Simplify (-root3+3i)^31
Pls ans I'll mark u as brainliest
Answers
Answered by
0
Answer:
We have:
(
−
√
3
−
i
)
4
First, let's consider the complex number
z
=
−
√
3
−
i
.
In order to apply De Moivre's theorem, we need to evaluate the modulus and argument of this
z
:
⇒
|
z
|
=
√
(
−
√
3
2
+
(
−
1
)
2
)
⇒
|
z
|
=
√
3
+
1
⇒
|
z
|
=
√
4
⇒
|
z
|
=
2
⇒
θ
=
arctan
(
−
1
−
√
3
)
⇒
θ
=
arctan
(
√
3
3
)
⇒
θ
=
π
6
Then,
z
is located in the third quadrant:
⇒
a
r
g
(
z
)
=
π
6
−
π
=
−
5
π
6
So,
z
=
2
(
cos
(
−
5
π
6
)
+
i
sin
(
−
5
π
6
)
)
Now, using De Moivre's theorem:
⇒
z
4
=
2
4
(
cos
(
4
⋅
−
5
π
6
)
+
i
sin
(
4
⋅
−
5
π
6
)
)
⇒
z
4
=
16
(
cos
(
−
10
π
3
)
+
i
sin
(
−
10
π
3
)
)
⇒
z
4
=
16
(
−
1
2
+
√
3
2
i
)
⇒
z
4
=
16
(
−
1
2
(
1
−
√
3
i
)
)
⇒
z
4
=
−
8
(
1
−
√
3
i
)
∴
z
4
=
−
8
+
8
√
3
i
Therefore,
(
−
√
3
−
i
)
4
=
−
8
+
8
√
3
i
.
Step-by-step explanation:
MARK AS A BRAINLISEST ANSWER
Similar questions