document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=122065>Question 122065</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find (1 / sq. rt. 2 + i / sq. rt. 2) ^ 8 \r<BR>\n" );
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document.write( "a.)     -8\r<BR>\n" );
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document.write( "b.)     256\r<BR>\n" );
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document.write( "c.)     8\r<BR>\n" );
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document.write( "d.)     1\r<BR>\n" );
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document.write( "e.)     None of these<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 #89603 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=89603\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8 ALIGN=MIDDLE  ALT=\"%281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
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document.write( "It's very useful to note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F4%29=1%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"cos%28pi%2F4%29=1%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28pi%2F4%29=1%2Fsqrt%282%29 ALIGN=MIDDLE  ALT=\"sin%28pi%2F4%29=1%2Fsqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . So the expression is equivalent to \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29%29%5E8 ALIGN=MIDDLE  ALT=\"%28cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29%29%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now we're going to use De Moivre's theorem to solve this problem.\r<BR>\n" );
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document.write( "Remember, De Moivre's theorem states: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29%2Bi%2Asin%28x%29%29%5En=cos%28n%2Ax%29%2Bi%2Asin%28n%2Ax%29 ALIGN=MIDDLE  ALT=\"%28cos%28x%29%2Bi%2Asin%28x%29%29%5En=cos%28n%2Ax%29%2Bi%2Asin%28n%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So using De Moivre's theorem we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%288%2Api%2F4%29%2Bi%2Asin%288%2Api%2F4%29 ALIGN=MIDDLE  ALT=\"cos%288%2Api%2F4%29%2Bi%2Asin%288%2Api%2F4%29\"   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=cos%282pi%29%2Bi%2Asin%282pi%29 ALIGN=MIDDLE  ALT=\"cos%282pi%29%2Bi%2Asin%282pi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F4 ALIGN=MIDDLE  ALT=\"pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8pi%2F4 ALIGN=MIDDLE  ALT=\"8pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now reduce to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   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=1%2B0i ALIGN=MIDDLE  ALT=\"1%2B0i\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the cosine of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 1. Take the sine of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 0.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Remove the zero term\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8 ALIGN=MIDDLE  ALT=\"%281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > simplifies to 1. \r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8=1 ALIGN=MIDDLE  ALT=\"%281%2Fsqrt%282%29%2Bi%2Fsqrt%282%29%29%5E8=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the answer is D) </SPAN>\n" );
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