document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1144148>Question 1144148</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the roots of : a) z3 = 1 + i b) z5 = −i c) z7 = −1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #765172 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=765172\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Use the \"^\" character (shift-6) to represent exponentiation (e.g., \"z^3\", not just \"z3\").  I assume the roots to be found are for the equations<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E3+=+1%2Bi ALIGN=MIDDLE  ALT=\"z%5E3+=+1%2Bi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E5+=+-i ALIGN=MIDDLE  ALT=\"z%5E5+=+-i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E7+=+-1 ALIGN=MIDDLE  ALT=\"z%5E7+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Finding roots of complex numbers is made easy with deMoivre's Theorem.<br><BR>\n" );
document.write( "If a complex number is represented in the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+r%2Acis%28theta%29 ALIGN=MIDDLE  ALT=\"x+=+r%2Acis%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "where r is a magnitude and theta is an angle of the complex number in the complex plane, then deMoivre's Theorem says that the n n-th roots of the complex number are found as follows:<br><BR>\n" );
document.write( "(1) the magnitude of each root is the n-th root of the magnitude of the given complex number; and<BR>\n" );
document.write( "(2) the \"primary\" n-th root has an angle that is (1/n) times the angle of the given complex number; and the other n-th roots are distributed around the circle at increments of 360/n degrees.<br><BR>\n" );
document.write( "Here is an example....<br><BR>\n" );
document.write( "Find the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E4+=+i ALIGN=MIDDLE  ALT=\"z%5E4+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The given complex number in the required form is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Acis%2890%29 ALIGN=MIDDLE  ALT=\"1%2Acis%2890%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "The magnitude of the given complex number is 1, so the magnitude of each of the 4-th roots is 1^(1/4)=1.<br><BR>\n" );
document.write( "The angle of the given complex number in the complex plane is 90 degrees, so the \"primary\" 4-th root of i is at an angle of 90/4 = 22.5 degrees.  The other 4-th roots of i are then distributed around the plane at increments of 360/4 = 90 degrees.<br><BR>\n" );
document.write( "So the solutions to the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E4=i ALIGN=MIDDLE  ALT=\"z%5E4=i\"   DISABLED_style= \"max-width:90%; height:auto;\" > are<BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Acis%2822.5%29 ALIGN=MIDDLE  ALT=\"1%2Acis%2822.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Acis%28112.5%29 ALIGN=MIDDLE  ALT=\"1%2Acis%28112.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(3) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Acis%28202.5%29 ALIGN=MIDDLE  ALT=\"1%2Acis%28202.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(4) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Acis%28292.5%29 ALIGN=MIDDLE  ALT=\"1%2Acis%28292.5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Use the theorem to find the solutions to your examples.<br><BR>\n" );
document.write( "To get you started -- in case you need help with this part -- the complex numbers in your examples are<BR>\n" );
document.write( "(1) magnitude sqrt(2), angle 45 degrees<BR>\n" );
document.write( "(2) magnitude 1, angle 90 degrees<BR>\n" );
document.write( "(3) magnitude 1, angle 180 degrees<BR>\n" );
document.write( " </SPAN>\n" );
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