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Section 7.3 
Simplifying & Combining Radical Expressions 
 Product & Quotient Rules for Radicals 
 Simplifying Radical Expressions 
 Adding & Subtracting Radical Expressions 
index radicand
The Product Rule for Radicals 
n n n ab  a  b 
100  425  4  25  25 10 
1000 8 125 8 125 2 5 10 3 3 3 3        
3 3 4 3 3 3 3 240  2 35  2 235  2 30
The Quotient Rule for Radicals 
n 
n 
n 
a 
b 
a 
 
b 
8 
11 
64 
121 
64 
121 
  
27 3 
y x 
3 4 
x 
3 3 3 
y x 
3 4 
x 
y x 
27 (4 ) 
x x 
y 
x 
2 
2 
2 (4 ) 
2 
3 3 3 
3 
2 
3 
3 
2 
3 
  
A Radical Expression is Simplified When: 
1. Each factor in the radicand is to a power less than the 
index of the radical 
2. The radicand contains no fractions or negative numbers 
3. No radicals appear in the denominator of a fraction 
3 2 4 
16 
x y 
3 2 
y x y 
2 2 
5 4 
a a 
5 4 
2 
2 
a a 
 
 
2 
2 
2 
a 
a 
32 
2 
x 
x 
x 
3 2 
3 
3 
2 2
Remove all perfect nth root factors 
from the radicand (numbers) 
12  4  3  4 3  
2 3 
98  49  2  49 2  
7 2 
3 3 3 3 3 
54  27  2  27 2  
3 2 
5 5 7 5 5 5 2 5 
        
128 ( 2) ( 2) ( 2) 2 4
Remove all perfect nth root factors from the 
radicand (variables) 
9 8 8 4 
m  m  m  m m  
m m 
5 4 2 
a a a a a 
128  64  2  
8 2 
3 5 3 3 2 3 2 
x x x x x 
24  8  3  
2 3 
5 9 5 5 5 5 4 5 4 
a b  a b  a  
ab a
Leave no fractions or negative 
numbers in the radicand 
x x 2 
2 
x 2 
x 
    
3 4 3 3 3 2 3 
y y y y y y y 
y x 
3 5 3 5 3 5 
3 4 
x 
     
3 3 3 
y x 
3 4 
x 
y x 
27 (4 ) 
x x 
y 
27 
x 
2 
2 
2 (4 ) 
2 
2 
4 
2 
2 2 
3 
3 3 3 
3 
2 
3 
3 
2 
3 
  
No radicals remain in the denominator 
of a fraction 
5 
xy 
432 
45 
3 3 4 3 3 3 
45 
xy 
3 5 
432 
3 
2 
2 2 
54 27 2 3 2 
8 
8 
9 3 
5 
5 
x x x x x 
x 
x 
x 
x 
y y 
x 
x 
       
 
 
 
  
Adding & Subtracting Like Radicals 
 Each term must have a radical with identical 
index and radicand 
 Law of distribution allows combining or 
factoring 
 Like radicals: 
4 4 3 2 and 7 2 2x 3x and 11 3x 
 Unlike radicals (cannot combine) 
5 4 3 5 and 2 3 2y 3x and 4 3x
Simplify before Trying to Combine 
2 12 3 48 3 3 
   
2 4 3 3 16 3 3 3 
   
2(2) 3 3(4) 3 3 3 
   
4 3  12 3  3 3   
5 3 
3 3 3 
16 54 24 
   
3 3 3 3 3 3 
8 2 27 2 8 3 
   
3 3 3 
2 2 3 2 2 3 
   
3 3 
  
2 2 3 
16 54 128 ( ) 3 4 3 2 3 4 x  x   x board
 More on addition and subtraction of radicals
What Next? Multiplication & Division! 
 Present Section 7.4 
Multiply & Divide Radical Expressions 
radicand index root index th 

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Lecture 05 a radicals

  • 1. Section 7.3 Simplifying & Combining Radical Expressions  Product & Quotient Rules for Radicals  Simplifying Radical Expressions  Adding & Subtracting Radical Expressions index radicand
  • 2. The Product Rule for Radicals n n n ab  a  b 100  425  4  25  25 10 1000 8 125 8 125 2 5 10 3 3 3 3        3 3 4 3 3 3 3 240  2 35  2 235  2 30
  • 3. The Quotient Rule for Radicals n n n a b a  b 8 11 64 121 64 121   27 3 y x 3 4 x 3 3 3 y x 3 4 x y x 27 (4 ) x x y x 2 2 2 (4 ) 2 3 3 3 3 2 3 3 2 3   
  • 4. A Radical Expression is Simplified When: 1. Each factor in the radicand is to a power less than the index of the radical 2. The radicand contains no fractions or negative numbers 3. No radicals appear in the denominator of a fraction 3 2 4 16 x y 3 2 y x y 2 2 5 4 a a 5 4 2 2 a a   2 2 2 a a 32 2 x x x 3 2 3 3 2 2
  • 5. Remove all perfect nth root factors from the radicand (numbers) 12  4  3  4 3  2 3 98  49  2  49 2  7 2 3 3 3 3 3 54  27  2  27 2  3 2 5 5 7 5 5 5 2 5         128 ( 2) ( 2) ( 2) 2 4
  • 6. Remove all perfect nth root factors from the radicand (variables) 9 8 8 4 m  m  m  m m  m m 5 4 2 a a a a a 128  64  2  8 2 3 5 3 3 2 3 2 x x x x x 24  8  3  2 3 5 9 5 5 5 5 4 5 4 a b  a b  a  ab a
  • 7. Leave no fractions or negative numbers in the radicand x x 2 2 x 2 x     3 4 3 3 3 2 3 y y y y y y y y x 3 5 3 5 3 5 3 4 x      3 3 3 y x 3 4 x y x 27 (4 ) x x y 27 x 2 2 2 (4 ) 2 2 4 2 2 2 3 3 3 3 3 2 3 3 2 3   
  • 8. No radicals remain in the denominator of a fraction 5 xy 432 45 3 3 4 3 3 3 45 xy 3 5 432 3 2 2 2 54 27 2 3 2 8 8 9 3 5 5 x x x x x x x x x y y x x             
  • 9. Adding & Subtracting Like Radicals  Each term must have a radical with identical index and radicand  Law of distribution allows combining or factoring  Like radicals: 4 4 3 2 and 7 2 2x 3x and 11 3x  Unlike radicals (cannot combine) 5 4 3 5 and 2 3 2y 3x and 4 3x
  • 10. Simplify before Trying to Combine 2 12 3 48 3 3    2 4 3 3 16 3 3 3    2(2) 3 3(4) 3 3 3    4 3  12 3  3 3   5 3 3 3 3 16 54 24    3 3 3 3 3 3 8 2 27 2 8 3    3 3 3 2 2 3 2 2 3    3 3   2 2 3 16 54 128 ( ) 3 4 3 2 3 4 x  x   x board
  • 11.  More on addition and subtraction of radicals
  • 12. What Next? Multiplication & Division!  Present Section 7.4 Multiply & Divide Radical Expressions radicand index root index th 