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If z = 1- √3i, then z^6 equals
Related Topics
Wize University Linear Algebra Textbook > Complex Numbers
Powers and Roots
5 Activities
If
z
=
1
−
3
i
z = 1- \sqrt{3}i
z
=
1
−
3
i
, then
z
6
z^6
z
6
equals
−
64
-64
−
64
64
64
64
−
64
i
-64i
−
64
i
64
i
64i
64
i
None of the above.
I don't know
Check Submission
More Powers and Roots Questions:
Suppose
f
(
x
)
f(x)
f
(
x
)
is a polynomial of degree
n
n
n
with coefficients in
C
\mathbb{C}
C
. Consider the following statements:
(i) If
z
=
a
+
b
i
z = a + bi
z
=
a
+
bi
is a root of
f
(
x
)
f(x)
f
(
x
)
, then so is
z
=
a
−
b
i
z = a - bi
z
=
a
−
bi
.
(ii) If
n
=
2
n = 2
n
=
2
and
f
(
x
)
f(x)
f
(
x
)
has a nonzero linear term, then
f
(
x
)
f(x)
f
(
x
)
can only have roots if its coefficients are real.
Practice: Powers and Roots of Complex Numbers
(a)
If
u
=
−
3
2
−
1
2
i
u=-\frac{\sqrt{3}}{2}-\frac{1}{2}i
u
=
−
2
3
−
2
1
i
, find
u
60
u^{60}
u
60
(b)
Find all three (complex) cube roots of 8
If
z
=
4
−
4
i
z = 4-4i
z
=
4
−
4
i
, find
z
8
z^8
z
8
.
If
w
=
2
−
2
i
w=\sqrt{2}-\sqrt{2}i
w
=
2
−
2
i
, find
w
8
w^8
w
8
.
Which of the following is a 5th root of
−
i
-i
−
i
?
If
z
=
3
2
+
1
2
i
z=\frac{\sqrt{3}}{2}+\frac{1}{2}i
z
=
2
3
+
2
1
i
, then find
z
9
z^9
z
9
Which of the following is a fourth root of the complex number
z
=
−
1
+
i
z = -1 + i
z
=
−
1
+
i
?
Complex roots
Find all of the solutions (real and complex) to the equation
z
6
−
729
=
0
z^6-729=0
z
6
−
729
=
0