Name _________________________________________
Objectives:
HA2 6.2 Multiplying/Dividing Radical
Expressions
To multiply and divide radical
expressions
ESSENTIAL UNDERSTANDING: A radical expression can be simplified when the exponent of one
factor of the radicand is a multiple of the radical’s index. Products of radicals that have the same index can
be simplified. You can write a quotient of roots as a root of a quotient…and vise versa! Remember, a radical
expression can be simplified when the exponent of the radicand is a multiple of the radical’s index.
KEY TERMS_:
__
_____
simplest form of a radical expression
quotient
rationalize the denominator
Remember, we are working with REAL NUMBER Radical Expressions!!
This means the radicand MUST be POSITIVE!
Example 1
Multiplying Radical Expressions
Can you simplify the product of these radical expressions? Explain your reasoning.
a) 3 6 2
b) 4 125 4 405
c)
2
8
Got It? Can you simplify the product of the radical expression? Explain.
4
7
5
5
7
Example 2
5
5
2
Simplifying a Radical Expression
What is the simplest form of 3 135x5 ?
Got It? What is the simplest form of
What is the simplest form of
3
128x 7 ?
4
48y 9 ?
Example 3
Simplifying a Product of Radical Expressions
What is the simplified form of each radical expression?
a.
12 x12
9 x3
b.
48x5 y 2
Solve 75x10 12 x7 in the following two ways:
a) Simplify each radical expression, then multiply
50 x 2 y 4
b) Multiply radicands, then simplify
Why does the order of the process not matter?
Got It? What is the simplest form of
45 x5 y 3
35 xy 4 ?
Practice
1) Can you simplify the product of the radical expressions? Explain.
a)
3
6 2
2) What is the simplest form of
b)
3
54x5 ?
3
4
2
2
3) What is the simplest form of 3 128x7 ?
4) What is the simplest form of
72 x 3 y 2
10 xy 3 ?
5) What is the simplest form of
45 x5 y 3
6) What is the simplest form of the quotient ?
18 x5
a)
3
b)
2 x3
162 y 5
3
7) What is the simplest form of
50 x 6
2x4
3y2
?
Recall that when working with radicals you will sometimes need to rationalize the denominator.
For example:
1
1  2
=

  ____________________
2
2  2 
8) Rationalize the denominator of each expression:
a)
3
7
2x
3
b)
7x
3 5y
35 xy 4 ?
Example 4
Dividing Radical Expressions
What is the simplest form of the quotient? Explain your reasoning.
b)
3
12 x11
b)
3x5
Why does the answer to
18 x5
2 x3
189 x 7
3
7 x2
c)
7
x5
4
x2
 3x not include absolute value symbols?
Compare the Combining Radicals: Quotients Property to the Combining Radicals: Products Property.
How are they alike and how are they different?
Got It? What is the simplest form of
50x 6
2x 4
?
When you rewrite a radical expression so that there are no radicals in any denominator, you apply the
concept that a radical expression can be simplified when the exponent of the radicand is a multiple of the
radical’s index.
Example 5
Rationalizing the Denominator – The Basics
Rationalize the denominator for each of the following quotients
3
5
x
x
a) 3
b)
5
2
9
.
3
c)
3
7
16x
Example 6
Rationalizing the Denominator
What is the simplest form of each radical expression?
a)
3
4 x4
32 yz 3
b)
18 x 5 y 2
48 x 2 y 3
 What is an advantage to writing the numerator and denominator using prime factors?
How do you choose what to multiply by when rationalizing the denominator of a radical quotient?
3
Got It? What is the simplest form of
3
7x
5y 2
?