document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=231057>Question 231057</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I'm having problems simplifying this problem:\r<BR>\n" );
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document.write( " (x^{a+b})^{a-b}<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "(x^{a-2b})^{a+2b}\r<BR>\n" );
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document.write( "Some of the things I have tried:\r<BR>\n" );
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document.write( "x^{2a-2b}           <BR>\n" );
document.write( "----------            --->  x^2b <BR>\n" );
document.write( "x^{2a-4b}\r<BR>\n" );
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document.write( "I'm also beginning to think it isn't possible to simplify any further than the original question. Help?\r<BR>\n" );
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document.write( "Thanks in advance. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #171072 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=171072\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll take a crack at it.\r<BR>\n" );
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document.write( "Your problem is:\r<BR>\n" );
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document.write( " (x^{a+b})^{a-b}<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "(x^{a-2b})^{a+2b}\r<BR>\n" );
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document.write( "A couple of basic concepts should help you.  There is no relationship between any of the variables shown below and the ones in your problem.\r<BR>\n" );
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document.write( "Concept 1.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5Ea%29%5Eb+=+x%5E%28a%2Ab%29 ALIGN=MIDDLE  ALT=\"%28x%5Ea%29%5Eb+=+x%5E%28a%2Ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Concept 2.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%28a-b%29+=+x%5Ea%2Fx%5Eb ALIGN=MIDDLE  ALT=\"x%5E%28a-b%29+=+x%5Ea%2Fx%5Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Concept 3.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Fb%29+%2F+%28c%2Fd%29+=+%28a%2Fb%29+%2A+%28d%2Fc%29+=+%28a%2Ad%29+%2F+%28b%2Ac%29 ALIGN=MIDDLE  ALT=\"%28a%2Fb%29+%2F+%28c%2Fd%29+=+%28a%2Fb%29+%2A+%28d%2Fc%29+=+%28a%2Ad%29+%2F+%28b%2Ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Back to your problem which is to simplify:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x%5E%28a%2Bb%29%29%5E%28a-b%29%29+%2F+%28%28x%5E%28a-2b%29%29%5E%28a%2B2b%29%29 ALIGN=MIDDLE  ALT=\"%28%28x%5E%28a%2Bb%29%29%5E%28a-b%29%29+%2F+%28%28x%5E%28a-2b%29%29%5E%28a%2B2b%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Use concept 1 to make your equation equal to:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%28%28a%2Bb%29%2A%28a-b%29%29%29+%2F+%28x%5E%28%28a-2b%29%2A%28a%2B2b%29%29%29 ALIGN=MIDDLE  ALT=\"%28x%5E%28%28a%2Bb%29%2A%28a-b%29%29%29+%2F+%28x%5E%28%28a-2b%29%2A%28a%2B2b%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "When you multiply (a+b) * (a-b) you get a^2 - b^2\r<BR>\n" );
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document.write( "When you multiply (a+2b) * (a-2b) you get a^2 - 4b^2\r<BR>\n" );
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document.write( "Your equation becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%28a%5E2-b%5E2%29%29+%2F+%28x%5E%28a%5E2-4b%5E2%29%29 ALIGN=MIDDLE  ALT=\"%28x%5E%28a%5E2-b%5E2%29%29+%2F+%28x%5E%28a%5E2-4b%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Using concept 2, we change this equation to be equivalent to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%28a%5E2%29%2Fx%5E%28b%5E2%29%29+%2F+%28x%5E%28a%5E2%29%2Fx%5E%284b%5E2%29%29 ALIGN=MIDDLE  ALT=\"%28x%5E%28a%5E2%29%2Fx%5E%28b%5E2%29%29+%2F+%28x%5E%28a%5E2%29%2Fx%5E%284b%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Using concept 3, we change this equation to be equivalent to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%28a%5E2%29+%2A+x%5E%284b%5E2%29%29+%2F+%28x%5E%28b%5E2%29+%2A+x%5E%28a%5E2%29%29 ALIGN=MIDDLE  ALT=\"%28x%5E%28a%5E2%29+%2A+x%5E%284b%5E2%29%29+%2F+%28x%5E%28b%5E2%29+%2A+x%5E%28a%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The x^(a^2) cancels out and the x^(4b^2) / x^(b^2) becomes:\r<BR>\n" );
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document.write( "x^(4b^2 - b^2) which becomes x^(3b^2) using concept 2.\r<BR>\n" );
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document.write( "You are left with:\r<BR>\n" );
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document.write( "as your answer.\r<BR>\n" );
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document.write( "To see if this is correct, we assign some values at random to the original equation and the final equation to see if they are equivalent.\r<BR>\n" );
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document.write( "I assigned values of:\r<BR>\n" );
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document.write( "a = 5, b = 2<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "a = 2, b = 2<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "a = 2, b = 3\r<BR>\n" );
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document.write( "and confirmed that the original equation and the final equation are equivalent.\r<BR>\n" );
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document.write( "I think the answer is good.\r<BR>\n" );
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document.write( "The answer is:\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );