express in a+ib form using De moivres theorem (1+i)
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Answer:
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Step-by-step explanation:
Explanation:
De Moivre's theorem states if
z
=
r
(
cos
(
θ
)
+
i
sin
(
θ
)
)
,
then
z
n
=
r
n
(
cos
(
n
θ
)
+
i
sin
(
n
θ
)
)
First step convert from complex form to trig form
a
+
b
i
→
r
(
cos
(
θ
)
+
i
sin
(
θ
)
)
By using
r
=
√
a
2
+
b
2
and
θ
=
arctan
(
b
a
)
We have the number
z
=
1
+
i
, thus
r
=
√
1
2
+
1
2
=
√
2
θ
=
arctan
(
1
1
)
=
π
4
z
=
√
2
(
cos
(
π
4
)
+
i
sin
(
π
4
)
)
←
Trig form
Second step apply De Moivre's Theorem
z
8
=
(
√
2
(
cos
(
π
4
)
+
i
sin
(
π
4
)
)
)
8
=
√
2
8
(
cos
(
8
π
4
)
+
i
sin
(
8
π
4
)
)
=
16
(
cos
(
2
π
)
+
i
sin
(
2
π
)
=
16
(
cos
(
2
π
)
+
i
sin
(
2
π
)
=
16
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