document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=624840>Question 624840</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ((5x)((3y^2)/2)^(1/2))-((3y)((8x^2)/3)^(1/2))+((2)((3x^2y^2)/2)^(1/2)) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #393191 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=393191\">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=%28%285x%29%28%283y%5E2%29%2F2%29%5E%281%2F2%29%29-%28%283y%29%28%288x%5E2%29%2F3%29%5E%281%2F2%29%29%2B%28%282%29%28%283x%5E2y%5E2%29%2F2%29%5E%281%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since exponents of 1/2 mean the same thing as square roots, I am going to rewrite the expression with square roots:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%29sqrt%28%283y%5E2%29%2F2%29-%283y%29sqrt%28%288x%5E2%29%2F3%29%2B%282%29sqrt%28%283x%5E2y%5E2%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we will simplify each square root. Part of simplifying square roots is rationalizing the denominators. There are a variety of ways to go about doing this. I like to start by making each denominator a perfect square:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%29sqrt%28%28%283y%5E2%29%2F2%29%282%2F2%29%29-%283y%29sqrt%28%28%288x%5E2%29%2F3%29%283%2F3%29%29%2B%282%29sqrt%28%28%283x%5E2y%5E2%29%2F2%29%282%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which leads to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%29sqrt%28%286y%5E2%29%2F4%29-%283y%29sqrt%28%2824x%5E2%29%2F9%29%2B%282%29sqrt%28%286x%5E2y%5E2%29%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we use the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28a%2C+p%2Fq%29+=+root%28a%2C+p%29%2Froot%28a%2C+q%29 ALIGN=MIDDLE  ALT=\"root%28a%2C+p%2Fq%29+=+root%28a%2C+p%29%2Froot%28a%2C+q%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > property of radicals to split each square root:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%29sqrt%286y%5E2%29%2Fsqrt%284%29-%283y%29sqrt%2824x%5E2%29%2Fsqrt%289%29%2B%282%29sqrt%286x%5E2y%5E2%29%2Fsqrt%284%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Because of our earlier work, each denominator will simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%29sqrt%286y%5E2%29%2F2-%283y%29sqrt%2824x%5E2%29%2F3%2B%282%29sqrt%286x%5E2y%5E2%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we simplify the square roots in the numerators. Each one happens to have one or more prefect square factors (which I like to put first using the Commutative Property of Multiplication):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%29sqrt%28y%5E2%2A6%29%2F2-%283y%29sqrt%284%2Ax%5E2%2A6%29%2F3%2B%282%29sqrt%28x%5E2%2Ay%5E2%2A6%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we use another property of radicals, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29 ALIGN=MIDDLE  ALT=\"root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to split the square roots so that each perfect square factor is in its own square root:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%29%28sqrt%28y%5E2%29%2Asqrt%286%29%29%2F2-%283y%29%28sqrt%284%29%2Asqrt%28x%5E2%29%2Asqrt%286%29%29%2F3%2B%282%29%28sqrt%28x%5E2%29%2Asqrt%28y%5E2%29%2Asqrt%286%29%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Each of the square roots of the perfect squares simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285x%29%28y%2Asqrt%286%29%29%2F2-%283y%29%282%2Ax%2Asqrt%286%29%29%2F3%2B%282%29%28x%2Ay%2Asqrt%286%29%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies further to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285xy%2Asqrt%286%29%29%2F2-%286xy%2Asqrt%286%29%29%2F3%2B%282xy%2Asqrt%286%29%29%2F2 ALIGN=MIDDLE  ALT=\"%285xy%2Asqrt%286%29%29%2F2-%286xy%2Asqrt%286%29%29%2F3%2B%282xy%2Asqrt%286%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The last two fractions reduce:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285xy%2Asqrt%286%29%29%2F2-%282xy%2Asqrt%286%29%29%2B%28xy%2Asqrt%286%29%29 ALIGN=MIDDLE  ALT=\"%285xy%2Asqrt%286%29%29%2F2-%282xy%2Asqrt%286%29%29%2B%28xy%2Asqrt%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And last of all, these are all like terms! They are all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%2Asqrt%286%29 ALIGN=MIDDLE  ALT=\"xy%2Asqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms. So we can add/subtract them. Just add/subtract the coefficients. To see this more easily, I'm going to rewrite the first and third terms so we can see the coefficient we should add:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F2%29xy%2Asqrt%286%29-%282xy%2Asqrt%286%29%29%2B%281xy%2Asqrt%286%29%29 ALIGN=MIDDLE  ALT=\"%285%2F2%29xy%2Asqrt%286%29-%282xy%2Asqrt%286%29%29%2B%281xy%2Asqrt%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29xy%2Asqrt%286%29 ALIGN=MIDDLE  ALT=\"%283%2F2%29xy%2Asqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F2%29-2%2B1+=+3%2F2 ALIGN=MIDDLE  ALT=\"%285%2F2%29-2%2B1+=+3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%285x%29%28%283y%5E2%29%2F2%29%5E%281%2F2%29%29-%28%283y%29%28%288x%5E2%29%2F3%29%5E%281%2F2%29%29%2B%28%282%29%28%283x%5E2y%5E2%29%2F2%29%5E%281%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29xy%2Asqrt%286%29 ALIGN=MIDDLE  ALT=\"%283%2F2%29xy%2Asqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or, if you prefer the 1/2 exponents:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F2%29xy%2A%286%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"%283%2F2%29xy%2A%286%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );