document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=506736>Question 506736</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is my question, with 'i' being an imaginary number. The question is to express the complex numbers in simplified cartesian form.\r<BR>\n" );
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document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F5 ALIGN=MIDDLE  ALT=\"3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4i%2F5 ALIGN=MIDDLE  ALT=\"4i%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)^39 x (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3i%2F5+%2B+4%2F5 ALIGN=MIDDLE  ALT=\"3i%2F5+%2B+4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)^39\r<BR>\n" );
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document.write( "I have tried finding the modulus, and that gives me the square root of (3/5)^2 + (4/5)^2, which equals 1. I don't know where to go from there, or if I've made a mistake already.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #340239 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=340239\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{3}{5} + \frac{4i}{5} = e^{i \theta}\"> where theta is the arc-cosine of 3/5. Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{4}{5} + \frac{3i}{5} = e^{i(\pi - \theta)}\">\r<BR>\n" );
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document.write( "because the arguments of both complex numbers are complementary angles. Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (\frac{3}{5} + \frac{4i}{5})(\frac{4}{5} + \frac{3i}{5}) = e^{i \theta}e^{i(\pi - \theta)} = e^{i \pi} = -1\">\r<BR>\n" );
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document.write( "What we want is this number raised to the 39th power. Since (-1)^39 = -1, the answer is -1 (or -1 + 0i). </SPAN>\n" );
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