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 #73988 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=73988\">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=root%283%2C%282z%2B3%29%5E2%29%2Broot%283%2C%282z%2B3%29%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C%282z%2B3%29%5E2%29%2Broot%283%2C%282z%2B3%29%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2z%2B3 ALIGN=MIDDLE  ALT=\"y=2z%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%5E2%29%2Broot%283%2Cy%29=6 ALIGN=MIDDLE  ALT=\"root%283%2Cy%5E2%29%2Broot%283%2Cy%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > So now we get this equation in terms of y\r<BR>\n" );
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document.write( "Here's an important side note: Take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%5E2%29%2Broot%283%2Cy%29=6 ALIGN=MIDDLE  ALT=\"root%283%2Cy%5E2%29%2Broot%283%2Cy%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%29 ALIGN=MIDDLE  ALT=\"root%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%5E2%29=6-root%283%2Cy%29 ALIGN=MIDDLE  ALT=\"root%283%2Cy%5E2%29=6-root%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will be useful later\r<BR>\n" );
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document.write( "Now let's move back to the main problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%2Ay%29%2Broot%283%2Cy%29=6 ALIGN=MIDDLE  ALT=\"root%283%2Cy%2Ay%29%2Broot%283%2Cy%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%29%2Aroot%283%2Cy%29%2Broot%283%2Cy%29=6 ALIGN=MIDDLE  ALT=\"root%283%2Cy%29%2Aroot%283%2Cy%29%2Broot%283%2Cy%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break down the root\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%29%28root%283%2Cy%29%2B1%29=6 ALIGN=MIDDLE  ALT=\"root%283%2Cy%29%28root%283%2Cy%29%2B1%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%29 ALIGN=MIDDLE  ALT=\"root%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28root%283%2Cy%29%2B1%29=6%2Froot%283%2Cy%29 ALIGN=MIDDLE  ALT=\"%28root%283%2Cy%29%2B1%29=6%2Froot%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%29 ALIGN=MIDDLE  ALT=\"root%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28root%283%2Cy%29%2B1%29%5E3=%286%2Froot%283%2Cy%29%29%5E3 ALIGN=MIDDLE  ALT=\"%28root%283%2Cy%29%2B1%29%5E3=%286%2Froot%283%2Cy%29%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Cube both sides\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B3root%283%2Cy%29%2B3root%283%2Cy%5E2%29%2B1=216%2Fy ALIGN=MIDDLE  ALT=\"y%2B3root%283%2Cy%29%2B3root%283%2Cy%5E2%29%2B1=216%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > Expand the left side\r<BR>\n" );
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document.write( "Remember, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%5E2%29=6-root%283%2Cy%29 ALIGN=MIDDLE  ALT=\"root%283%2Cy%5E2%29=6-root%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so replace <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2Cy%5E2%29 ALIGN=MIDDLE  ALT=\"root%283%2Cy%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6-root%283%2Cy%29 ALIGN=MIDDLE  ALT=\"6-root%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B3root%283%2Cy%29%2B3%2Ahighlight%28%286-root%283%2Cy%29%29%29%2B1=216%2Fy ALIGN=MIDDLE  ALT=\"y%2B3root%283%2Cy%29%2B3%2Ahighlight%28%286-root%283%2Cy%29%29%29%2B1=216%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B3root%283%2Cy%29%2B18-3root%283%2Cy%29%2B1=216%2Fy ALIGN=MIDDLE  ALT=\"y%2B3root%283%2Cy%29%2B18-3root%283%2Cy%29%2B1=216%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B18%2Bcross%283root%283%2Cy%29-3root%283%2Cy%29%29%2B1=216%2Fy ALIGN=MIDDLE  ALT=\"y%2B18%2Bcross%283root%283%2Cy%29-3root%283%2Cy%29%29%2B1=216%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice the terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3root%283%2Cy%29 ALIGN=MIDDLE  ALT=\"3root%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3root%283%2Cy%29 ALIGN=MIDDLE  ALT=\"-3root%283%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > cancel to zero\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B18%2B1=216%2Fy ALIGN=MIDDLE  ALT=\"y%2B18%2B1=216%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B19=216%2Fy ALIGN=MIDDLE  ALT=\"y%2B19=216%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2B19y%5E2=216 ALIGN=MIDDLE  ALT=\"y%5E2%2B19y%5E2=216\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by the LCD y to clear any fractions\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2B19y%5E2-216=0 ALIGN=MIDDLE  ALT=\"y%5E2%2B19y%5E2-216=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Get all terms to one side\r<BR>\n" );
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document.write( "Now simply use any technique to solve for y\r<BR>\n" );
