document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=639861>Question 639861</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi<BR>\n" );
document.write( "This question has to do with negative exponents. I am trying to rewrite this expression without negative exponents.\r<BR>\n" );
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document.write( "a^-1 - b^-1<BR>\n" );
document.write( "___________<BR>\n" );
document.write( "a^-3 - b^-3\r<BR>\n" );
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document.write( "I believe the answer is:\r<BR>\n" );
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document.write( "a^2b^2<BR>\n" );
document.write( "_________________<BR>\n" );
document.write( "a^2+ab+b^2\r<BR>\n" );
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document.write( "I just don't know the steps on how to get there.\r<BR>\n" );
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document.write( "Thank you<BR>\n" );
document.write( "Regards<BR>\n" );
document.write( "Mike </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #402956 by </B><A HREF=/tutors/aboutme.mpl?userid=DrBeeee><B>DrBeeee(684)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=DrBeeee><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=402956\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Your answer is correct. Here is how I get there.<BR>\n" );
document.write( "The first operation we need to do is get rid of the negative exponents.<BR>\n" );
document.write( "This is done by replacing them with their inverse (also called the reciprocal). <BR>\n" );
document.write( "For example,<BR>\n" );
document.write( "a^(-1) = 1/a<BR>\n" );
document.write( "Likewise<BR>\n" );
document.write( "b^(-1) = 1/b<BR>\n" );
document.write( "and <BR>\n" );
document.write( "a^(-3) = 1/a^3<BR>\n" );
document.write( "and<BR>\n" );
document.write( "b^(-3) = 1/b^3<BR>\n" );
document.write( "Make these sustitutions in your expression yields<BR>\n" );
document.write( "(1) (1/a - 1/b)/(1/a^3 - 1/b^3)<BR>\n" );
document.write( "Now let's get rid of the fractions.<BR>\n" );
document.write( "Multiply (1) by (a^3*b^3)/(a^3*b^3), which changes nothing because we are multiplying by one, and obtain<BR>\n" );
document.write( "(2) (a^2*b^2)((b - a)/(b^3 - a^3))<BR>\n" );
document.write( "The denominator of (2) factors into<BR>\n" );
document.write( "(3) (b - a)*(b^2 + ab + a^2)<BR>\n" );
document.write( "You can FOIL (3) to show that it is equal to the denominator of (2).<BR>\n" );
document.write( "Substituting (3) into (2) which simplifies to<BR>\n" );
document.write( "(4) (a^2*b^2)/(b^2 + ab + a^2)<BR>\n" );
document.write( "The answer (4) is the same as your \"belief\", but now you know the rest of the story.<BR>\n" );
document.write( "We can also go from (1) to (2) by cross multiplication to subtract the fractions in the numerator and denominator of (1).<BR>\n" );
document.write( "The numerator of (1) is<BR>\n" );
document.write( "(5) (1/a - 1/b) = (b-a)/(ab)<BR>\n" );
document.write( "and the denominator of (1) is<BR>\n" );
document.write( "(6) (1/a^3 - 1/b^3) = (b^3 - a^3)/(a^3*b^3)<BR>\n" );
document.write( "Taking the ratio of (5)/(6) yields (2).\r<BR>\n" );
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document.write( " </SPAN>\n" );
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