document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=148114>Question 148114</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify:\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">        \r\n" );
document.write( "        ((x^(3/2))y^-3)^1/3\r\n" );
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document.write( "        ((x^(1/2))y^-6)^-1\r\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #108498 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=108498\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\left(x^{\frac{3}{2}}y^{-3}\right)^{\frac{1}{3}}}{\left(x^{\frac{1}{2}}y^{-6}\right)^{-1}}\"> Start with the given expression.\r<BR>\n" );
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document.write( "Distribute the outer exponents to the inner exponents. Remember, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E%28y%29%29%5Ez=x%5E%28y%2Az%29 ALIGN=MIDDLE  ALT=\"%28x%5E%28y%29%29%5Ez=x%5E%28y%2Az%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So in our case the numerator goes from  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(x^{\frac{3}{2}}y^{-3}\right)^{\frac{1}{3}}\"> to  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{\left(\frac{3}{2}\right)\left(\frac{1}{3}\right)}y^{\left(-3\right)\left(\frac{1}{3}\right)}\">and the denominator goes from  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(x^{\frac{1}{2}}y^{-6}\right)^{-1}\"> to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{\left(\frac{1}{2}\right)\left(-1\right)}y^{\left(-6\right)\left(-1\right)}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{x^{\left(\frac{3}{2}\right)\left(\frac{1}{3}\right)}y^{\left(-3\right)\left(\frac{1}{3}\right)}}{x^{\left(\frac{1}{2}\right)\left(-1\right)}y^{\left(-6\right)\left(-1\right)}}\">  Use the technique described above to distribute the exponents.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{x^{\frac{3}{6}}y^{-\frac{3}{3}}}{x^{-\frac{1}{2}}y^{6}}\"> Multiply the exponents.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{x^{\frac{1}{2}}y^{-1}}{x^{-\frac{1}{2}}y^{6}}\"> Reduce. \r<BR>\n" );
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document.write( "Now when we divide monomials, we simply subtract the exponents. So for example <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E5%2Fx%5E2=x%5E%285-2%29=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E5%2Fx%5E2=x%5E%285-2%29=x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's apply this technique to the problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{\frac{1}{2}-\left(-\frac{1}{2}\right)}y^{-1-6}\"> Subtract the exponents.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{\frac{1}{2}+\frac{1}{2}}y^{-1-6}\"> Rewrite <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{2}-\left(-\frac{1}{2}\right)\"> as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{2}+\frac{1}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{\frac{2}{2}}y^{-7}\"> Combine the exponents.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x^{1}y^{-7}\"> Reduce.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE xy^{-7}\"> Simplify\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{x}{y^{7}}\"> Now rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%28-7%29 ALIGN=MIDDLE  ALT=\"y%5E%28-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fy%5E7 ALIGN=MIDDLE  ALT=\"1%2Fy%5E7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\left(x^{\frac{3}{2}}y^{-3}\right)^{\frac{1}{3}}}{\left(x^{\frac{1}{2}}y^{-6}\right)^{-1}}\"> simplifies to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{x}{y^{7}}\"> \r<BR>\n" );
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document.write( "In other words, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{\left(x^{\frac{3}{2}}y^{-3}\right)^{\frac{1}{3}}}{\left(x^{\frac{1}{2}}y^{-6}\right)^{-1}}=\frac{x}{y^{7}}\"> where every variable is positive and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C%3E0 ALIGN=MIDDLE  ALT=\"x%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C%3E0 ALIGN=MIDDLE  ALT=\"y%3C%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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