9-5 Complex Numbers and De Moivre's Theorem
Find each power and express it in rectangular form.
37. (12i – 5)3
SOLUTION: First, write 12i – 5 in polar form.
The polar form of 12i – 5 is
Therefore,
39. (
– i)
. Now use De Moivre’s Theorem to find the third power.
.
3
SOLUTION: First, write
– i in polar form.
The polar form of
Therefore,
– i is
.
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41. (2 + 4i)
4
. Now use De Moivre’s Theorem to find the third power.
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9-5 Therefore,
Complex Numbers and De. Moivre's Theorem
39. (
– i)
3
SOLUTION: – i in polar form.
First, write
The polar form of
Therefore,
– i is
. Now use De Moivre’s Theorem to find the third power.
.
41. (2 + 4i)4
SOLUTION: First, write 2 + 4i in polar form.
The polar form of 2 + 4i is Therefore,
43. (2 + 3i)2
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SOLUTION: First, write 2 + 3i in polar form.
. Now use De Moivre’s Theorem to find the fourth power.
.
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9-5 Therefore,
Complex Numbers and De. Moivre's Theorem
43. (2 + 3i)2
SOLUTION: First, write 2 + 3i in polar form.
The polar form of 2 + 3i is
Therefore,
. Now use De Moivre’s Theorem to find the second power.
.
45. SOLUTION: is already in polar form. Use De Moiver’s Theorem to find the fourth power.
Find all of the distinct p th roots of the complex number.
47. sixth roots of i
SOLUTION: First, write i in polar form.
TheManual
polar form
of iby
is Cognero
1(cos
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+ i sin
). Now write an expression for the sixth roots.
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9-5 Complex Numbers and De Moivre's Theorem
The polar form of i is 1(cos
+ i sin
). Now write an expression for the sixth roots.
Let n = 0, 1, 2, 3, 4 and 5 successively to find the sixth roots.
Let n = 0.
Let n = 1.
Let n = 2.
Let n = 3.
Let n = 4.
Let n = 4
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The sixth roots of i are approximately 0.97 + 0.26i, 0.26 + 0.97i, –0.71 + 0.71i, –0.97 – 0.26 i, – 0.26 – 0.97i, 0.71 –
9-5 Complex Numbers and De Moivre's Theorem
Let n = 4
The sixth roots of i are approximately 0.97 + 0.26i, 0.26 + 0.97i, –0.71 + 0.71i, –0.97 – 0.26 i, – 0.26 – 0.97i, 0.71 –
0.71i.
49. fourth roots of 4
– 4i
SOLUTION: First, write 4
– 4i in polar form.
The polar form of 4
– 4i is
. Now write an expression for the fourth roots.
Let n = 0, 1, 2 and 3 successively to find the fourth roots.
Let n = 0.
Let n = 1.
Let n = 2.
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9-5 Complex Numbers and De Moivre's Theorem
Let n = 2.
Let n = 3.
– 4i are approximately 0.22 + 1.67i, −1.67 + 0.22i, –0.22– 1.67i, 1.67 – 0.22i.
The fourth roots of 4
51. fifth roots of –1 + 11
i
SOLUTION: First, write −1 + 11
i in polar form.
The polar form of −1 + 11
i is
(cos 1.63 + i sin 1.63). Now write an expression for the fifth roots.
Let n = 0, 1, 2, 3, and 4 successively to find the fifth roots.
Let n = 0.
Let n = 1.
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9-5 Complex Numbers and De Moivre's Theorem
Let n = 1.
Let n = 2.
Let n = 3.
Let n = 4.
The fifth roots of −1 + 11
– 1.39i.
i are approximately 1.64 + 0.55i, –0.02 + 1.73i, −1.65 + 0.52i, –1.00 – 1.41i, and 1.03
53. find the square roots of unity
SOLUTION: First, write 1 in polar form.
The polar form of 1 is 1 · (cos 0 + i sin 0). Now write an expression for the square roots.
Let n = 0 to find the first root of 1.
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i areDe
fifth rootsNumbers
of −1 + 11 and
approximately
+ 0.55i, –0.02 + 1.73i, −1.65 + 0.52i, –1.00 – 1.41i, and 1.03
9-5 The
Complex
Moivre's1.64
Theorem
– 1.39i.
53. find the square roots of unity
SOLUTION: First, write 1 in polar form.
The polar form of 1 is 1 · (cos 0 + i sin 0). Now write an expression for the square roots.
Let n = 0 to find the first root of 1.
Notice that the modulus of each complex number is 1. The arguments are found by nπ, resulting in increasing by nπ for each successive root. Therefore, we can calculate the remaining root by adding nπ to the previous .
n =1
The square roots of 1 are ±1.
Use the Distinct Roots Formula to find all of the solutions of each equation. Express the solutions in
rectangular form.
67. x3 = i
SOLUTION: Solve for x.
Find the cube roots of i. First, write i in polar form.
The polar form of i is cos
+ i sin
. Now write an expression for the cube roots.
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Let n = 0, 1, and 2 successively to find the cube roots.
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9-5 Complex Numbers and De Moivre's Theorem
The polar form of i is cos
+ i sin
. Now write an expression for the cube roots.
Let n = 0, 1, and 2 successively to find the cube roots.
Let n = 0.
Let n = 1.
Let n = 2.
The cube roots of i are
,
,and i. Thus, the solutions to the equation are
,
,and i.
69. x4 = 81i
SOLUTION: Solve for x.
Find the fourth roots of 81i. First, write 81i in polar form.
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SOLUTION: Solve for x.
9-5 Complex Numbers and De Moivre's Theorem
Find the fourth roots of 81i. First, write 81i in polar form.
The polar form of 81i is
. Now write an expression for the fourth roots.
Let n = 0, 1, 2, and 3 successively to find the fourth roots.
Let n = 0.
Let n = 1.
Let n = 2.
Let n = 3.
Thus, the solutions to the equation are 2.77 + 1.15i, −1.15 + 2.77i, −2.77 − 1.15i, and 1.15 − 2.77i.
3
71. x +Manual
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SOLUTION: Page 10
9-5 Complex Numbers and De Moivre's Theorem
Thus, the solutions to the equation are 2.77 + 1.15i, −1.15 + 2.77i, −2.77 − 1.15i, and 1.15 − 2.77i.
71. x3 + 1 = i
SOLUTION: Solve for x.
Find the cube roots of −1 + i. First, write −1 + i in polar form.
The polar form of −1 + i is
. Now write an expression for the cube roots.
Let n = 0, 1, and 2 successively to find the cube roots.
Let n = 0.
Let n = 1.
Let n = 2.
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Thus, the solutions to the equation are 0.79 + 0.79i, −1.08 + 0.29i, and 0.29 − 1.08i.
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9-5 Complex Numbers and De Moivre's Theorem
Thus, the solutions to the equation are 0.79 + 0.79i, −1.08 + 0.29i, and 0.29 − 1.08i.
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