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Question
Mathematics
If z = 1 + i√3, then z6 equals
Q. If
z
=
1
+
i
3
, then
z
6
equals
2571
202
Complex Numbers and Quadratic Equations
Report Error
A
32
B
−
32
C
64
D
None of these
Solution:
z
=
1
+
i
3
∴
z
6
=
(
1
+
i
3
)
6
...
(
i
)
Using binomial theorem, (i) becomes
=
6
C
0
+
6
C
1
(
i
3
)
+
6
C
2
(
i
3
)
2
+
6
C
3
(
i
3
)
3
+
6
C
4
(
i
3
)
4
+
6
C
5
(
i
3
)
5
+
6
C
6
(
i
3
)
6
=
(
6
C
0
−
6
C
2
(
3
)
+
6
C
4
(
9
)
−
6
C
6
(
27
))
+
i
3
(
6
C
1
−
3
6
C
3
+
9
6
C
5
)
=
(
1
−
15
×
3
+
135
−
27
)
+
i
3
(
60
−
60
)
=
(
136
−
72
)
+
i
3
(
0
)
=
64
Short Cut Method (i) :
z
=
1
+
i
3
=
−
(
2
−
1
−
i
3
)
×
2
=
−
2
ω
2
∴
z
6
=
(
−
2
)
6
(
ω
2
)
6
=
2
6
ω
12
2
6
=
2
6
=
64
Short Cut Method (ii) :
z
=
1
+
i
3
z
=
2
(
2
1
+
i
2
3
)
⇒
z
=
2
(
cos
3
π
+
i
s
in
3
π
)
⇒
z
=
2
e
iπ
/3
∴
z
6
=
2
6
e
2
iπ
=
2
6
=
64
(
∵
cos
2
π
=
1
)