document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=101751>Question 101751</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321057635/algebrahomeworkh>Precalculus with Limits</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Please help us solve...\r<BR>\n" );
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document.write( "(2z+3)^(2/3) + (2z+3)^(1/3) = 6\r<BR>\n" );
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document.write( "We've tried cubing each side which gets rather complicated, and we've tried factoring out a (2z+3)^(1/3), but we're not making any progress.  Dad is helping, but he's having a tough time remembering complex radical problems! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #73986 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=73986\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Interesting problem. You can make the problem a little easier to work by making the following<BR>\n" );
document.write( "substitution: let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+%282z+%2B+3%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"A+=+%282z+%2B+3%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "When you make that substitution the problem becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+%2B+A+=+6 ALIGN=MIDDLE  ALT=\"A%5E2+%2B+A+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get this into the standard quadratic form by subtracting 6 from both sides. When you do that<BR>\n" );
document.write( "the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E2+%2B+A+-+6+=+0 ALIGN=MIDDLE  ALT=\"A%5E2+%2B+A+-+6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This equation can be solved by factoring. When factored you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28A+%2B+3%29%2A%28A+-+2%29=+0 ALIGN=MIDDLE  ALT=\"%28A+%2B+3%29%2A%28A+-+2%29=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This can be solved for A by setting each of the factors equal to zero, which is one technique<BR>\n" );
document.write( "for solving a quadratic equation that can be factored.  Setting each of the factors equal<BR>\n" );
document.write( "to zero results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+%2B+3+=+0 ALIGN=MIDDLE  ALT=\"A+%2B+3+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which, after subtracting 3 from both sides, gives <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+-3 ALIGN=MIDDLE  ALT=\"A+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+-+2+=+0 ALIGN=MIDDLE  ALT=\"A+-+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > which, after adding 2 to both sides, results in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+2 ALIGN=MIDDLE  ALT=\"A+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So there are two answers for A. And at this time we can return to the original definition<BR>\n" );
document.write( "of A and substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282z+%2B+3%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"%282z+%2B+3%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into the two answers that we got for A. When<BR>\n" );
document.write( "we do we get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282z+%2B+3%29%5E%281%2F3%29+=+-3 ALIGN=MIDDLE  ALT=\"%282z+%2B+3%29%5E%281%2F3%29+=+-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Cube both sides of this to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z+%2B+3+=+-27 ALIGN=MIDDLE  ALT=\"2z+%2B+3+=+-27\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 3 from both sides and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z+=+-30 ALIGN=MIDDLE  ALT=\"2z+=+-30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and solve for z by dividing both sides by 2 to find that:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-15 ALIGN=MIDDLE  ALT=\"z+=+-15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can next use that same process on the second answer that you got for A, which was<BR>\n" );
document.write( "A = 2. Substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282z+%2B+3%29%5E%281%2F3%29 ALIGN=MIDDLE  ALT=\"%282z+%2B+3%29%5E%281%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for A and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282z+%2B+3%29%5E%281%2F3%29+=+2 ALIGN=MIDDLE  ALT=\"%282z+%2B+3%29%5E%281%2F3%29+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Cube both sides and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z+%2B+3+=+8 ALIGN=MIDDLE  ALT=\"2z+%2B+3+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 3 from both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2z+=+5 ALIGN=MIDDLE  ALT=\"2z+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide both sides by 2 and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+5%2F2 ALIGN=MIDDLE  ALT=\"z+=+5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "So you have two answers for z ... <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+-15 ALIGN=MIDDLE  ALT=\"z+=+-15\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+5%2F2 ALIGN=MIDDLE  ALT=\"z+=+5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope this helps you to see how you can get answers for z. </SPAN>\n" );
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