document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=709334>Question 709334</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ((5x^-2)+(5y^-2))/((8x^-1)+(8y^-1))\r<BR>\n" );
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document.write( "Simplify the compound fractional expression.  Give your answer in factored form. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #436559 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=436559\">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=%28%285x%5E%28-2%29%29%2B%285y%5E%28-2%29%29%29%2F%28%288x%5E%28-1%29%29%2B%288y%5E%28-1%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is a compound fraction because of the negative exponents. Let's rewrite the expression with positive exponents so we can see it better:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%285%2Fx%5E2%29%2B%285%2Fy%5E2%29%29%2F%28%288%2Fx%29%2B%288%2Fy%29%29 ALIGN=MIDDLE  ALT=\"%28%285%2Fx%5E2%29%2B%285%2Fy%5E2%29%29%2F%28%288%2Fx%29%2B%288%2Fy%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Different methods are taught on how to simplify this expression. One method is to<ol><li>Combine the terms in both the numerator and the denominator of the \"big\" fraction</li><li>Turn the division into a multiplication by the reciprocal of the denominator</li><li>Simplify</li></ol><BR>\n" );
document.write( "But I prefer a different method:<ol><li>Find the lowest common denominator (LCD) of all the \"little\" fractions.</li><li>Multiply the numerator and denominator of the \"big\" fraction by the LCD from step one.</li></ol>The first method requires addition/subtraction of fractions. The second one does not which is a major reason I prefer it.<br><BR>\n" );
document.write( "Using the second method...<BR>\n" );
document.write( "The \"little\" denominators are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, x and y. The LCD of these 4 is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Multiplying the numerator and denominator by this LCD:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2y%5E2%2Fx%5E2y%5E2%29%28%28%285%2Fx%5E2%29%2B%285%2Fy%5E2%29%29%2F%28%288%2Fx%29%2B%288%2Fy%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To multiply correctly we need to use the Distributive Property:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2y%5E2%285%2Fx%5E2%29%2Bx%5E2y%5E2%285%2Fy%5E2%29%29%2F%28x%5E2y%5E2%288%2Fx%29%2Bx%5E2y%5E2%288%2Fy%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In each \"little\" fraction the denominator cancels with all or some part of the LCD leaving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285y%5E2%2B5x%5E2%29%2F%288xy%5E2%2B8x%5E2y%29 ALIGN=MIDDLE  ALT=\"%285y%5E2%2B5x%5E2%29%2F%288xy%5E2%2B8x%5E2y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are no like terms here so we cannot simplify this further. But we might be able to reduce the fraction. To see if the fraction will reduce we must first factor the numerator and denominator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%28y%5E2%2Bx%5E2%29%29%2F%288xy%28y%2Bx%29%29 ALIGN=MIDDLE  ALT=\"%285%28y%5E2%2Bx%5E2%29%29%2F%288xy%28y%2Bx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are no factors in common between the numerator and denominator so the fraction will not reduce.<br><BR>\n" );
document.write( "The problem asks that the answer be left in factored form so our answer is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%28y%5E2%2Bx%5E2%29%29%2F%288xy%28y%2Bx%29%29 ALIGN=MIDDLE  ALT=\"%285%28y%5E2%2Bx%5E2%29%29%2F%288xy%28y%2Bx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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