document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=119631>Question 119631</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0618317953/algebrahomeworkh>Trigonometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Please solve:\r<BR>\n" );
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document.write( "x^6-64 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #87728 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=87728\">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=x%5E6-64=0 ALIGN=MIDDLE  ALT=\"x%5E6-64=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2-4%29%28x%5E4%2B4x%5E2%2B16%29=0 ALIGN=MIDDLE  ALT=\"%28x%5E2-4%29%28x%5E4%2B4x%5E2%2B16%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor the left side by using the difference of cubes formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%28x-2%29%28x%5E4%2B4x%5E2%2B16%29=0 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28x-2%29%28x%5E4%2B4x%5E2%2B16%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-4 ALIGN=MIDDLE  ALT=\"x%5E2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%28x-2%29 ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now set each factor equal to zero:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2=0 ALIGN=MIDDLE  ALT=\"x%2B2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-2=0 ALIGN=MIDDLE  ALT=\"x-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B4x%5E2%2B16=0 ALIGN=MIDDLE  ALT=\"x%5E4%2B4x%5E2%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now solve for x for the first two factors:\r<BR>\n" );
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document.write( "So the first two roots are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "-------------\r<BR>\n" );
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document.write( "Now let's solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E4%2B4x%5E2%2B16=0 ALIGN=MIDDLE  ALT=\"x%5E4%2B4x%5E2%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=x%5E2 ALIGN=MIDDLE  ALT=\"w=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's use the quadratic formula to solve for w:\r<BR>\n" );
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document.write( "Starting with the general quadratic\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=aw%5E2%2Bbw%2Bc=0 ALIGN=MIDDLE  ALT=\"aw%5E2%2Bbw%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "the general solution using the quadratic equation is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29 ALIGN=MIDDLE  ALT=\"w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w%5E2%2B4%2Aw%2B16=0 ALIGN=MIDDLE  ALT=\"w%5E2%2B4%2Aw%2B16=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ( notice <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4 ALIGN=MIDDLE  ALT=\"b=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=16 ALIGN=MIDDLE  ALT=\"c=16\"   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=w+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A1%2A16+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"w+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A1%2A16+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a=1, b=4, and c=16\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-4+%2B-+sqrt%28+16-4%2A1%2A16+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"w+=+%28-4+%2B-+sqrt%28+16-4%2A1%2A16+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 4 to get 16  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-4+%2B-+sqrt%28+16%2B-64+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"w+=+%28-4+%2B-+sqrt%28+16%2B-64+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2A16%2A1 ALIGN=MIDDLE  ALT=\"-4%2A16%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-64 ALIGN=MIDDLE  ALT=\"-64\"   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=w+=+%28-4+%2B-+sqrt%28+-48+%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"w+=+%28-4+%2B-+sqrt%28+-48+%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms in the radicand (everything under the square root)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-4+%2B-+4%2Ai%2Asqrt%283%29%29%2F%282%2A1%29 ALIGN=MIDDLE  ALT=\"w+=+%28-4+%2B-+4%2Ai%2Asqrt%283%29%29%2F%282%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify the square root (note: If you need help with simplifying the square root, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+%28-4+%2B-+4%2Ai%2Asqrt%283%29%29%2F%282%29 ALIGN=MIDDLE  ALT=\"w+=+%28-4+%2B-+4%2Ai%2Asqrt%283%29%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and 1 to get 2\r<BR>\n" );
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document.write( "After simplifying, the quadratic has roots of\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=-2+%2B+2%2Asqrt%283%29i ALIGN=MIDDLE  ALT=\"w=-2+%2B+2%2Asqrt%283%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=-2+-+2%2Asqrt%283%29i ALIGN=MIDDLE  ALT=\"w=-2+-+2%2Asqrt%283%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Remember, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w=x%5E2 ALIGN=MIDDLE  ALT=\"w=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=-2+%2B+2%2Asqrt%283%29i ALIGN=MIDDLE  ALT=\"x%5E2=-2+%2B+2%2Asqrt%283%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=-2+-+2%2Asqrt%283%29i ALIGN=MIDDLE  ALT=\"x%5E2=-2+-+2%2Asqrt%283%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now take the square root of both sides for each case\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0%2B-sqrt%28-2+%2B+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=0%2B-sqrt%28-2+%2B+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0%2B-sqrt%28-2+-+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=0%2B-sqrt%28-2+-+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now break up the plus/minus\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28-2+%2B+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28-2+%2B+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%28-2+%2B+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%28-2+%2B+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28-2+-+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28-2+-+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%28-2+-+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%28-2+-+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E6-64=0 ALIGN=MIDDLE  ALT=\"x%5E6-64=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > are \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-2 ALIGN=MIDDLE  ALT=\"x=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28-2+%2B+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28-2+%2B+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%28-2+%2B+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%28-2+%2B+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=sqrt%28-2+-+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=sqrt%28-2+-+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-sqrt%28-2+-+2%2Asqrt%283%29i%29 ALIGN=MIDDLE  ALT=\"x=-sqrt%28-2+-+2%2Asqrt%283%29i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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