Beginning Algebra, 11th Edition

SECTION 8.4 Rationalizing the Denominator 521

Complete solution available
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8.4 EXERCISES

Rationalize each denominator. See Examples 1 and 2.

1. 
   
   
   
   2. 
      
      
      
      3. 
         
         
         
         4. 
            
         
         
         
         
      
      
      
      
   
   
   
   


2. 
   
   
   
   6. 
      
      
      
      7. 
         
         
         
         8. 
            
         
         
         
         
      
      
      
      
   
   
   
   


3. 
   
   
   
   10. 
       
       
       
       11. 
           
           
           
           12. 
               
           
           
           
           
       
       
       
       
   
   
   
   


4. 
   
   
   
   14. 
       
       
       
       15. 
           
           
           
           16. 
               
           
           
           
           
       
       
       
       
   
   
   
   


5. 
   
   
   
   18. 
       
       
       
       19. 
           
           
           
           20. 
               
           
           
           
           
       
       
       
       
   
   
   
   


6. 
   
   
   
   22. 
       
       
       
       23. 
           
           
           
           24. 
               
           
           
           
           
       
       
       
       
   
   
   
   


7. 
   
   
   
   26. 
       
       
       
       27. 
           
           
           
           28. 
               
               
               
               29. 
                   
               
               
               
               
           
           
           
           
       
       
       
       
   
   
   
   


8. 
   
   
   
   31. 
       
       
       
       32. 
           
           
           
           33. 
               
               
               
               34. 
                   
               
               
               
               
           
           
           
           
       
       
       
       
   
   
   
   

35.Concept Check To rationalize the denominator of an expression such as , we multiply

both the numerator and denominator by. By what number are we actually multiplying

the given expression, and what property of real numbers justifies the fact that our result is

equal to the given expression?

36.In Example 1(a),we showed algebraically that. Support this result numeri-

cally by finding the decimal approximation of on your calculator and then finding the

decimal approximation of. What do you notice?

Simplify. See Example 3.

37. 
    
    
    
    38. 
        
        
        
        39. 
            
            
            
            40. 
                
            
            
            
            
        
        
        
        
    
    
    
    


38. 
    
    
    
    42. 
        
        
        
        43. 
            
            
            
            44. 
                
            
            
            
            
        
        
        
        
    
    
    
    


39. 
    
    
    
    46. 
        
        
        
        47. 
            
            
            
            48. 
                
            
            
            
            
        
        
        
        
    
    
    
    


40. 
    
    
    
    50. 
        
        
        
        51. 
            
            
            
            52. B
            
            
            
            
        
        
        
        
    
    
    
    

256

125

#

B

1

B 16

16

27

#

B

1

B 9

9

8

#

B

7

B 16

2

5

#

B

3

10

B

1

11

#

B

33

B^16

17

3

#

B

17

B^6

1

10

#

B

10

B^3

3

4

#

B

1

5

B

4

3

#

B

3

B 4

2

9

#

B

9

B 2

1

8

#

B

1

B 2

1

12

#

B

1

3

B

5

8

#

B

5

B^6

21

7

#

B

21

B^8

19

20

#

B

20

B^3

7

13

#

B

13

3

326

2

9

26

326

= 2

9

26

23

4

23

B

17

B 11

13

5

• B

1

6

• B

1

5

25

210

28

224

27

232

63

245

• 5
  275


• 3
  250

B

16

B^7

9

B^5

1

B^8

1

32

B

5

B 8

40

3

26

23

210

25

21

245

12

272

10

2300

6

2200

12

218

8

227

9215

622

12210

823

926

25

823

25

15

210

4

26

15

215

5

25

3

22

6

25