Download 10.6 Roots of Complex Numbers

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 = rcisθ,
the n distinct nth roots of z are the complex numbers
𝑛
√𝑟𝑐𝑖𝑠
𝜃+2𝑘𝜋
𝑛
for k= 0, 1, 2, . . ., n-1
Example 1:
2𝜋
a) Find the four fourth roots of 81𝑐𝑖𝑠
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b) Find the three cube roots of 125𝑐𝑖𝑠
LESSON 10.6 ROOTS OF COMPLEX NUMBERS
𝜋
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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
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
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
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 other real numbers.
a) 125
b) 216
c) Find the fifth roots of 243
HOMEWORK: Page 525 Class Ex. #1-10 odd; 12
Page 526 Practice Ex. #1-10 even; 12-14; 16-20
LESSON 10.6 ROOTS OF COMPLEX NUMBERS
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LESSON 10.6 ROOTS OF COMPLEX NUMBERS
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