document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=632217>Question 632217</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please tell how to graphically locate the nth roots of a complex number, given the location of one of the nth roots. Thanks!!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #398374 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=398374\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The n roots of a number will be located in a circle, equally spaced at angles of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2Fn ALIGN=MIDDLE  ALT=\"2pi%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For example, if A is one of the 6th roots of a number, the other 5 roots are at the ends of the green spokes, all spaced at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F3 ALIGN=MIDDLE  ALT=\"pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > angles<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1.5%2C1.5%2C-1.5%2C1.5%2C%0D%0Aarrow%28-1.4%2C0%2C1.4%2C0%29%2Carrow%280%2C-1.4%2C0%2C1.4%29%2C%0D%0Alocate%281.4%2C0%2Cx%29%2Clocate%280.1%2C1.48%2Cy%29%2C%0D%0Acircle%28cos%28pi%2F7%29%2Csin%28pi%2F7%29%2C0.04%29%2C+locate%280.95%2C0.5%2CA%29%2C%0D%0Agreen%28circle%280%2C0%2C1%29%29%2C+green%28line%280%2C0%2Ccos%28pi%2F7%29%2Csin%28pi%2F7%29%29%29%2C%0D%0Agreen%28line%280%2C0%2Ccos%2810pi%2F21%29%2Csin%2810pi%2F21%29%29%29%2C%0D%0Agreen%28line%280%2C0%2Ccos%2817pi%2F21%29%2Csin%2817pi%2F21%29%29%29%2C%0D%0Agreen%28line%280%2C0%2Ccos%2824pi%2F21%29%2Csin%2824pi%2F21%29%29%29%2C%0D%0Agreen%28line%280%2C0%2Ccos%2831pi%2F21%29%2Csin%2831pi%2F21%29%29%29%2C%0D%0Agreen%28line%280%2C0%2Ccos%2838pi%2F21%29%2Csin%2838pi%2F21%29%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >. That is because<BR>\n" );
document.write( "if two of the roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28cos%28theta%29%2Bisin%28theta%29%29 ALIGN=MIDDLE  ALT=\"r%28cos%28theta%29%2Bisin%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%28cos%28alpha%29%2Bisin%28alpha%29%29 ALIGN=MIDDLE  ALT=\"r%28cos%28alpha%29%2Bisin%28alpha%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the nth power of that is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%5En%29%28cos%28n%2Atheta%29%2Bisin%28n%2Atheta%29%29=%28r%5En%29%28cos%28n%2Aalpha%29%2Bisin%28n%2Aalpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means that the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Atheta ALIGN=MIDDLE  ALT=\"n%2Atheta\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Aalpha ALIGN=MIDDLE  ALT=\"n%2Aalpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > are co-terminal, meaning that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Aalpha=n%2Atheta%2Bk%2A%282pi%29 ALIGN=MIDDLE  ALT=\"n%2Aalpha=n%2Atheta%2Bk%2A%282pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some k integer.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Aalpha=n%2Atheta%2Bk%2A%282pi%29 ALIGN=MIDDLE  ALT=\"n%2Aalpha=n%2Atheta%2Bk%2A%282pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2Aalpha-n%2Atheta=k%2A%282pi%29 ALIGN=MIDDLE  ALT=\"n%2Aalpha-n%2Atheta=k%2A%282pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2A%28alpha-theta%29=k%2A%282pi%29 ALIGN=MIDDLE  ALT=\"n%2A%28alpha-theta%29=k%2A%282pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha-theta=k%2A%282pi%29%2Fn ALIGN=MIDDLE  ALT=\"alpha-theta=k%2A%282pi%29%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha-theta=k%2A%282pi%2Fn%29 ALIGN=MIDDLE  ALT=\"alpha-theta=k%2A%282pi%2Fn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so the angles are spaced <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2Fn ALIGN=MIDDLE  ALT=\"2pi%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" > apart,<BR>\n" );
document.write( "like spokes of a wheel with n spokes. </SPAN>\n" );
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