document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=932083>Question 932083</A>: <I><SPAN STYLE=\"word-break: break-all;\"> FIND THE VALUES OF i^-i<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #565954 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=565954\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The imaginary number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > is defined solely by the property that its square is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   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=i%5E2+=+-1 ALIGN=MIDDLE  ALT=\"i%5E2+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i+ ALIGN=MIDDLE  ALT=\"i+\"   DISABLED_style= \"max-width:90%; height:auto;\" > defined this way, it follows directly from algebra that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-i+ ALIGN=MIDDLE  ALT=\"-i+\"   DISABLED_style= \"max-width:90%; height:auto;\" > are both square roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Making use of Euler's formula, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5Ei ALIGN=MIDDLE  ALT=\"i%5Ei\"   DISABLED_style= \"max-width:90%; height:auto;\" > is\r<BR>\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5Ei+=+%28+e%5E%28i%28pi%2F2+%2B2k%2Api%29%29%29%5Ei+=+e%5E%28i%5E2+%28pi%2F2+%2B+2k%2Api%29%29+=+e%5E%28-%28pi%2F2+%2B+2k%2Api%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > is element <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Z ALIGN=MIDDLE  ALT=\"Z\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the set of integers.\r<BR>\n" );
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document.write( "The principal value (for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+0 ALIGN=MIDDLE  ALT=\"k+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-pi%2F2%29 ALIGN=MIDDLE  ALT=\"e%5E%28-pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.207879576 ALIGN=MIDDLE  ALT=\"0.207879576\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "in your case, we have:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E%28-i%29+=+%28e%5E%28i%28pi%2F2+%2B+2k%2Api%29%29%29%5E%281%2Fi%29+=+e%5E%28%281%2Fi%5E2%29%28pi%2F2+%2B+2k%2Api%29%29+=+e%5E%28%28-1%2F-1%29%28pi%2F2+%2B+2k%2Api%29%29++=+e%5E%28pi%2F2+%2B+2kpi%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "The principal value (for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+0 ALIGN=MIDDLE  ALT=\"k+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28pi%2F2%29 ALIGN=MIDDLE  ALT=\"e%5E%28pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4.81047738 ALIGN=MIDDLE  ALT=\"4.81047738\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
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