Math Skill: Multiplying and Dividing Radical Expressions
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Multiplying and Dividing Radical Expressions


Knowing how to simplify radical expressions will make multiplying and dividing radical expression easier.
To review radical expression simplification, see here.

Multiplying Radical Expressions

From simplification, we know that √ab = √a • √b

Examples

Solve and simplify.
a.
√3 • √7 = √3 • 7   Multiply
= √21
b.
√5 • √15 = √5 • 15   Multiply
= √5 • 5 • 3   Factor to find perfect square factors
= 5√3
c.
√6x2y • √27x6 = √6x2y • 27x6   Multiply
= √6 • 27 • √x2 • x6 • √ y   Separate integers and variables to make simplifying easier
= √2 • 3 • 3 • 9 • √x8 • √ y   Factor to find perfect square factors
= 9√2 • x4 • √ y   Simplify
= 9x4√2y   Combine

Dividing Radical Expressions

For any nonnegative numbers a and b, where b ≠ 0,
=
    

Examples

Solve and simplify.
a.
=
    
 = √25 = 5
b.
=
    
 = √6b

Rationalizing the denominator

A simplified radical expression does not contain radicals in the denominator.
Removing a radical from the denominator is called rationalizing the denominator.

Examples

Solve and simplify.
a.
=
•
  Multiply by
√2
√2
= 1
=
b.
=
•
  Multiply by
√2
√2
= 1
=
c.
=
•
  Multiply by
√2y
√2y
= 1
=
=

Complexity=1, Mode=mul

Solve and simplify.
1.  
√3 • √15 =    
2.  
√7 • √35 =    

Complexity=2, Mode=mul

Solve and simplify.
1.  
   9e 2
 •
   9e 10
 =    
2.  
   5a 9
 •
   4a 2
 =    

Complexity=1, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1.  
    
 
=
   
2.  
=
   

Complexity=2, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1.  
=
   
2.  
=
   

Complexity=3, Mode=mul

Solve and simplify.
1.  
√21r •
   14r 8s 7
 =    
2.  
   10mn 9
 •
   30m 3n
 =    

Complexity=3, Mode=div

Solve and simplify. Leave fractions as improper fractions.
1.  
    
 
=
   
2.  
=
   

Answers

Complexity=1, Mode=mul

Solve and simplify.
#Correct AnswerYour Answer
1
√3 • √15 =    
Solution
   3
•
   15
 =
   3 • 15
=
   3 • 3 • 5
=3
   5
#Correct AnswerYour Answer
2
√7 • √35 =    
Solution
   7
•
   35
 =
   7 • 35
=
   7 • 7 • 5
=7
   5

Complexity=2, Mode=mul

Solve and simplify.
#Correct AnswerYour Answer
1
   9e 2
 •
   9e 10
 =    
Solution
   9e 2
 •
   9e 10
 =
   9 • e 2
 •
   9 • e 10
=
   9 • 9
•
    e 2 • e 10
=
9•
    e 12
=
9•e 6
=9e 6
#Correct AnswerYour Answer
2
   5a 9
 •
   4a 2
 =    
Solution
   5a 9
 •
   4a 2
 =
   5 • a 9
 •
   4 • a 2
=
   4 • 5
•
    a 9 • a 2
=
2
   5
•
    a 11
=
2
   5
•a 5
    a
=2a 5
   5a

Complexity=1, Mode=div

Solve and simplify. Leave fractions as improper fractions.
#Correct AnswerYour Answer
1
    
 
=
   
Solution
    
 
=
=
#Correct AnswerYour Answer
2
=
   
Solution
=
=

Complexity=2, Mode=div

Solve and simplify. Leave fractions as improper fractions.
#Correct AnswerYour Answer
1
=
   
Solution
=
=
•
=
#Correct AnswerYour Answer
2
=
   
Solution
The numerator and denominator are both divisible by 2
=
    
 
=
    
 
=
=
•
=

Complexity=3, Mode=mul

Solve and simplify.
#Correct AnswerYour Answer
1
√21r •
   14r 8s 7
 =    
Solution
   21r
 •
   14r 8s 7
 =
   7 • 3 • r
 •
   7 • 2 • r 8 • s 7
=
   2 • 3 • 7 • 7
•
    r • r 8
•
    s 7
=
7
   2 • 3
•
    r 9
•s 3
    s
=
7
   6
•r 4
    r
•s 3
    s
=7r 4s 3
   6rs
#Correct AnswerYour Answer
2
   10mn 9
 •
   30m 3n
 =    
Solution
   10mn 9
 •
   30m 3n
 =
   10 • m • n 9
 •
   10 • 3 • m 3 • n
=
   3 • 10 • 10
•
    m • m 3
•
    n 9 • n
=
10
   3
•
    m 4
•
    n 10
=
10
   3
•m 2•n 5
=10m 2n 5
   3

Complexity=3, Mode=div

Solve and simplify. Leave fractions as improper fractions.
#Correct AnswerYour Answer
1
    
 
=
   
Solution
The numerator and denominator are both divisible by 2s
    
 
=
   2s 9
=s 4√2s
#Correct AnswerYour Answer
2
=
   
Solution
The numerator and denominator are both divisible by u 5
=
    
 
=
    
 
=
=
•
=
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