document.write( "Question 4138: Z=1+i√3. Find the smallest positive integer n for which z^n is real and evaluate z^n for this value of n. Show that there is no integral value of n for which z^n is imaginary. \n" ); document.write( "
Algebra.Com's Answer #1832 by khwang(438)\"\" \"About 
You can put this solution on YOUR website!
Z=1+i√3, note r = √(a^2+b^2) = √(1+3) = 2,
\n" ); document.write( " and theta = Arc Tan(b/a) = Arc Tan(√3) = pi/3 (set theta = x)
\n" ); document.write( " in polar coordinates, z = r(cos x + i sin x)
\n" ); document.write( " = 2(cos pi/3 + i sin pi/3)
\n" ); document.write( " By DeMoivre Theorem,
\n" ); document.write( " z^n = 2^n(cos pi/3 + i sin pi/3)^n = 2^3(cos n pi/3 + i sin n pi/3).
\n" ); document.write( " If z^n is real, then sin npi/3 = 0 , equivalently n pi /3 must
\n" ); document.write( " be multiple of pi. We see that when n = 3 , n pi /3 = pi.
\n" ); document.write( " Hence,the smallest positive integer n such that z^n is real is 3 and
\n" ); document.write( " we have z^3 = 2^3(cos pi + i sin pi) = 8(-1+ i * 0) = -8.
\n" ); document.write( "
\n" ); document.write( " Note that if z^n is imaginary then cos n pi/3 should be 0 and so
\n" ); document.write( " n pi/3 must be equal to (2k +1)pi/2 for some integer k.
\n" ); document.write( " But n pi/3 = (2k +1)pi/2 implies 2n = 3(2k+1), which is impossible
\n" ); document.write( " for any integer because the left side is even while the right hand side is
\n" ); document.write( " odd. Hence,z^n cannot be iaginary.
\n" ); document.write( " This completes the proof of the two requirements.\r
\n" ); document.write( "\n" ); document.write( " Kenny
\n" ); document.write( " \n" ); document.write( "
\n" );