document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1095680>Question 1095680</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If x+y=4,xy=2 THEN X^6+Y^6=?<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 #710201 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=710201\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Given:<BR>\n" );
document.write( "x+y = 4<BR>\n" );
document.write( "xy = 2\r<BR>\n" );
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document.write( "We'll call the equations shown above to be <font color=blue>equation (1)</font> and <font color=blue>equation (2)</font> (in that order)\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "Multiply both sides of <font color=blue>equation (2)</font> by 2\r<BR>\n" );
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document.write( "xy = 2\r<BR>\n" );
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document.write( "2*xy = 2*2\r<BR>\n" );
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document.write( "2xy = 4\r<BR>\n" );
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document.write( "We'll refer to this as <font color=blue>equation (3)</font>.\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "Square both sides of <font color=blue>equation (1)</font>\r<BR>\n" );
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document.write( "x+y = 4\r<BR>\n" );
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document.write( "(x+y)^2 = 4^2\r<BR>\n" );
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document.write( "x^2+2xy+y^2 = 16\r<BR>\n" );
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document.write( "x^2+4+y^2 = 16 ....note how the \"2xy\" term has been replaced with 4. This is valid due to <font color=blue>equation (3)</font>\r<BR>\n" );
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document.write( "x^2+4+y^2-4 = 12-4 ... subtract 4 from both sides\r<BR>\n" );
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document.write( "x^2+y^2 = 12\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "We will refer to x^2+y^2 = 12 as <font color=blue>equation (4)</font>.\r<BR>\n" );
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document.write( "Cube both sides of <font color=blue>equation (4)</font> to get the following:\r<BR>\n" );
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document.write( "x^2+y^2 = 12\r<BR>\n" );
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document.write( "(x^2+y^2)^3 = 12^3\r<BR>\n" );
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document.write( "(x^2)^3+3(x^2)^2(y^2)+3(x^2)(y^2)^2+(y^2)^3 = 1728\r<BR>\n" );
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document.write( "x^6+3x^4y^2+3x^2y^4+y^6 = 1728\r<BR>\n" );
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document.write( "(x^6+y^6)+3x^4y^2+3x^2y^4 = 1728\r<BR>\n" );
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document.write( "(x^6+y^6)+3x^2y^2(x^2+y^2) = 1728\r<BR>\n" );
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document.write( "(x^6+y^6)+3(xy)^2(x^2+y^2) = 1728\r<BR>\n" );
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document.write( "-----------------------------\r<BR>\n" );
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document.write( "Now make substitutions. We'll use <font color=blue>equation (2)</font> and <font color=blue>equation (4)</font>\r<BR>\n" );
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document.write( "(x^6+y^6)+3(xy)^2(x^2+y^2) = 1728\r<BR>\n" );
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document.write( "(x^6+y^6)+3(2)^2(x^2+y^2) = 1728 ... Use <font color=blue>equation (2)</font> to make the first substitution. Replace xy with 2.\r<BR>\n" );
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document.write( "(x^6+y^6)+3(2)^2(12) = 1728 ... Use <font color=blue>equation (4)</font> to make the next substitution. Replace x^2+y^2 with 12\r<BR>\n" );
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document.write( "(x^6+y^6)+144 = 1728\r<BR>\n" );
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document.write( "(x^6+y^6)+144-144 = 1728-144 ... Subtract 144 from both sides\r<BR>\n" );
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document.write( "x^6+y^6 = <font color=red>1584</font>\r<BR>\n" );
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document.write( "Final Answer: <font color=red>1584</font>\r<BR>\n" );
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