document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=420170>Question 420170</A>: <I><SPAN STYLE=\"word-break: break-all;\"> to take the 4th root of a -16 isn't possible right? and if it is possible then what would it be? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #293679 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=293679\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The fourth roots of -16 are definitely possible, using complex numbers. Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" > be a complex number such that z^4 = -16 (we know the magnitude of z must be 2 since 2^4 = 16). We can express z in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2e%5E%28i%2Atheta%29 ALIGN=MIDDLE  ALT=\"2e%5E%28i%2Atheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2e%5E%28i%2Atheta%29+=+2%28cos%28theta%29+%2B+i%2Asin%28theta%29%29 ALIGN=MIDDLE  ALT=\"z+=+2e%5E%28i%2Atheta%29+=+2%28cos%28theta%29+%2B+i%2Asin%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (by Euler's formula)\r<BR>\n" );
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document.write( "Hence,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z%5E4+=+2%5E4e%5E%284i%2Atheta%29+=+16e%5E%284i%2Atheta%29+=+-16 ALIGN=MIDDLE  ALT=\"z%5E4+=+2%5E4e%5E%284i%2Atheta%29+=+16e%5E%284i%2Atheta%29+=+-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%284i%2Atheta%29+=+-1 ALIGN=MIDDLE  ALT=\"e%5E%284i%2Atheta%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. From DeMoivre's theorem, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%284i%2Atheta%29+=+cos%284%2Atheta%29+%2B+i%2Asin%284%2Atheta%29 ALIGN=MIDDLE  ALT=\"e%5E%284i%2Atheta%29+=+cos%284%2Atheta%29+%2B+i%2Asin%284%2Atheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This number must equal -1, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%284%2Atheta%29+=+-1 ALIGN=MIDDLE  ALT=\"cos%284%2Atheta%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%284%2Atheta%29+=+0 ALIGN=MIDDLE  ALT=\"sin%284%2Atheta%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This implies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Atheta+=+pi+%2B+k%2A2pi ALIGN=MIDDLE  ALT=\"4%2Atheta+=+pi+%2B+k%2A2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+%28pi%2F4%29+%2B+k%2A%28pi%2F2%29 ALIGN=MIDDLE  ALT=\"theta+=+%28pi%2F4%29+%2B+k%2A%28pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where k is any integer. For now, we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+pi%2F4 ALIGN=MIDDLE  ALT=\"theta+=+pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the fourth root of -16 is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28i%2Api%2F4%29 ALIGN=MIDDLE  ALT=\"e%5E%28i%2Api%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In a+bi form, this is equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28pi%2F4%29+%2B+i%2Asin%28pi%2F4%29+=+%28sqrt%282%29%2F2%29+%2B+%28sqrt%282%29%2F2%29i ALIGN=MIDDLE  ALT=\"cos%28pi%2F4%29+%2B+i%2Asin%28pi%2F4%29+=+%28sqrt%282%29%2F2%29+%2B+%28sqrt%282%29%2F2%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
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