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document.write( "When you solve for y (I simply used a calculator), you get \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=8 ALIGN=MIDDLE  ALT=\"y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-27 ALIGN=MIDDLE  ALT=\"y=-27\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug in the first answer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=8 ALIGN=MIDDLE  ALT=\"y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > into  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2z%2B3 ALIGN=MIDDLE  ALT=\"y=2z%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now solve for z\r<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;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z=2.5 ALIGN=MIDDLE  ALT=\"z=2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug in the second answer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-27 ALIGN=MIDDLE  ALT=\"y=-27\"   DISABLED_style= \"max-width:90%; height:auto;\" > into  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2z%2B3 ALIGN=MIDDLE  ALT=\"y=2z%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-27=2z%2B3 ALIGN=MIDDLE  ALT=\"-27=2z%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now solve for z\r<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;\" > \r<BR>\n" );
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document.write( "Check:\r<BR>\n" );
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document.write( "Let's check the answer <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%282z%2B3%29%5E2%29%2Broot%283%2C%282z%2B3%29%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C%282z%2B3%29%5E2%29%2Broot%283%2C%282z%2B3%29%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%282%285%2F2%29%2B3%29%5E2%29%2Broot%283%2C%282%285%2F2%29%2B3%29%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C%282%285%2F2%29%2B3%29%5E2%29%2Broot%283%2C%282%285%2F2%29%2B3%29%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%285%2B3%29%5E2%29%2Broot%283%2C%285%2B3%29%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C%285%2B3%29%5E2%29%2Broot%283%2C%285%2B3%29%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2 ALIGN=MIDDLE  ALT=\"5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 5\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C8%5E2%29%2Broot%283%2C8%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C8%5E2%29%2Broot%283%2C8%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 5 and 3 to get 8\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C64%29%2Broot%283%2C8%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C64%29%2Broot%283%2C8%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square 8 to get 64\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B2=6 ALIGN=MIDDLE  ALT=\"4%2B2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the cube root of 64 to get 4 and take the cube root of 8 to get 2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6=6 ALIGN=MIDDLE  ALT=\"6=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since the two sides of the equation are equal, this verifies our answer.\r<BR>\n" );
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document.write( "Now let's check the answer <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%282z%2B3%29%5E2%29%2Broot%283%2C%282z%2B3%29%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C%282z%2B3%29%5E2%29%2Broot%283%2C%282z%2B3%29%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%282%28-15%29%2B3%29%5E2%29%2Broot%283%2C%282%28-15%29%2B3%29%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C%282%28-15%29%2B3%29%5E2%29%2Broot%283%2C%282%28-15%29%2B3%29%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%28-30%2B3%29%5E2%29%2Broot%283%2C%28-30%2B3%29%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C%28-30%2B3%29%5E2%29%2Broot%283%2C%28-30%2B3%29%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply 2 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15 ALIGN=MIDDLE  ALT=\"-15\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get -30\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C%28-27%29%5E2%29%2Broot%283%2C-27%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C%28-27%29%5E2%29%2Broot%283%2C-27%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add -30 and 3 to get -27\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C729%29%2Broot%283%2C-27%29=6 ALIGN=MIDDLE  ALT=\"root%283%2C729%29%2Broot%283%2C-27%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square -27 to get 729\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-3=6 ALIGN=MIDDLE  ALT=\"9-3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the cube root of 729 to get 9 and take the cube root of -27 to get -3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6=6 ALIGN=MIDDLE  ALT=\"6=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since the two sides of the equation are equal, this verifies our answer.\r<BR>\n" );
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