50 Chapter 1
for z #- 1 and derive the formulas~ k() _ ~ sin(n + 1/i)()
6 cos - 2 + 2. 1; ()
k=O Sln 2~. k() _ ~ ~ () _ cos(n +
1
/ 2 )()
6 sm - 2 cot 2 2 sin 1/i()
k=O
where 0 < () < 27r.- Compute algebraically the square roots of the following complex
numbers.
(a)--i (b)35+12i
(c) 24 - lOi (cl) -39 - 80i
- Compute algebraically the cube roots of:
(a)l (b)2-lli - Find and represent geometrically the roots indicated.
(a) V-1 (b) V"i
(c) *{/16e 4 i,,./ 9 (cl) *{/~34_3_e_s,-.,,.-/ 1 - 2- Find the roots of the equation 16z^4 = ( z - 1 )^4 and show that all the
roots lie on or inside the circle - Show that all roots of the equation (z + 1)^3 + z^3 = 0 lie on the line
x -- -1; 2· - Let fJ be one of the imaginary cubic roots of unity. Verify the following
identities.
(a) 1 + fJ + (3^2 = 0
(b) (1 - fJ + (3^2 )(1 + fJ - (3^2 ) = 4
(c) (2 + 5(3 + 2(3^2 )^6 = 729
(cl) (1 + fJ)(l - (3^5 )(1 + (3^8 )(1 - (3^10 ) = 3
(e) (a+ b)(a(J + bfJ^2 )(afJ^2 + bfJ) = a^3 + b^3
(f) (a+ b )^2 + ( a(J + b(J^2 )^2 + ( a(J^2 + b(J)^2 = 6ab
(g) (a+ b + c)(a + b(J + c(J^2 )(a + b(J^2 + c(J) = a^3 + b^3 + c^3 - 3abc
In the last three identities, a, b, and c denote real numbers.- Prove that (x + y)n - xn - yn = 0 for x = (Jy whenever n is an odd
integer relatively prime to 3.
19. If 'Y = e^2 ,,.i/n and k is an. integer, prove:
(a) 1+'Yk+1^2 k +. · · + 'Y(n-l)k = 0 if k #-multiple of n
(b) l-1k+'Y2k_,,,+(-1r-1,(n-1)k= l+(-l)n-1
1 + 'Yk
if k #-1/i(2h + l)n.