document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=544045>Question 544045</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the fourth roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Z%5E4=8%28-1%2Bsqrt%283%29i%29+ ALIGN=MIDDLE  ALT=\"+Z%5E4=8%28-1%2Bsqrt%283%29i%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving your answers in the form a+bi where a and b are real numbers.\r<BR>\n" );
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document.write( "I have gotten to this point,Z^4=16(COS 5/3 PIE+i SIN 5/3 PIE) i am not sure if am right. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #354999 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=354999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> find the fourth roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+Z%5E4=8%28-1%2Bsqrt%283%29i%29+ ALIGN=MIDDLE  ALT=\"+Z%5E4=8%28-1%2Bsqrt%283%29i%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving your answers in the form a+bi where a and b are real numbers.\r<BR>\n" );
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document.write( "z^4 = -8+8sqrt(3)i<BR>\n" );
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document.write( "r = sqrt[8^2 (8sqrt(3))^2] = sqrt[64 + 64*3] = 2^4<BR>\n" );
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document.write( "theta = tan^-1(8sqrt(3)/(-8)) = -60 degrees or 120 degrees<BR>\n" );
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document.write( "since cos(-60) = cos(60) and since sin(-60) = -sin(60),<BR>\n" );
document.write( "z^4 = 2^4(cos(120)+isin(120))<BR>\n" );
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document.write( "4th Roots:<BR>\n" );
document.write( "z = 2(cos((120+360n)/4) + isin((120+360n)/4))/4<BR>\n" );
document.write( "n= 0: z = 2(cis(30) = 2(cos(30)+isin(30)) = 2(sqrt(3)/2 + i/2) = sqrt(3)+i<BR>\n" );
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document.write( "n= 1: z = 2(cis(120))= 2(cos(120)+isin(120)) = 2(-sqrt(3)/2+i/2)= -sqrt(3)+i<BR>\n" );
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document.write( "n= 2: z = 2(cis(210)) = 2(cos(210)+isin(210)) = -sqrt(3)-i<BR>\n" );
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document.write( "n= 3: z = 2(cis(300)) = sqrt(3)-i<BR>\n" );
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document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
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document.write( "z = 2(cos(30)+ isin(30)) </SPAN>\n" );
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