1. Complex Roots Solve for z given that z 4 =81cis60°

2. If z n = r cis , then one solution will be z= r 1/n cis  /n De Moivre’s Theorem can be used to solve equations involving a complex numbers & powers However, the Fundamental Theorem of Algebra states that there must be n roots… and these are equally spaced around the argand diagram

3. Any complex number can be written in a more general form as z= rcis(Ө+2πk) or z= rcis(Ө+360k) Eg: 81cis60 could also be written as: –81cis420, 81cis780, 81cis1140 etc –Or more generally, 81cis(60+360k) Applying DeMoivres theorem, z 4 =81cis(60°+360k) kZ -23cis(-165) 3cis(-75) 03cis(15) 13cis(105) Substitute in k values:

4. Notice that the roots are always symmetrical around the origin 3cis(-165) 3cis(-75) 3cis(15) 3cis(105) They are spread at angles of 360/n (in this case 90°)

5. To find ALL complex roots we must apply De Moivre’s Thm to the general expression r cis (  + k2  ) or r cis (  + 360k) e.g.Solve z 4 = 4 +3i Step 1:Write in polar form Step 2:Express in general form Step 3:Use De Moivre’s Theorum Step 4:Substitute values of k up to n to generate solutions. Step 5: Give your solutions in the same form they were asked (in this case, rectangular).

6. More practice: 32.4 p.299