http://www.ck12.org Chapter 11. Complex Numbers
(√
2 −
√
2 i) 6
= (2 cis 315◦)^6
= 26 ·cis( 6 · 315 ◦)
= 64 ·cis( 1890 ◦)
= 64 ·cis( 1890 ◦)
= 64 ·cis( 90 ◦)
= 64 (cos 90◦+i·sin 90◦)
= 64 ( 0 +i)
= 64 iPractice
Use De Moivre’s Theorem to evaluate each expression. Write your answers in rectangular form.
1.( 1 +i)^5
(
1 −√ 3 i) 3
3.( 1 + 2 i)^6
(√
3 −i) 5
5.
(
(^12) +i
√ 3
2) 4
- Find the cube roots of 3+ 4 i.
- Find the 5throots of 32i.
- Find the 5throots of 1+
√
5 i.- Find the 6throots of - 64 and plot them on the complex plane.
- Use your answers to #9 to help you solvex^6 + 64 =0.
For each equation: a) state the number of roots, b) calculate the roots, and c) represent the roots graphically.
11.x^3 = 1
12.x^8 = 1
13.x^12 = 1
14.x^4 = 16
15.x^3 = 27
You learned the motivation for complex numbers by studying the Fundamental Theorem of Algebra. Then you
learned how complex numbers are used in common operations. You learned different ways of representing complex
numbers. Finally, you took powers of roots of complex numbers using De Moivre’s Theorem.