the square root in the denominator to find
the appropriate form of 1 for multiplication.
11-4
Multiplying and Dividing Radical Expressions
Check It Out! Example 4a
Simplify the quotient.
Multiply by a form of 1 to get a perfectsquare radicand in the denominator.
Simplify the square root in denominator.
11-4
Multiplying and Dividing Radical Expressions
Check It Out! Example 4b
Simplify the quotient.
Multiply by a form of 1 to get a perfectsquare radicand in the denominator.
Simplify the square root in denominator.
11-4
Multiplying and Dividing Radical Expressions
Check It Out! Example 4c
Simplify the quotient.
Multiply by a form of 1 to get a perfectsquare radicand in the denominator.
Simplify the square root in denominator.
Factor and simplify the square root in
the numerator.
11-4
Multiplying and Dividing Radical Expressions
Lesson Quiz
Multiply. Write each product in simplest form.
All variables represent nonnegative numbers.
1.
2.
3.
4.
5.
6.
7.
Simplify each quotient. All variables represent
nonnegative numbers.
8.
9.