Question

Find the values of (1+i✓3)3​

Answer 1

Given, z
5
=(1+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.