document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=211188>Question 211188</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780198534464/algebrahomeworkh>Visual Complex Analysis</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> How does one determine the number of roots of unity?  The solution for the following problems must be made using the radian not polar rectangular approach:\r<BR>\n" );
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document.write( "(1-i)^1/4\r<BR>\n" );
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document.write( "(1+i*sqrt(3)^7/2\r<BR>\n" );
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document.write( "(-i)^1/2\r<BR>\n" );
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document.write( "These problems I have to work are not from my text, but rather from the instructor.  My textbook is:<BR>\n" );
document.write( "Tristan Needham's  - Visual Complex Analysis ISBN 978 0 19 853446 4 (pbk) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #159633 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=159633\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> determine the number of roots of unity? The solution for the following problems must be made using the radian not polar rectangular approach: <BR>\n" );
document.write( "(1-i)^1/4<BR>\n" );
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document.write( "On a coordinate system, plot the point (1,-1)<BR>\n" );
document.write( "Notice that it is in the 4th quadrant.<BR>\n" );
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document.write( "r = sqrt(1 + 1) = sqrt(2)<BR>\n" );
document.write( "theta = arctan(-1/1) = 315 degrees in the 4th quadrant.<BR>\n" );
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document.write( "(1-i)^(1/4) = r^(1/4)(cis[(315+(n(360)]/4) where n = 0,1,2,3<BR>\n" );
document.write( "= 2^(1/8)cis(78.75 degrees) when n=0<BR>\n" );
document.write( "= 2^(1/8)cis(78.75+90) when n = 1<BR>\n" );
document.write( "= 2^(1/8)cis(78.75+180) when n = 2<BR>\n" );
document.write( "= 2^(1/8)cis(78.75+270) when n = 3<BR>\n" );
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document.write( "(1+i*sqrt(3)^7/2<BR>\n" );
document.write( "Plot the point; it is in the 1st quadrant.<BR>\n" );
document.write( "r = sqrt(1+3) = 2<BR>\n" );
document.write( "theta = arctan(sqrt(3)) = 60 degrees<BR>\n" );
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document.write( "(1+i*sqrt(3))^(7/2) = 2^(7/2)cis[(60 + n*360)]/(7/2) when n = 0,1. etc.<BR>\n" );
document.write( "= 2^(7/2)cis(17.14 degrees) when n = 0<BR>\n" );
document.write( "= 2^(7/2)cis(17.14+102.86) when n = 1<BR>\n" );
document.write( "etc.<BR>\n" );
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document.write( "(-i)^1/2 <BR>\n" );
document.write( "Plot the point (0,-1)<BR>\n" );
document.write( "r = sqrt(0+1) = 1<BR>\n" );
document.write( "theta = arctan(-1/0) which is undefined but you can see that theta = 270<BR>\n" );
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document.write( "(-i)^(1/2)= 1cis(270+360n)/2<BR>\n" );
document.write( "= cis(135) when n=0<BR>\n" );
document.write( "= cis(135+180) when n = 1<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
document.write( " </SPAN>\n" );
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