10.5 Powers of Complex Numbers and De Moivre

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10.5 Powers of Complex Numbers and De Moivres
Theorem(de moi-yay)
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• DeMoivres Theorem is used to raise complex
  numbers to integer powers.
• If z abi is any complex number with polar form
  rcis? and n is any positive integer then the nth
  power of z is given by

The proof of this is simple, we remember that if
(rcis?)2 (rcis?)(rcis?) And then by multiplying
2 complex numbers in polar form we get rrcis(?
?) Which is the same as r2(cis2?) Now if we
wanted to find (rcis ?)3. We would simply take
(rcis ?)2(rcis ?). Which is the same as r2(cis2
?)(rcis ?) Which is the same as r3cis3 ?. Thus a
pattern has developed that is consistent with de
Moivres Theorem.
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Example
And in rectangular form that would be 064i Draw
a picture to convince ourselves of this
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• Rewrite this in rectangular form.

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Write in
the form a bi
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• Let z1-i Express z3, z5, z7 in polar form. Then
  in rectangular form.
• In order to complete this you must first put z
  into polar form.
• So now we have

In rectangular form that would be -8 8i
On your own find the other 2
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HWK pg. 410 1,2,3,5
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11.4 Roots of Complex numbers

• De Moivres Theorem helps as well to find roots.
• The n nth roots of z rcis? are

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Change to polar form, identify r and
?. Replace z in Rectangular form with z in
polar form Apply DeMoivres Theorem involving
complex roots.
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• Evaluate

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• Find the cube roots of 16i

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• We can also take nth roots of complex numbers, so
  maybe we want to find the square root or the cube
  root of (23i), DeMoivres Theorem is beneficial
  in helping us do that.
• Something to remember, if I want to find the
  fourth root of (23i), there will be in fact 4
  different solutions that I could raise to the 4th
  power to get (23i), so when you are asked for
  the nth roots of something you will have n
  solutions.

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The following comes directly from de moivres
Theorem
Find the cube roots of 16i. First understand
that there will be a z1, z2, and z3. To find z1
you will use k0 To find z2 you will use k1 To
find z3 you will use k2