Polar Coordinates And Complex Numbers Worksheets With Answers - Math 611b Page 28
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50Powers and Roots of Complex Numbers Worksheet (9.8)
1. Evaluate the product (1 + i)(1 + i)(1 + i)(1 + i)(1 + i) by traditional multiplication. Compare the results
5
with the results using De Moivre’s Theorem on (1 + i)
. Which method do you prefer?
2. Explain how to use De Moivre’s Theorem to find the reciprocal of a complex number in polar form.
3. Find each power. Express the result in rectangular form.
π
π
+ i
− i
3
3
b) (3 ! 5i)
(
3
)
4
[ (cos
3
sin
)]
a)
c)
6
6
π
π
+ i
+
5
4
e) ( ! 2 + 2i)
3
[ (cos
2
sin
)]
(
1
3
i
)
d)
f)
4
4
g) (3 ! 6i)
!2
4
h) (2 + 3i)
i) Raise 2 + 4i to the fourth power.
4. Find all roots of and write answers in a + bi form.
a)the cube roots of ! 8
b) the fifth roots of 32
c) the fourth roots of 16i
5. Find each principal root. Express the result in the form a + bi with a and b rounded to the nearest
hundredth.
π
π
1
1
1
2
2
− −
+ i
6
3
5
i
(
2
i
)
[
32
(cos
sin
)]
a)
b)
c)
3
3
1
1
− +
− i
4
3
(
2
i
)
(
4
)
d)
e)
f) Find the principal square root of i.
6. Solve each equation. Then graph the roots in the complex plane.
! 1 = 0
4
3
3
a) x
+ i = 0
b) 2x
+ 4 + 2i = 0
c) x
+
+
=
4
! (1 + i) = 0
4
4
2
x
2
2 3
i
0
d) 3x
+ 48 = 0
e) x
f)
7. Use a graphing calculator to find all of the indicated roots.
a) fifth roots of 10 ! 9i
b) sixth roots of 2 + 4i
c) eighth roots of 36 + 20i
8. Gloria works for an advertising firm. She must incorporate a hexagon design into a logo for one of the
ads she is working on. She can locate the vertices of a regular hexagon by graphing the solutions to the
! 1 = 0 in the complex plane. What are the solutions to this equation?
6
equation x
28
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