Find all possible values of (1+i)1/3 and obtain
their product.
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0
Answer:
+i=
3
)
=2
∣
∣
∣
∣
∣
∣
2
1
+i
2
3
∣
∣
∣
∣
∣
∣
=2[cos(
3
π
)+isin(
3
π
)]
Here, r=2,ϕ=arg(z)=π/3,n=5
n
th
roots are given by r
1/5
[cos(
n
ϕ+2kπ
)+isin(
n
ϕ+2kπ
)]
z
0
=2
1/5
⎣
⎢
⎡
5
cos
3
π
+2(0)π
⎦
⎥
⎤
+i
⎣
⎢
⎡
5
sin
3
π
+2(0)π
⎦
⎥
⎤
z
0
=2
1/5
(cos
15
π
+isin
15
π
)
z
1
=2
1/5
⎣
⎢
⎡
5
cos
3
π
+2π
+
5
isin
3
π
+2π
⎦
⎥
⎤
=2
1/5
∣
∣
∣
∣
∣
cos
15
7π
+isin
15
7π
]
z
2
=2
1/5
⎣
⎢
⎡
5
cos
3
π
+4π
+
5
isin
3
π
+4π
⎦
⎥
⎤
=2
1/5
∣
∣
∣
∣
∣
cos
15
13π
+isin
15
3π
]
z
3
=2
1/5
[cos
15
19π
+isin
15
19π
] [ putting k=3 ]
z
4
=2
1/5
[cos
15
25π
+isin
15
25π
] [ putting k=4 ]
z
4
=2
1/5
∣
∣
∣
∣
∣
cos
3
5π
+isin
3
5π
∣
∣
∣
∣
∣
z
0
,z
1
,z
2
,z
3
,z
4
are the fifth roots of given complex no.
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