The nth roots of unity are the solutions to the equation zn = ±1. When placed on an Argand diagram, the nth roots of unity form the vertices of a regular n-sided polygon on the unit circle. For example, the fifth roots of unity are the solutions to z5 = 1, which are the five regularly spaced points forming a pentagon on the unit circle: 1, cis(2π/5), cis(-2π/5), cis(4π/5), cis(-4π/5).

1. nth Roots Of Unity  z n
 1

2. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity

3. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.

4. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1

5. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 

6. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis 
5 5 5 5

7. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis  y
5 5 5 5
x

8. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis  y
5 5 5 5
cis0
x

9. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis  y 2
5 5 5 5 cis
5
cis0
x

10. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis  y 2
5 5 5 5 cis
5
cis0
x
2
cis 
5

11. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis  4 y 2
5 5 5 5 cis cis
5 5
cis0
x
2
cis 
5

12. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis  4 y 2
5 5 5 5 cis cis
5 5
cis0
x
4
cis  2
5 cis 
5

13. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis  4 y 2
5 5 5 5 cis cis
5 5
cis0
x
4
cis  2
5 cis 
5

14. nth Roots Of Unity  z n
 1
The solutions of equations of the form, z n  1, are the nth roots of unity
When placed on an Argand Diagram, they form a regular n sided
polygon, with vertices on the unit circle.
e.g . z 5  1
 2 k  0 
z  cis   k  0, 1, 2
 5 
2 2 4 4
z  cis 0, cis , cis  , cis , cis  4 y 2
5 5 5 5 cis cis
5 5
cis0
x
4
cis  2
5 cis 
5

15. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.

16. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1

17. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1

18. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
1

19. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5

20. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2

21. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
 12
1

22. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
 12
1
 2 is a solution

23. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
    
3 5 5 3
 12
1
 2 is a solution

24. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
    
3 5 5 3
 12  13
1 1
 2 is a solution

25. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
    
3 5 5 3
 12  13
1 1
 2 is a solution  3 is a solution

26. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
    
3 5 5 3
    
4 5 5 4
 12  13
1 1
 2 is a solution  3 is a solution

27. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
    
3 5 5 3
    
4 5 5 4
 12  13  14
1 1 1
 2 is a solution  3 is a solution

28. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
    
3 5 5 3
    
4 5 5 4
 12  13  14
1 1 1
 2 is a solution  3 is a solution  4 is a solution

29. b) i  If  is a complex root of z 5  1  0, show that  2 ,  3 ,  4 and  5 are
the other roots.
z5  1 If  is a solution then  5  1
    15
5 5
 1  is a solution
5
    
2 5 5 2
    
3 5 5 3
    
4 5 5 4
 12  13  14
1 1 1
 2 is a solution  3 is a solution  4 is a solution
Thus if  is a root then  2 ,  3 ,  4 and  5 are also roots

30. ii  Prove that 1     2   3   4  0

31. ii  Prove that 1     2   3   4  0
b
  2  3  4  5  
a

32. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a

33. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0

34. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  5
 1

35. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  5
 1
OR   1  0
5

36. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  5
 1
OR   1  0
5
  11     2   3   4   0

37. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0   1
5
OR   1  0
5
 NOTE : 
 n 
  11     2   3   4   0  1  0 
  11     2     n1   0
 

38. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0   1
5
OR   1  0
5
 NOTE : 
 n 
  11     2   3   4   0  1  0 
1     2   3   4  0   1   11     2     n1   0
 

39. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0   1
5
OR   1  0
5
 NOTE : 
 n 
  11     2   3   4   0  1  0 
1     2   3   4  0   1   11     2     n1   0
 
OR
1   2  3  4

40. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0   1
5
OR   1  0
5
 NOTE : 
 n 
  11     2   3   4   0  1  0 
1     2   3   4  0   1   11     2     n1   0
 
OR
1   2  3  4 GP: a  1, r   , n  5

41. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  5
 1
OR   1  0
5
 NOTE : 
 n 
  11     2   3   4   0  1  0 
1     2   3   4  0   1   11     2     n1   0
 
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1

42. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  5
 1
OR   1  0
5
 NOTE : 
 n 
  11     2   3   4   0  1  0 
1     2   3   4  0   1   11     2     n1   0
 
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1

 1

43. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  5
 1
OR   1  0
5
 NOTE : 
 n 
  11     2   3   4   0  1  0 
1     2   3   4  0   1   11     2     n1   0
 
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1
 0
 1

44. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  5
 1
OR   1  0
5
 NOTE : 
 n 
  11     2   3   4   0  1  0 
1     2   3   4  0   1   11     2     n1   0
 
