document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118008>Question 118008</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are two problems that i have submitted and i do not understand the answer can you explain how to do these two problems to me.\r<BR>\n" );
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document.write( "1.     Perform the indicated operations:<BR>\n" );
document.write( "     sqrt[50] + 2sqrt [32] - sqrt [8]\r<BR>\n" );
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document.write( "and the second one is \r<BR>\n" );
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document.write( "2.  Rationalize the denominator\r<BR>\n" );
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document.write( "    2 / sqrt[6] - sqrt [5]\r<BR>\n" );
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document.write( "thank you i really appreciate this i really do not understand this and the two tutors got these answers \r<BR>\n" );
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document.write( "julie<BR>\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 #86050 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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=86050\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1) Perform the indicated operations:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2850%29%2B2sqrt%2832%29-sqrt%288%29 ALIGN=MIDDLE  ALT=\"sqrt%2850%29%2B2sqrt%2832%29-sqrt%288%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The key here is to factor the integers under the radical (the radicands) and see if you have any perfect squares that can be taken outside the radical by taking the square root.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2850%29+=+sqrt%282%2A25%29 ALIGN=MIDDLE  ALT=\"sqrt%2850%29+=+sqrt%282%2A25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 25 is a perfect square, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25+=+5%5E2 ALIGN=MIDDLE  ALT=\"25+=+5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%5E2%29+=+5 ALIGN=MIDDLE  ALT=\"sqrt%285%5E2%29+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2850%29+=+sqrt%282%2A25%29 ALIGN=MIDDLE  ALT=\"sqrt%2850%29+=+sqrt%282%2A25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2Asqrt%2825%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2Asqrt%2825%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2A5 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"5%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Similarly for:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%2832%29+=+2%2Asqrt%282%2A16%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%2832%29+=+2%2Asqrt%282%2A16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%282%29%2Asqrt%2816%29+=+2%2Asqrt%282%29%2A4 ALIGN=MIDDLE  ALT=\"2%2Asqrt%282%29%2Asqrt%2816%29+=+2%2Asqrt%282%29%2A4\"   DISABLED_style= \"max-width:90%; height:auto;\" >or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A4%2Asqrt%282%29+=+8%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"2%2A4%2Asqrt%282%29+=+8%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and for:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%29+=+sqrt%282%2A4%29 ALIGN=MIDDLE  ALT=\"sqrt%288%29+=+sqrt%282%2A4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29%2Asqrt%284%29+=+sqrt%282%29%2A2 ALIGN=MIDDLE  ALT=\"sqrt%282%29%2Asqrt%284%29+=+sqrt%282%29%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "OK, now let's put all of this together:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2850%29%2B2sqrt%2832%29-sqrt%288%29+=+5%2Asqrt%282%29%2B8%2Asqrt%282%29-2%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%2850%29%2B2sqrt%2832%29-sqrt%288%29+=+5%2Asqrt%282%29%2B8%2Asqrt%282%29-2%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now we can combine all of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >'s<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Asqrt%282%29%2B8%2Asqrt%282%29-2%2Asqrt%282%29+=+%285%2B8-2%29%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"5%2Asqrt%282%29%2B8%2Asqrt%282%29-2%2Asqrt%282%29+=+%285%2B8-2%29%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11sqrt%282%29 ALIGN=MIDDLE  ALT=\"11sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "2) Rationalize the denominator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%28sqrt%286%29-sqrt%285%29%29 ALIGN=MIDDLE  ALT=\"2%2F%28sqrt%286%29-sqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To rationalize the denominator, you will multiply both the numerator and the denominator by the \"conjugate\" of the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29-sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29-sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this is just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29%2Bsqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29%2Bsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The conjugate of (a+b) is (a-b) and the conjugate of (a-b) is (a+b), see the idea?<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F%28sqrt%286%29-sqrt%285%29%29%29%2A%28%28sqrt%286%29%2Bsqrt%285%29%29%2F%28sqrt%286%29%2Bsqrt%285%29%29%29=2%2A%28sqrt%286%29%2Bsqrt%285%29%29%2F%28%28sqrt%286%29-sqrt%285%29%29%2A%28sqrt%286%29%2Bsqrt%285%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Let's look at the denominator which is in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-b%29%28a%2Bb%29+=+a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"%28a-b%29%28a%2Bb%29+=+a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So after multiplying, as indicated, the denominator will look like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%286%29%29%5E2+-+%28sqrt%285%29%29%5E2+=+6-5 ALIGN=MIDDLE  ALT=\"%28sqrt%286%29%29%5E2+-+%28sqrt%285%29%29%5E2+=+6-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so all we have left is the numerator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%28sqrt%286%29%2Bsqrt%285%29%29 ALIGN=MIDDLE  ALT=\"2%2A%28sqrt%286%29%2Bsqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );