document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=470932>Question 470932</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do we calculate the real and imaginary parts of (sqrt3 + i)^100 ?<BR>\n" );
document.write( "I have tried and my working are as follows:<BR>\n" );
document.write( "r = sqrt ((sqrt3)^2 + 1^2) = 2<BR>\n" );
document.write( "alpha = arctan (1/sqrt3) = pi/6<BR>\n" );
document.write( "Z = r (cos alpha + i sin alpha)<BR>\n" );
document.write( "= 2 (cos pi/6 + i sin pi/6)<BR>\n" );
document.write( "= 2 e^i(pi/6)<BR>\n" );
document.write( "(sqrt3 + i)^100 = (2e^i(pi/6)^100<BR>\n" );
document.write( "= 2^100 x e^i(100pi/6)<BR>\n" );
document.write( "= 2^100 x e^i(50pi/3)<BR>\n" );
document.write( "= ?<BR>\n" );
document.write( "How do I solve from here ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #323031 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=323031\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> How do we calculate the real and imaginary parts of (sqrt3 + i)^100 ?<BR>\n" );
document.write( "I have tried and my working are as follows:<BR>\n" );
document.write( "r = sqrt ((sqrt3)^2 + 1^2) = 2<BR>\n" );
document.write( "alpha = arctan (1/sqrt3) = pi/6<BR>\n" );
document.write( "Z = r (cos alpha + i sin alpha)<BR>\n" );
document.write( "Z = 2 (cos pi/6 + i sin pi/6)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28100%2CZ%29+=+root%28100%2C2%29%2A%28cos%28pi%2F600%29+%2B+i%2Asin%28pi%2F600%29%29 ALIGN=MIDDLE  ALT=\"root%28100%2CZ%29+=+root%28100%2C2%29%2A%28cos%28pi%2F600%29+%2B+i%2Asin%28pi%2F600%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the other 99 roots add 2pi and divide by 600<BR>\n" );
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document.write( "= 2 e^i(pi/6)<BR>\n" );
document.write( "(sqrt3 + i)^100 = (2e^i(pi/6)^100<BR>\n" );
document.write( "= 2^100 x e^i(100pi/6)<BR>\n" );
document.write( "= 2^100 x e^i(50pi/3)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );