document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=972923>Question 972923</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello Tutor, I really need your help to answer this question.\r<BR>\n" );
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document.write( "How are polar coordinates useful for working with complex numbers (hint: think about powers and roots)? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #595165 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=595165\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Calculating the nth power or the nth root of a complex number in the rectangular form is a complicated, messy business.<BR>\n" );
document.write( "Calculate powers and roots of complex numbers in polar form is simple:<BR>\n" );
document.write( "For a power: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28r+%3C%22 ALIGN=MIDDLE  ALT=\"%22%28r+%3C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22%5En=r%5En ALIGN=MIDDLE  ALT=\"%22%29%22%5En=r%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%3C%22 ALIGN=MIDDLE  ALT=\"%22%3C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2Atheta%29 ALIGN=MIDDLE  ALT=\"%28n%2Atheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <BR>\n" );
document.write( "for an nth root: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28r+%3C%22 ALIGN=MIDDLE  ALT=\"%22%28r+%3C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%29%22%5E%28%221+%2F+n%22%29=r%5E%28%221+%2F+n%22%29 ALIGN=MIDDLE  ALT=\"%22%29%22%5E%28%221+%2F+n%22%29=r%5E%28%221+%2F+n%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%3C%22 ALIGN=MIDDLE  ALT=\"%22%3C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta%2Fn=root%28n%2Cr%29 ALIGN=MIDDLE  ALT=\"theta%2Fn=root%28n%2Cr%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%3C%22 ALIGN=MIDDLE  ALT=\"%22%3C%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta%2Fn ALIGN=MIDDLE  ALT=\"theta%2Fn\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Transforming between polar and rectangular coordinates is also simple,<BR>\n" );
document.write( "so if you are given the rectangular form and need to calculate a power or root,<BR>\n" );
document.write( "you can easily transform to polar form, and then calculate the desired power or root in polar form.<BR>\n" );
document.write( "If the result must be expressed in rectangular form, you just transform your polar form result to rectangular form.<BR>\n" );
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document.write( "If you have a complex number  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bi%2Ab ALIGN=MIDDLE  ALT=\"a%2Bi%2Ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bj%2Ab ALIGN=MIDDLE  ALT=\"a%2Bj%2Ab\"   DISABLED_style= \"max-width:90%; height:auto;\" > (whichever letter you use for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ),<BR>\n" );
document.write( "with real part <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and imaginary part <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "you can calculate the argument, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > and modulus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28theta%29=b%2Fa ALIGN=MIDDLE  ALT=\"tan%28theta%29=b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=sqrt%28a%5E2%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"r=sqrt%28a%5E2%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If you know the argument, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > and modulus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > of a complex number,<BR>\n" );
document.write( "you can calculate its real part <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and imaginary part <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=r%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"a=r%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=r%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"a=r%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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