document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1134034>Question 1134034</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the fourth roots of 16(cos 200° + i sin 200°). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #751329 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13214)</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=751329\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "deMoivre's Theorem (unfortunately not in the curriculum in many high schools....):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%28cos%28b%29%2Bi%2Asin%28b%29%29%29%5En+=+%28a%5En%29%2A%28cos%28n%2Ab%29%2Bi%2Asin%28n%2Ab%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In your problem, a=16, b=200, n = 1/4, so<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816%28cos%28200%29%2Bi%2Asin%28200%29%29%29%5E%281%2F4%29+=+%2816%5E%281%2F4%29%29%2A%28cos%28%281%2F4%29%2A200%29%2Bi%2Asin%28%281%2F4%29%2A200%29%29+=+2%2A%28cos%2850%29%2Bi%2Asin%2850%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A result of deMoivre's Theorem is that the n n-th roots of any complex number all have the same magnitude and are distributed around a circle in increments of 360/n degrees.  So the four 4th roots of your number are<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28cos%2850%29%2Bi%2Asin%2850%29%29 ALIGN=MIDDLE  ALT=\"2%2A%28cos%2850%29%2Bi%2Asin%2850%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28cos%28140%29%2Bi%2Asin%28140%29%29 ALIGN=MIDDLE  ALT=\"2%2A%28cos%28140%29%2Bi%2Asin%28140%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28cos%28230%29%2Bi%2Asin%28230%29%29 ALIGN=MIDDLE  ALT=\"2%2A%28cos%28230%29%2Bi%2Asin%28230%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28cos%28320%29%2Bi%2Asin%28320%29%29 ALIGN=MIDDLE  ALT=\"2%2A%28cos%28320%29%2Bi%2Asin%28320%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note we can use deMoivre's Theorem to verify that each of those is a 4th root of the given complex number.  For example,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2A%28cos%28230%29%2Bi%2Asin%28230%29%29%29%5E4+=+%282%5E4%29%2A%28cos%284%2A230%29%2Bi%2Asin%284%2A230%29%29+=+16%2A%28cos%28920%29%2Bi%2Asin%28920%29%29+=+16%2A%28cos%28200%29%2Bi%2Asin%28200%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );