Final answer:
The question asks to identify the square roots of the complex number -8 + 8i√3 from given options. The answer involves checking if squaring the options results in the initial complex number. option D is correct.
Step-by-step explanation:
The question revolves around finding the square roots of a complex number, specifically -8 + 8i√3. To determine which of the given options are correct, one would typically use the method of equating real and imaginary parts after squaring the potential square roots. However, since all options are of the form 2 ± 2√3i/3, one can quickly verify by squaring any of these options to see if they result in the initial complex number -8 + 8i√3.
To find the square roots of the complex number
−
8
+
8
�
3
−8+8i
3
, we can use the fact that the square roots of a complex number can be found using the square root property. Let
�
=
−
8
+
8
�
3
z=−8+8i
3
, and we want to find
�
w such that
�
2
=
�
w
2
=z.
The complex square root property states that if
�
2
=
�
w
2
=z, then
�
=
±
�
w=±
z
.
First, find the magnitude and argument of
�
z. The magnitude,
�
r, is given by
�
=
∣
�
∣
=
(
−
8
)
2
+
(
8
3
)
2
=
16
r=∣z∣=
(−8)
2
+(8
3
)
2
=16, and the argument,
�
θ, is given by
�
=
arctan
(
8
3
−
8
)
=
−
�
3
θ=arctan(
−8
8
3
)=−
3
π
.
Now,
�
=
�
⋅
cis
(
�
+
2
�
�
2
)
w=
r
⋅cis(
2
θ+2kπ
) for
�
=
0
,
1
k=0,1.
Substitute
�
r and
�
θ to get the possible square roots.
In this case, the square roots are
�
=
±
4
cis
(
−
�
6
)
w=±4cis(−
6
π
). Thus, the correct answers are
4
cis
(
−
�
6
)
4cis(−
6
π
) and
−
4
cis
(
−
�
6
)
−4cis(−
6
π
).