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10.6 Roots of Complex Numbers
For complex numbers r and z and for any positive
integer n, r is an nth root of z if and only if rn = z
Complex Roots Theorem
For any positive integer n and any complex number 𝑧 = 𝑟 𝑐𝑖𝑠 𝜃,
the n distinct nth roots of z are the complex numbers
𝜃 + 2𝑘𝜋
𝑛
for 𝑘 = 0, 1, 2, … , 𝑛 − 1
√𝑟 𝑐𝑖𝑠
𝑛
Example 1:
(a) Find the four fourth roots of 81 𝑐𝑖𝑠
(b) Find the three cube roots of 125 𝑐𝑖𝑠
2𝜋
3
𝜋
4
Advanced Mathematics/Trigonometry: Lesson 10.6 Roots of Complex Numbers
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Example 2:
(a) Find the fifth roots of -32i
(b) Find the cube roots of 1000i
(c) Find the fifth roots of -3 + 3i
Advanced Mathematics/Trigonometry: Lesson 10.6 Roots of Complex Numbers
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Example 3:
(a) Graph the four fourth roots of 81.
imaginary
real
(b) Graph the five fifth roots of 32.
imaginary
real
Advanced Mathematics/Trigonometry: Lesson 10.6 Roots of Complex Numbers
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Example 4:
(a) Find the five fifth roots of 1 and locate them on the complex plane.
imaginary
real
(b) Find the three cube roots of 1 and locate them on the complex plane.
imaginary
real
Advanced Mathematics/Trigonometry: Lesson 10.6 Roots of Complex Numbers
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Example 5: Use the cube roots of unity to find the cube roots of each.
(a) 125
(b) 216
(c) Use the fifth roots of unity to find the fifth roots of 243.
Homework: p. 525 Class Exercises 1 – 10; 12
p. 526 Practice Exercises 1 – 10; 12 – 14; 16 – 20
Advanced Mathematics/Trigonometry: Lesson 10.6 Roots of Complex Numbers
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