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10.5 Powers of Complex Numbers

10.5 Powers of Complex Numbers. Exploration! Let z = r ( cos θ + i sin θ ) z 2 = z • z = [ r ( cos θ + i sin θ )] • [ r ( cos θ + i sin θ )] from yesterday… = r 2 ( cos 2 θ + i sin 2 θ )

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10.5 Powers of Complex Numbers

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  1. 10.5 Powers of Complex Numbers

  2. Exploration! Let z = r(cosθ + isinθ) z2 = z • z = [r(cosθ + isinθ)] • [r(cosθ + isinθ)] from yesterday… = r2(cos 2θ + isin 2θ) how about z3 = z • z2 = [r(cosθ + isinθ)] • [r2(cos 2θ + isin 2θ)] = r3(cos 3θ + isin 3θ) this pattern keeps happening… let’s generalize! DeMoivre’s Theorem If z = r(cosθ + isinθ) is a complex number in polar form, then for any integer n, zn = rn(cosnθ + isinnθ)

  3. Ex 1) Evaluate each power and express answer in rectangular form. A) try on your own B) 1 0

  4. Ex 2) Evaluate z7 for . Express answer in rectangular form. *Hint: Convert to polar  do DeMoivre’s  convert back 

  5. A nautilus shell has been traced on the complex plane. z (1, 1)  1 + i rect complex Ex 3) Calculate z2, z3, and z4 and see if they lie on the curve of the shell wall. (divide class into 3 groups, just do one) (1, 1)  polar Yes, they lie on the sketch of the shell wall!

  6. Ex 4) Evaluate z4 for z = 2(cos 40° – isin 40°) Do DeMoivre’s … WAIT is something wrong? it has to be in complex polar  can’t subtract! change to z = 2(cos (–40°) + isin (–40°)) z4 = (2)4(cos (–160°) + isin (–160°)) z4 ≈ –15.0351 – 5.4723i (calculator! )

  7. *Remember: z0 = 1 and *DeMoivre’s Theorem doesn’t just have to be positive powers. Ex 5) Use DeMoivre’sThm to evaluate and express in rectangular form. A) B) try on your own

  8. Homework #1006 Pg 520 #1, 3, 5, 9, 11, 13, 16, 20, 21, 23, 29, 31, 32

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