Intermediate Algebra (11th edition)

Substitute for

= 1 - i

=

- i+ 1

1

= - 1 i^2.

- i- 1 - 12

- 1 - 12

SECTION 8.7 Complex Numbers 479

(b)

= Distributive property; multiply.

- i-i^2

- i^2

=

11 + i 21 - i 2

i 1 - i 2

1 +i

i

Multiply numerator and denominator

by -i,the conjugate of i.

OBJECTIVE 6 Find powers of i.Because is defined to be we can find

greater powers of ias shown in the following examples.

Notice that the powers of irotate through the four numbers i, and 1. Greater

powers of ican be simplified by using the fact that

Simplifying Powers of i

Find each power of i.

(a)

(b)

(c)

(d)i-^1 = NOW TRY

1

i

=

11 - i 2

i 1 - i 2

=

- i

- i^2

=

- i

- 1 - 12

=

- i

1

=-i

i-^2 =

1

i^2

=

1

- 1

=- 1

i^39 =i^36 #i^3 = 1 i^429 #i^3 = 19 # 1 - i 2 =-i

i^12 = 1 i^423 = 13 = 1

EXAMPLE 8

i^4 =1.

- 1,-i,

i^5 =i#i^4 =i# 1 = i i^8 =i^4 #i^4 = 1 # 1 = 1

i^4 =i^2 #i^2 = 1 - 121 - 12 = 1 i^7 =i^3 #i^4 = 1 - i 2 # 1 =-i

i^3 =i#i^2 =i 1 - 12 =-i i^6 =i^2 #i^4 = 1 - 12 # 1 =- 1

i^2 - 1,

NOW TRY

Use parentheses

to avoid errors.

NOW TRY
EXERCISE 8
Find each power of i.

(a) (b)

(c)i-^6 (d)i-^13

i^16 i^21

NOW TRY
EXERCISE 7
Find each quotient.

(a) (b)

5 - 4 i

i

4 + 2 i

1 + 3 i

NOW TRY ANSWERS

7. (a) (b)


8. (a) 1 (b)i (c) - 1 (d)-i

1 - i - 4 - 5 i

Complete solution available
on the Video Resources on DVD

8.7 EXERCISES

Concept Check Decide whether each expression is equal to 1, i,or

1. 
   
   
   
   2. 
      
      
      
      3. 
         
         
         
         4. 
            
            
            
            5. 
               
               
               
               6. 
                  
               
               
               
               
            
            
            
            
         
         
         
         
      
      
      
      
   
   
   
   

Write each number as a product of a real number and i. Simplify all radical expressions. See

Example 1.

7. 
   
   
   
   8. 
      
      
      
      9. 
         
         
         
         10. 
             
         
         
         
         
      
      
      
      
   
   
   
   


8. 2 - 5 12. 2 - 21 13. 2 - 48 14. 2 - 96

2 - 169 2 - 225 - 2 - 144 - 2 - 196

1 - i 22

1

i

2 - 1 - 2 - 1 i^2 - i^2

• 1, -i.