1. 5.4 – Complex Numbers

2. √ -1

3. √ -1 = i

4. √ -1 = i
-1 = i2

5. √ -1 = i
-1 = i2
Ex. 1 a) Simplify √ -18

6. √ -1 = i
-1 = i2
Ex. 1 a) Simplify √ -18
√ -18 = √ -1 · √ 18

7. √ -1 = i
-1 = i2
Ex. 1 a) Simplify √ -18
√ -18 = √ -1 · √ 18
= √ -1 · √ 9 · √ 2

8. √ -1 = i
-1 = i2
Ex. 1 a) Simplify √ -18
√ -18 = √ -1 · √ 18
= √ -1 · √ 9 · √ 2
= i · 3 · √ 2

9. √ -1 = i
-1 = i2
Ex. 1 a) Simplify √ -18
√ -18 = √ -1 · √ 18
= √ -1 · √ 9 · √ 2
= i · 3 · √ 2
= 3i√ 2

10. b) Simplify √ -125x2

11. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2

12. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2
= √ -1 · √ 53 · √ x2

13. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2
= √ -1 · √ 53 · √ x2
= i · 5√ 5 · x

14. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2
= √ -1 · √ 53 · √ x2
= i · 5√ 5 · x
= 5ix√ 5

15. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2
= √ -1 · √ 53 · √ x2
= i · 5√ 5 · x
= 5ix√ 5
Ex. 2 a) Simplify –2i · 7i

16. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2
= √ -1 · √ 53 · √ x2
= i · 5√ 5 · x
= 5ix√ 5
Ex. 2 a) Simplify –2i · 7i
–2i · 7i = -2 · 7 · i · i

17. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2
= √ -1 · √ 53 · √ x2
= i · 5√ 5 · x
= 5ix√ 5
Ex. 2 a) Simplify –2i · 7i
–2i · 7i = -2 · 7 · i · i
= -14 · i2

18. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2
= √ -1 · √ 53 · √ x2
= i · 5√ 5 · x
= 5ix√ 5
Ex. 2 a) Simplify –2i · 7i
–2i · 7i = -2 · 7 · i · i
= -14 · i2
= -14 · -1

19. b) Simplify √ -125x2
√ -125x2 = √ -1 · √ 125 · √ x2
= √ -1 · √ 53 · √ x2
= i · 5√ 5 · x
= 5ix√ 5
Ex. 2 a) Simplify –2i · 7i
–2i · 7i = -2 · 7 · i · i
= -14 · i2
= -14 · -1
= 14

20. b) √ -10 · √ -15

21. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15

22. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15

23. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15
= i2 · √ 150

24. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15
= i2 · √ 150
= i2 · √ 25 · √ 6

25. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15
= i2 · √ 150
= i2 · √ 25 · √ 6
= -1 · 5 · √ 6

26. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15
= i2 · √ 150
= i2 · √ 25 · √ 6
= -1 · 5 · √ 6
= -5√ 6

27. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15
= i2 · √ 150
= i2 · √ 25 · √ 6
= -1 · 5 · √ 6
= -5√ 6
c) i44

28. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15
= i2 · √ 150
= i2 · √ 25 · √ 6
= -1 · 5 · √ 6
= -5√ 6
c) i44
i44 = (i2)22

29. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15
= i2 · √ 150
= i2 · √ 25 · √ 6
= -1 · 5 · √ 6
= -5√ 6
c) i44
i44 = (i2)22
= (-1)22

30. b) √ -10 · √ -15
√ -10 · √ -15 = i√ 10 · i√ 15
= i · i · √ 10 · √ 15
= i2 · √ 150
= i2 · √ 25 · √ 6
= -1 · 5 · √ 6
= -5√ 6
c) i44
i44 = (i2)22
= (-1)22
= 1

31. Ex. 3 Solve 3x = 0.

32. Ex. 3 Solve 3x = 0.
3x = 0

33. Ex. 3 Solve 3x = 0.
3x = 0
3x2 = -48

34. Ex. 3 Solve 3x = 0.
3x = 0
3x2 = -48

35. Ex. 3 Solve 3x = 0.
3x = 0
3x2 = -48
x2 = -16

36. Ex. 3 Solve 3x = 0.
3x = 0
3x2 = -48
x2 = -16
√ x2 = √ -16

37. Ex. 3 Solve 3x = 0.
3x = 0
3x2 = -48
x2 = -16
√ x2 = √ -16
x = √ -1 · √ 16

38. Ex. 3 Solve 3x = 0.
3x = 0
3x2 = -48
x2 = -16
√ x2 = √ -16
x = √ -1 · √ 16
x = 4i

39. Ex. 3 Solve 3x = 0.
3x = 0
3x2 = -48
x2 = -16
√ x2 = √ -16
x = √ -1 · √ 16
x = ±4i

40. Ex. 3 Solve 3x = 0.
3x = 0
3x2 = -48
x2 = -16
√ x2 = √ -16
x = √ -1 · √ 16
x = ±4i