document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170262>Question 1170262</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the three cube roots of 6-2i and express the roots in polar form.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #795124 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=795124\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> find the three cube roots of 6-2i and express the roots in polar form. \r<BR>\n" );
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document.write( "First we plot the point 6-2i which is the point (6,-2), x=6, y=-2<BR>\n" );
document.write( "We draw a vector from(0,0) to (6,-2).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%283200%2F7%2C400%2C-7%2C9%2C-7%2C7%2Cgraph%283200%2F7%2C400%2C-7%2C9%2C-7%2C7%29%2Clocate%283%2C.71%2Cx=6%29%2C%0D%0Alocate%286.1%2C-.7%2Cy=-2%29%2C+green%28line%286%2C0%2C6%2C-2%29%2Cline%280%2C0%2C6%2C0%29%29%2Cline%280%2C0%2C6%2C-2%29%2C%0D%0Ared%28arc%280%2C0%2C2.3%2C-2.3%2C0%2C342%29%29%2Clocate%282%2C-1.5%2Cr=2sqrt%2810%29%29%2Clocate%28-1.2%2C1.2%2Ctheta%29%29%0D%0A%0D%0A+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The hypotenuse is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=sqrt%28x%5E2%2By%5E2%29=sqrt%286%5E2%2B%28-2%29%5E2%29=sqrt%2836%2B4%29=sqrt%2840%29=sqrt%284%2A10%29=2sqrt%2810%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >, but we'll leave it <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2840%29 ALIGN=MIDDLE  ALT=\"sqrt%2840%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for now.\r<BR>\n" );
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document.write( "Calculate &theta; by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28reference%29=2%2F6=1%2F3%29 ALIGN=MIDDLE  ALT=\"tan%28reference%29=2%2F6=1%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=tan%5E%28-1%29%281%2F3%29 ALIGN=MIDDLE  ALT=\"theta=tan%5E%28-1%29%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So 6-2i in polar form is \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6-2i=r%28cos%28theta%29%2Bi%2Asin%28theta%29%5E%22%22%29=sqrt%2840%29%28cos%28tan%5E%28-1%29%281%2F3%29%29%2Bi%2Asin%28tan%5E%28-1%29%281%2F3%29%29%5E%22%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We can add any integer times 360° to the angle without changing the value:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6-2i=sqrt%2840%29%28cos%28theta%2B360%5Eo%2An%29%2Bi%2Asin%28theta%2B360%5Eo%2An%29%5E%22%22%29=sqrt%2840%29%28cos%28tan%5E%28-1%29%281%2F3%29%2B360%5Eo%2An%2B360%5Eo%2An%29%2Bi%2Asin%28tan%5E%28-1%29%281%2F3%29%2B360%5Eo%2An%29%5E%22%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6-2i=sqrt%2840%29%28cos%28tan%5E%28-1%29%281%2F3%29%2B360%5Eo%2An%2B360%5Eo%2An%29%2Bi%2Asin%28tan%5E%28-1%29%281%2F3%29%2B360%5Eo%2An%29%5E%22%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We raise both side to the 1/3 power, the same as taking cube root, and write<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2840%29 ALIGN=MIDDLE  ALT=\"sqrt%2840%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the 1/2 power of 40\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C1%2C%22%22%2C%286-2i%29%5E%281%2F3%29=%2840%5E%281%2F2%29%28cos%28tan%5E%28-1%29%281%2F3%29%2B360%5Eo%2An%29%2Bi%2Asin%28tan%5E%28-1%29%281%2F3%29%2B360%5Eo%2An%29%5E%22%22%29%5E%22%22%29%5E%281%2F3%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Next we use DeMoivre's theorem, which is to raise the modulus to the 1/3<BR>\n" );
document.write( "power and multiply the argument by 1/3:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C1%2C%22%22%2C%286-2i%29%5E%281%2F3%29=%2840%5E%28%281%2F2%29%281%2F3%29%29%28cos%28expr%281%2F3%29tan%5E%28-1%29%281%2F3%29%2B120%5Eo%2An%29%2Bi%2Asin%28expr%281%2F3%29tan%5E%28-1%29%281%2F3%29%2B120%5Eo%2An%29%5E%22%22%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%282%2C1%2C%22%22%2C%286-2i%29%5E%281%2F3%29=%2840%5E%281%2F6%29%28cos%28expr%281%2F3%29tan%5E%28-1%29%281%2F3%29%2B120%5Eo%2An%29%2Bi%2Asin%28expr%281%2F3%29tan%5E%28-1%29%281%2F3%29%2B120%5Eo%2An%29%5E%22%22%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%286%2C40%29%28cos%28expr%281%2F3%29tan%5E%28-1%29%281%2F3%29%2B120%5Eo%2An%5E%22%22%29+-+i%2Asin%28expr%281%2F3%29tan%5E%28-1%29%281%2F3%29%2B120%5Eo%2An%5E%22%22%29%5E%22%22%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > where n=0,1,2\r<BR>\n" );
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document.write( "[Write all three values out].\r<BR>\n" );
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document.write( "Or if you want it in radians, substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2F3 ALIGN=MIDDLE  ALT=\"2pi%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 120°\r<BR>\n" );
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document.write( "Edwin</pre>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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