document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209405>Question 1209405</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How many pairs of integers (a,b) satisfy the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%5Ea ALIGN=MIDDLE  ALT=\"ab%5Ea\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 648?<BR>\n" );
document.write( "CC11F #10 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848757 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848757\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%5Ea=648=%282%5E3%29%283%5E4%29 ALIGN=MIDDLE  ALT=\"ab%5Ea=648=%282%5E3%29%283%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The only prime factors of 648 are 2 and 3.<br><BR>\n" );
document.write( "a is an exponent in the expression; the exponents in the prime factorization of 648 are 3 and 4.  That means a can only be 1, 2, or 3.<br><BR>\n" );
document.write( "(1) a=1<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%28b%5E1%29=648 ALIGN=MIDDLE  ALT=\"%281%29%28b%5E1%29=648\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=648 ALIGN=MIDDLE  ALT=\"b=648\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "first solution: (a,b) = (1,648)<br><BR>\n" );
document.write( "(2) a=2<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%29%28b%5E2%29=648 ALIGN=MIDDLE  ALT=\"%282%29%28b%5E2%29=648\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=648%2F2=324 ALIGN=MIDDLE  ALT=\"b%5E2=648%2F2=324\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=18 ALIGN=MIDDLE  ALT=\"b=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-18 ALIGN=MIDDLE  ALT=\"b=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "second and third solutions: (a,b) = (2,18) and (a,b) = (2,-18)<br><BR>\n" );
document.write( "(3) a=3<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%29%28b%5E3%29=648 ALIGN=MIDDLE  ALT=\"%283%29%28b%5E3%29=648\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E3=216 ALIGN=MIDDLE  ALT=\"b%5E3=216\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=6 ALIGN=MIDDLE  ALT=\"b=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "fourth solution: (a,b) = (3,6)<br><BR>\n" );
document.write( "ANSWER: 4 pairs of integer (a,b) satisfy the equation ab^a = 648<br><BR>\n" );
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