document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=372457>Question 372457</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2^(2/3)x+1 - 3*2^(1/3)x - 20 = 0.\r<BR>\n" );
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document.write( "This would be read 2 to the (2/3)x+1[(2/3)x+1 is exponent], minus 3 times 2 to the (1/3)x [(1/3)x is exponent], minus 20 equals 0. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #265350 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=265350\">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=2%5E%28%282%2F3%29x%2B1%29+-+3%2A2%5E%28%281%2F3%29x%29+-+20+=+0 ALIGN=MIDDLE  ALT=\"2%5E%28%282%2F3%29x%2B1%29+-+3%2A2%5E%28%281%2F3%29x%29+-+20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is a tough one. The keys to solving this are:<ul><li>Recognizing that the first exponent is <i>almost</i> exactly twice the second exponent. (The +1 keeps it from being exactly twice.)</li><li>If the first exponent was twice the second one then this equation would be in \"quadratic form\" for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28%281%2F3%29x%29 ALIGN=MIDDLE  ALT=\"2%5E%28%281%2F3%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and quadratic form equations can be solved.</li><li>The +1 in the first exponent can be \"factored out\". Just like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%5E4+=+2%5E%284%2B1%29 ALIGN=MIDDLE  ALT=\"2%2A2%5E4+=+2%5E%284%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%5E%28%282%2F3%29x%29+=+2%5E%28%282%2F3%29x+%2B+1%29 ALIGN=MIDDLE  ALT=\"2%2A2%5E%28%282%2F3%29x%29+=+2%5E%28%282%2F3%29x+%2B+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li></ul><BR>\n" );
document.write( "So by factoring out a 2 from the first term we end up with a solvable quadratic form equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%5E%28%282%2F3%29x%29+-+3%2A2%5E%28%281%2F3%29x%29+-+20+=+0 ALIGN=MIDDLE  ALT=\"2%2A2%5E%28%282%2F3%29x%29+-+3%2A2%5E%28%281%2F3%29x%29+-+20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The next step is kind of a big one. If you have trouble following it, see \"Using a temporary variable\" below (where I use a lot of little steps instead of this one big one.) Next we factor the quadratic form equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2A2%5E%28%281%2F3%29x%29+%2B+5%29%282%5E%28%281%2F3%29x%29+-+4%29+=+0 ALIGN=MIDDLE  ALT=\"%282%2A2%5E%28%281%2F3%29x%29+%2B+5%29%282%5E%28%281%2F3%29x%29+-+4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From the Zero Product Property we know that this (or any) product can be zero <i>only</i> if one (or more of the factors is zero. So:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%5E%28%281%2F3%29x%29+%2B+5+=+0 ALIGN=MIDDLE  ALT=\"2%2A2%5E%28%281%2F3%29x%29+%2B+5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28%281%2F3%29x%29+-+4+=+0 ALIGN=MIDDLE  ALT=\"2%5E%28%281%2F3%29x%29+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting 5 from both sides of the first equation we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%5E%28%281%2F3%29x%29+=+-5 ALIGN=MIDDLE  ALT=\"2%2A2%5E%28%281%2F3%29x%29+=+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But a power of 2 cannot be negative (and neither can 2 times a power of 2. So this equation has not solutions. But we still have the other equation. Adding 4 to each side of that equation we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28%281%2F3%29x%29+=+4%29 ALIGN=MIDDLE  ALT=\"2%5E%28%281%2F3%29x%29+=+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The quick way to solve this is to realize that 4 is also a power of 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28%281%2F3%29x%29+=+2%5E2%29 ALIGN=MIDDLE  ALT=\"2%5E%28%281%2F3%29x%29+=+2%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In order for these powers of 2 to be equal , the exponents must be equal:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29x+=+2%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29x+=+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply both sides by 3 we get:<BR>\n" );
document.write( "x = 6<BR>\n" );
document.write( "which is our solution.<br><BR>\n" );
document.write( "<b>\"Using a temporary variable\"</b><BR>\n" );
document.write( "We have an equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%5E%28%282%2F3%29x%29+-+3%2A2%5E%28%281%2F3%29x%29+-+20+=+0 ALIGN=MIDDLE  ALT=\"2%2A2%5E%28%282%2F3%29x%29+-+3%2A2%5E%28%281%2F3%29x%29+-+20+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It often takes practice to see how to make the big step I used above. Until then you can use a temporary variable.<BR>\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q+=+2%5E%28%281%2F3%29x%29 ALIGN=MIDDLE  ALT=\"q+=+2%5E%28%281%2F3%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=q%5E2+=+%282%5E%28%281%2F3%29x%29%29%5E2+=+2%5E%28%282%2F3%29x%29 ALIGN=MIDDLE  ALT=\"q%5E2+=+%282%5E%28%281%2F3%29x%29%29%5E2+=+2%5E%28%282%2F3%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting these into the equation we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2q%5E2+-3q+-+20+-+0 ALIGN=MIDDLE  ALT=\"2q%5E2+-3q+-+20+-+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is clearly a quadratic equation and it is not very hard to factor:<BR>\n" );
document.write( "(2q + 5)(q - 4) = 0<BR>\n" );
document.write( "From the Zero Product Property we know that this (or any) product can be zero <i>only</i> if one (or more of the factors is zero. So:<BR>\n" );
document.write( "2q + 5 = 0 or q - 4 = 0<BR>\n" );
document.write( "Solving these we get:<BR>\n" );
document.write( "q = -5/2 or q = 4<BR>\n" );
document.write( "We have found q. But we want to find x. So at this point we substitute back in for q:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28%281%2F3%29x%29+=+%28-5%29%2F2 ALIGN=MIDDLE  ALT=\"2%5E%28%281%2F3%29x%29+=+%28-5%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28%281%2F3%29x%29+=+4 ALIGN=MIDDLE  ALT=\"2%5E%28%281%2F3%29x%29+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A power of 2 cannot be negative. So there are no solutions for the first equation. But we can find a solution to the second equation. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28%281%2F3%29x%29+=+4%29 ALIGN=MIDDLE  ALT=\"2%5E%28%281%2F3%29x%29+=+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The quick way to solve this is to realize that 4 is also a power of 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28%281%2F3%29x%29+=+2%5E2%29 ALIGN=MIDDLE  ALT=\"2%5E%28%281%2F3%29x%29+=+2%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In order for these powers of 2 to be equal, the exponents must be equal:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29x+=+2%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29x+=+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiply both sides by 3 we get:<BR>\n" );
document.write( "x = 6<BR>\n" );
document.write( "which is our solution.<br> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );