x2 t01 07 nth roots of unity
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nth Roots Of Unity 1nz
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
2 0 0, 1, 25kz cis k
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
y
x
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
y
x0cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
y
x0cis
52cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
y
x0cis
52cis
52
cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
y
x0cis
52cis
52
cis
54cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
y
x0cis
52cis
52
cis
54cis
54
cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
y
x0cis
52cis
52
cis
54cis
54
cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
nth Roots Of Unity 1nzunity of rootsnth theare ,1 form, theof equations of solutions The nz
1 .. 5 zge
When placed on an Argand Diagram, they form a regular n
sided polygon, with vertices on the unit circle.
y
x0cis
52cis
52
cis
54cis
54
cis
2 0 0, 1, 25kz cis k 2 2 4 40, , , ,5 5 5 5
z cis cis cis cis cis
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z 1hen solution tais If 5
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
15 z 1hen solution tais If 5
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roots.other he t
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15 z 1hen solution tais If 5
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solutionais5
roots.other he t
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15 z 1hen solution tais If 5
11555
solutionais5
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15 z 1hen solution tais If 5
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solutionais5
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112
roots.other he t
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15 z 1hen solution tais If 5
11555
solutionais5
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solutionais 2112
roots.other he t
are and ,, that show ,01 ofroot complex a is If b) 54325 zi
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solutionais5
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solutionais 2112
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solutionais5
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solutionais 2112
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roots.other he t
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2552
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15 z 1hen solution tais If 5
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roots.other he t
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roots.other he t
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roots also are and ,,root then a is if Thus 5432
0 1 that Prove 432 ii
0 1 that Prove 432 ii
ab
5432
0 1 that Prove 432 ii
ab
5432 rootstheof sum
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432
OR2 3 41
1
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432
OR2 3 41 GP: 1, , 5a r n
1
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
1
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
1
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0
1
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
1
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii324
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii324
1
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii324
1 324
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii324
1 324
7643
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii324
1 324
7643
1
243
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
0 1 that Prove 432 ii
ab
5432 rootstheof sum
01 432 15
OR 015 011 432
011
01:
12 n
n
NOTE
01 432 1
324 and are roots hoseequation w quadratic theFind iii324
1 324
7643
1
243
01 isequation 2 xx
OR2 3 41 GP: 1, , 5a r n 1
1
na rr
51 1
1
0 5 1 0
c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz
011 2 zzz
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz
011 2 zzz01223 zzzzz
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz
011 2 zzz01223 zzzzz0122 23 zzz
32( ) Evaluate 1i c) If is a complex cube root of unity, use the fact that to; 01 2 ωω
3
1
3
2
1 1( ) Evaluate 1 1
ii
11
2
21
1
2
2
(iii) Form the cubic equation with roots 21 ,1 ,1 0121 3222 zzz
011 2 zzz01223 zzzzz0122 23 zzz
Exercise 4I; 1, 3, 5 (need 4b), 7