Polar Form Of Complex Numbers Worksheet With Answer Key - Openstax College Page 13
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25OpenStax-CNX module: m49408
13
8 Finding Roots of Complex Numbers in Polar Form
th root of a complex number
De Moivre's
n
n
To nd the
in polar form, we use the
th Root Theorem or
Theorem
and raise the complex number to a power with a rational exponent. There are several ways to
n
represent a formula for nding
th roots of complex numbers in polar form.
n
a general note label:
To nd the
th root of a complex number in polar form, use the formula
given as
θ
2kπ
θ
2kπ
1
1
z
= r
cos
+
+ isin
+
(25)
n
n
n
n
2kπ
θ
k = 0, 1, 2, 3, . . . , n
1.
where
We add
to
n in order to obtain the periodic roots.
n
Example 11
Finding the nth Root of a Complex Number
2π
2π
z = 8 cos
+ isin
.
Evaluate the cube roots of
3
3
Solution
We have
2
2
1
1
2kπ
2kπ
z
= 8
cos
+
+ isin
+
3
3
3
3
3
3
3
3
(26)
1
2π
2kπ
2π
2kπ
z
= 2 cos
+
+ isin
+
3
9
3
9
3
k = 0, 1, 2.
k = 0,
There will be three roots:
When
we have
2π
2π
1
z
= 2 cos
+ isin
(27)
3
9
9
k = 1,
When
we have
1
2(1)π
2π
6π
2π
6π
z
= 2 cos
+
+ isin
+
Add
to each angle.
3
9
9
9
9
3
(28)
1
8π
8π
z
= 2 cos
+ isin
3
9
9
k = 2,
When
we have
1
2(2)π
2π
12π
2π
12π
z
= 2 cos
+
+ isin
+
Add
to each angle.
3
9
9
9
9
3
(29)
1
14π
14π
z
= 2 cos
+ isin
3
9
9
k =
Remember to nd the common denominator to simplify fractions in situations like this one. For
1,
the angle simpli cation is
2
2(1)π
2(1)π
2π
1
3
+
=
+
3
3
3
3
3
3
3
2π
6π
=
+
(30)
9
9
8π
=
9
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