document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=934104>Question 934104</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the constant term in the expansion of (-3x^5 + 4/x)^36. <BR>\n" );
document.write( "Please help me solve this, thanks. <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 #567244 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=567244\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the method is shown in the following video.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.youtube.com/watch?v=dANFpZ95kSs\" target = \"_blank\">https://www.youtube.com/watch?v=dANFpZ95kSs</a>\r<BR>\n" );
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document.write( "basically, you solve the exponent that allows x to be equal to 1.<BR>\n" );
document.write( "this occurs when the combined exponent of x is equal to 0.\r<BR>\n" );
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document.write( "to make a long story short, this occurs in the 30th term of the expansion.\r<BR>\n" );
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document.write( "here's why.\r<BR>\n" );
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document.write( "the binomial expansion formula is for (a + b)^n\r<BR>\n" );
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document.write( "the formula for the r term in the binomial expansion is c(n,r) * a^(n-r) * b^r.\r<BR>\n" );
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document.write( "our equation is (-3x^5 + 4/x)^36\r<BR>\n" );
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document.write( "in our equation, the formula for the r term in the expansion becomes:\r<BR>\n" );
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document.write( "r term = c(36,r) * (-3x^5)^(36-r) * (4/x)^r\r<BR>\n" );
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document.write( "4/x can be written as 4x^(-1), so our formula for the r term in our expansion becomes:\r<BR>\n" );
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document.write( "r term = c(36,r) * (-3x^5)^(36-r) * (4x^(-1))^r\r<BR>\n" );
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document.write( "by using the rules of exponent arithmetic, we can simplify this equation.<BR>\n" );
document.write( "the rules of exponents we will be using are:<BR>\n" );
document.write( "(a*b)^c = a^c * b^c<BR>\n" );
document.write( "(a^b)^c = a^(b*c)<BR>\n" );
document.write( "a^b * a^c = a^(b+c)\r<BR>\n" );
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document.write( "using these rules, we simplify our equation as follows:\r<BR>\n" );
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document.write( "r term = c(36,r) * (-3x^5)^(36-r) * (4x^(-1))^r becomes:\r<BR>\n" );
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document.write( "r term = c(36,r) * (-3)^(36-r) * (x^5)^(36-r) * 4^r * (x^(-1))^r which becomes:\r<BR>\n" );
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document.write( "r term = c(36,r) * (-3)^(36-r) * x^(5*(36-r) * 4^r * x^(-1*r) which becomes:\r<BR>\n" );
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document.write( "r term = c(36,r) * (-3)^(36-r) * x^(180-5r) * 4^r * x^(-r) which becomes:\r<BR>\n" );
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document.write( "r term = c(36,r) * (-3)^(36-r) * 4^r * x^(180-6r).\r<BR>\n" );
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document.write( "we need the exponent of x to be equal to 0 so we set 180 - 6r equal to 0 and solve for r to get r = 30.\r<BR>\n" );
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document.write( "the constant term in the expression will be when r = 30.\r<BR>\n" );
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document.write( "to see what the value of the constant term is, we need to evaluate the r term when r = 30.\r<BR>\n" );
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document.write( "we get r term = c(36,30) * (-3)^(36-30) * 4^30 * x^(180 - 180) which becomes:\r<BR>\n" );
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document.write( "r term = c(36,30) * (-3)^6 * 4^30 * x^0 which becomes:\r<BR>\n" );
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document.write( "r term = c(36,30) * (-3)^6 * 4^30\r<BR>\n" );
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document.write( "c(36,30) is the combination formula of 36! / (6! * 30!) which is equal to 1947792.\r<BR>\n" );
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document.write( "r term becomes 1947792 * (-3)^6 * 4^30).\r<BR>\n" );
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document.write( "use your calculator to evaluate to get:\r<BR>\n" );
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document.write( "r term = 1.637079786 * 10^27\r<BR>\n" );
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document.write( "that's your constant term.\r<BR>\n" );
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