Section 98 Powers and Roots of Complex Numbers

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Section 9-8Powers and Roots ofComplex Numbers
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DeMoivres Thm.

• r(cos?isin ?)n rn (cos n? isin n?)
• n is a pos. integer
• Find (1-i)8 .this would take forever to
  expandso
• r ?(12 (-1)2) ?2
  
• ? tan-1 (-1/1) -?/4 OR 7?/4
• ?2(cos 7?/4 isin 7?/4)8
• (?2)8 (cos 8 7?/4 isin 8 7?/4)
• POLAR 16(cos 14? isin 14?)
• WRITE IN RECT. 16(1 i(0)) 16

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(xyi)1/p r1/p (cos ?/p isin ?/p)

• Example
• 3vi i1/3
• (01i)1/3
• r v(02 12) v11
• T tan-1 (1/0) ?/2
• POLAR 1(cos ?/2 isin ?/2)1/3
• 1(cos ?/6 isin ?/6)
• 1(v3/2 i 1/2)
• RECT v3/2 1/2 i

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(1 i) 1/3

• r v(12 12) v2
• T tan -1 (1/1) ?/4
• v2(cos ?/4 isin ?/4) 1/3
• (21/2)1/3 (cos ?/12 isin ?/12)
• 21/6 (cos ?/12 isin ?/12)
• 6v2 (cos ?/12 isin ?/12 )
• 1.08 .29i

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Any complex has

• 2, square roots
• 3, cube roots
• 4, 4th roots
• 5, 5th roots
• P, Pth roots

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The p distinct pth roots of xyi

• (xyi)1/p r(cos(?2n?) isin(?2n?)1/p
• For n0,1,2,,p-1

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Find the 5,5th roots of -243

• (xyi)1/p r(cos(?2n?) isin(?2n?)1/p
• (-2430i)1/5
• r 243 and ? tan-1 (0/-243) ?
• 243(cos(?2n?)isin(?2n
  ?)1/5
• n0 3(cos(?/5) isin(?/5)) 2.43 1.76i
• n1 3(cos(3?/5) isin(3?/5)) -.93 2.85i
• n2 3(cos(5?/5) isin(5?/5)) -3 0i -3
• n3 3(cos(7?/5) isin(7?/5)) -.93 2.85i
• n4 3(cos(9?/5) isin(9?/5)) 2.43 1.76i

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Solve x3270

• X3 -27
• Find the 3 cube roots -27 0i
• r 27 , ? ?
• 27(cos (? 2n?) isin (? 2n?) 1/3
• n0 3(cos (?/3) isin (?/3)) (3/2)(3v3i)
  
  2
• n1 3(cos (3?/3) isin (3?/3)) (-3 0i)
• n2 3(cos (5?/3) isin (5?/3))
  (3/2)-(3v3i)
  2

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Homework

• P 516
• 11-21 odd, 22,24,35-39