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1  5
1  0
 0
 1

45. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  1
5
OR   1  0
5
 NOTE : 
 n 
  11           0 
2 3 4
 1  0 
1     2   3   4  0   11     2     n1   0
  1  
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1  5  1  0 
 0
 1
iii  Find the quadratic equation whose roots are    4 and  2   3

46. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  1
5
OR   1  0
5
 NOTE : 
 n 
  11           0 
2 3 4
 1  0 
1     2   3   4  0   11     2     n1   0
  1  
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1  5  1  0 
 0
 1
iii  Find the quadratic equation whose roots are    4 and  2   3
      4  2  3

47. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0  1
5
OR   1  0
5
 NOTE : 
 n 
  11           0 
2 3 4
 1  0 
1     2   3   4  0   11     2     n1   0
  1  
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1  5  1  0 
 0
 1
iii  Find the quadratic equation whose roots are    4 and  2   3
      4  2  3
 1

48. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0   1
5
OR   1  0
5
 NOTE : 
 n 
  11           0 
2 3 4
 1  0 
1     2   3   4  0   11     2     n1   0
  1  
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1  5  1  0 
 0
 1
iii  Find the quadratic equation whose roots are    4 and  2   3
       4 2 3      4  2   3 
 1

49. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0   1
5
OR   1  0
5
 NOTE : 
 n 
  11           0 
2 3 4
 1  0 
1     2   3   4  0   11     2     n1   0
  1  
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1  5  1  0 
 0
 1
iii  Find the quadratic equation whose roots are    4 and  2   3
       4 2 3      4  2   3 
 1  3  4  6  7

50. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0   1
5
OR   1  0
5
 NOTE : 
 n 
  11           0 
2 3 4
 1  0 
1     2   3   4  0   11     2     n1   0
  1  
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1  5  1  0 
 0
 1
iii  Find the quadratic equation whose roots are    4 and  2   3
       4 2 3      4  2   3 
 1  3  4  6  7
 3  4    2
 1

51. ii  Prove that 1     2   3   4  0
b
  2  3  4  5   sum of the roots 
a
1   2  3  4  0   1
5
OR   1  0
5
 NOTE : 
 n 
  11           0 
2 3 4
 1  0 
1     2   3   4  0   11     2     n1   0
  1  
OR a  r n  1
1   2  3  4  GP: a  1, r   , n  5
r 1
1 5  1  5  1  0 
 0
 1
iii  Find the quadratic equation whose roots are    4 and  2   3
       4 2 3      4  2   3 
 1  3  4  6  7
 3  4    2
 equation is x 2  x  1  0
 1

52. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;

53. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1   
2 3

54. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1   
2 3
   
3

55. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1   2 3

   
3
  3
 1

56. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1   2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
  3
 1

57. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1   2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1

58. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1   2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1 1 

 2

59. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1   2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1 1 

 2
2
 2

1

60. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1   2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1 1 

(iii) Form the cubic equation with roots 1, 1   , 1   2 2
2
 2

1

61. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1    2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1 1 

(iii) Form the cubic equation with roots 1, 1   , 1   2 2
 z  1z  2     z  1         0
2 2 2 3

2
2
1

62. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1    2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1 1 

(iii) Form the cubic equation with roots 1, 1   , 1   2 2
 z  1z  2     z  1         0
2 2 2 3

2
2
 z  1z 2  z  1  0
1

63. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1    2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1 1 

(iii) Form the cubic equation with roots 1, 1   , 1   2 2
 z  1z  2     z  1         0
2 2 2 3

2
2
 z  1z 2  z  1  0
1
z  z  z  z  z 1  0
3 2 2

64. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1    2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1 1 

(iii) Form the cubic equation with roots 1, 1   , 1   2 2
 z  1z  2     z  1         0
2 2 2 3

2
2
 z  1z 2  z  1  0
1
z  z  z  z  z 1  0
3 2 2
z3  2z 2  2z 1  0

65. c) If  is a complex cube root of unity, use the fact that 1  ω  ω 2  0 to;
(i ) Evaluate 1    2 3 1 1
(ii ) Evaluate 
1  1 2
   
3
1 1
 2 
  3  
 1 1 

(iii) Form the cubic equation with roots 1, 1   , 1   2 2
 z  1z  2     z  1         0
2 2 2 3

2
2
 z  1z 2  z  1  0
1
z  z  z  z  z 1  0
3 2 2
z3  2z 2  2z 1  0
Exercise 4I; 1, 3, 5 (need 4b), 7