Page 713 - Advanced_Engineering_Mathematics o'neil

19.5 Powers 693

SECTION 19.5 PROBLEMS

In each of Problems 1 through 14, determine all values 10. 6 −2−3i
w
of z . 1/4
11. (−16)
1. i 1+i 12. [(1 + i)/(1 − i)] 1/3
2. (1 + i) 2i 13. 1 1/6
3. i i 14. (7i) 3i
4. (1 + i) 2−i 15. Let u 1 ,··· ,u n be the nth roots of unity with n a pos-
5. (−1 + i) −3i itive integer and n ≥ 2. Prove that n j=1 u j = 0. Hint:
Write each nth root of unity as a power of e 2πi/n .A
6. (1 − i) 1/3
vector argument can also be made based on plotting
7. i 1/4 the nth roots as vertices of a polygon.
8. 16 1/4 16. Let n be a positive integer, and let ω =e 2πi/n . Evaluate
n−1 (−1) ω . j

j
9. (−4) 2−i j=0

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