Intermediate Algebra (11th edition)

(b)

Quotient rule

3

2 -

2

3 =

9

6 -

4

6 =

5

= x 6

5/6

= x3/2-2/3

=

x3/2

x2/3

2 x^3

23 x^2

SECTION 8.2 Rational Exponents 441

Convert to rational

exponents.

(c)

= z1/8 Power rule

= 1 z1/4 2 1/2

= 2 z1/4

324 z

Convert the inside

radical to rational

exponents.

Convert to rational

exponents.

NOW TRY

NOW TRY
EXERCISE 6
Write all radicals as exponen-
tials, and then apply the rules
for rational exponents. Leave
answers in exponential form.
Assume that all variables rep-
resent positive real numbers.

(a) (b)

(c)

NOW TRY ANSWERS

6. (a) (b) (c)y1/6
   1
   y7/4

y14 /15

323 y

24 y^3

2 y^5

25 y^3 # 23 y

Complete solution available
on the Video Resources on DVD

8.2 EXERCISES

Concept Check Match each expression from Column I with the equivalent choice from

Column II.

I

1. 
   
   
   
   2. 
      
   
   
   
   


2. 
   
   
   
   4. 
      
   
   
   
   


3. 
   
   
   
   6. 
      
   
   
   
   


4. 
   
   
   
   8. 
      
   
   
   
   


5. 
   
   
   
   10. 
       
   
   
   
   

II

A. B. 8

C. D.

E. F.

G. 4 H.

I. 6 J.Not a real number

- 2

- 3 26

23 - 26

- 4

- 6 2/4 36 0.5

4 3/2 6 2/4

1 - 322 1/5 1 - 322 2/5

- 16 1/2 1 - 252 1/2

3 1/2 1 - 272 1/3

Evaluate each exponential. See Examples 1–3.

11. 
    
    
    
    12. 
        
        
        
        13. 
            
            
            
            14. 
                
            
            
            
            
        
        
        
        
    
    
    
    


12. 
    
    
    
    16. 
        
        
        
        17. 
            
            
            
            18. 
                
            
            
            
            
        
        
        
        
    
    
    
    


13. 
    
    
    
    20. 
        
        
        
        21. 
            
            
            
            22. 
                
            
            
            
            
        
        
        
        
    
    
    
    


14. 
    
    
    
    24. 
        
        
        
        25. 
            
            
            
            26. 
                
            
            
            
            
        
        
        
        
    
    
    
    


15. 
    
    
    
    28. 
        
        
        
        29. 
            
            
            
            30. 
                
            
            
            
            
        
        
        
        
    
    
    
    


16. 
    
    
    
    32. 
        
        
        
        33. 
            
            
            
            34. 
                
            
            
            
            
        
        
        
        
    
    
    
    


17. 
    
    
    
    36. 
        
        
        
        37. 
            
            
            
            38. 
                
            
            
            
            
        
        
        
        
    
    
    
    

Write with radicals. Assume that all variables represent positive real numbers. See Example 4.

39. 
    
    
    
    40. 
        
        
        
        41. 
            
        
        
        
        
    
    
    
    


40. 
    
    
    
    43. 
        
        
        
        44. 
            
        
        
        
        
    
    
    
    


41. 
    
    
    
    46. 
        
        
        
        47. 
            
        
        
        
        
    
    
    
    


42. 
    
    
    
    49. 
        
        
        
        50. 
            
        
        
        
        
    
    
    
    

Simplify by first converting to rational exponents. Assume that all variables represent positive

real numbers. See Example 4.

51. 
    
    
    
    52. 
        
        
        
        53. 
            
            
            
            54. 
                
                
                
                55. 
                    
                
                
                
                
            
            
            
            
        
        
        
        
    
    
    
    

56. 57. 58. 59. 60.

24 w^3

26 w

23 t^4

25 t^4

2 r^5023 x# 2 x 24 y# 25 y^2

2212 2510 2349 2468 2 x^20

1 r+ 2 z 2 3/2 13 m^4 + 2 k^22 - 2/3 15 x^2 + 3 z^32 - 5/6

12 m 2 - 3/2 15 y 2 - 3/5 12 y+x 2 2/3

7 2/3 19 q 2 5/8- 12 x 2 2/3 13 p 2 3/4+ 14 x 2 1/3

10 1/2 3 1/2 8 3/4

a

729

64

b

• 5/6
  a

16

81

b

• 3/4
  a

64

125

b

• 2/3
  a

125

27

b

• 2/3

32 - 3/5 27 - 4/3 64 - 3/2 81 - 3/2

- 16 5/2 - 32 3/5 1 - 82 4/3 1 - 2432 2/5

100 3/2 64 3/2 81 3/4 216 2/3

1 - 272 1/3 1 - 322 1/5 1 - 1442 1/2 1 - 362 1/2

a

8

27

b

1/3

a

64

81

b

1/2

16 1/4 625 1/4

169 1/2 121 1/2 729 1/3 512 1/3

NOTE The ability to convert between radicals and rational exponents is important

in the study of exponential and logarithmic functions in Chapter 10.