document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=242673>Question 242673</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do you answer <BR>\n" );
document.write( "square root of 9 +4<BR>\n" );
document.write( "over<BR>\n" );
document.write( "square root of 8 +6\r<BR>\n" );
document.write( "\n" );
document.write( "I can't find the square root symbol on my keyboard,sorry!<BR>\n" );
document.write( "Thanks for your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #177661 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=177661\">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=%28sqrt%289%29%2B4%29%2F%28sqrt%288%29+%2B+6%29 ALIGN=MIDDLE  ALT=\"%28sqrt%289%29%2B4%29%2F%28sqrt%288%29+%2B+6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With a square root (which is irrational) in the denominator, the problem is to rationalize the denominator. But before we start we are going to simplify the square roots we do have.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%289%29+=+3 ALIGN=MIDDLE  ALT=\"sqrt%289%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288%29+=+sqrt%284%2A2%29+=+sqrt%284%29%2Asqrt%282%29+=+2sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%288%29+=+sqrt%284%2A2%29+=+sqrt%284%29%2Asqrt%282%29+=+2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting for these two we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283%29%2B4%29%2F%282sqrt%282%29+%2B+6%29 ALIGN=MIDDLE  ALT=\"%28%283%29%2B4%29%2F%282sqrt%282%29+%2B+6%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=7%2F%282sqrt%282%29+%2B+6%29 ALIGN=MIDDLE  ALT=\"7%2F%282sqrt%282%29+%2B+6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we can try to rationalize the denominator. With two terms, this denominator is a little harder to rationalize than if it was just one term. To rationalize two-term denominators we take advantage of the pattern: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2 ALIGN=MIDDLE  ALT=\"a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. All two-term expressions fit either the (a+b) or the (a-b) part of the pattern. And as we can see, the right side is a two-term expression <i>of perfect squares!</i> Expressions like (a+b) and (a-b) are called conjugates. So to rationalize a two-term denominator all we have to do is multiply the numerator and denominator by the conjugate of the denominator.<br><BR>\n" );
document.write( "Our denominator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%282%29+%2B+6 ALIGN=MIDDLE  ALT=\"2sqrt%282%29+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Its conjugate is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%282%29+-+6 ALIGN=MIDDLE  ALT=\"2sqrt%282%29+-+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this is what we will use to multiply the numerator and denominator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%2F%282sqrt%282%29+%2B+6%29%29%28%282sqrt%282%29+-+6%29%2F%282sqrt%282%29+-+6%29%29 ALIGN=MIDDLE  ALT=\"%287%2F%282sqrt%282%29+%2B+6%29%29%28%282sqrt%282%29+-+6%29%2F%282sqrt%282%29+-+6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2814sqrt%282%29+-+42%29%2F%28%282sqrt%282%29%29%5E2+-+%286%29%5E2%29 ALIGN=MIDDLE  ALT=\"%2814sqrt%282%29+-+42%29%2F%28%282sqrt%282%29%29%5E2+-+%286%29%5E2%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=%2814sqrt%282%29+-+42%29%2F%284%2A2+-+36%29 ALIGN=MIDDLE  ALT=\"%2814sqrt%282%29+-+42%29%2F%284%2A2+-+36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2814sqrt%282%29+-+42%29%2F%288+-+36%29 ALIGN=MIDDLE  ALT=\"%2814sqrt%282%29+-+42%29%2F%288+-+36%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2814sqrt%282%29+-+42%29%2F%28-24%29 ALIGN=MIDDLE  ALT=\"%2814sqrt%282%29+-+42%29%2F%28-24%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And last of all, we reduce the fraction. Factor out 2 in the numerator:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%287sqrt%282%29+-+21%29%29%2F%282%28-12%29%29 ALIGN=MIDDLE  ALT=\"%282%287sqrt%282%29+-+21%29%29%2F%282%28-12%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Cancel the 2's:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287sqrt%282%29+-+21%29%2F%28-12%29 ALIGN=MIDDLE  ALT=\"%287sqrt%282%29+-+21%29%2F%28-12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%287sqrt%282%29+-+21%29%2F12 ALIGN=MIDDLE  ALT=\"-%287sqrt%282%29+-+21%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-7sqrt%282%29+%2B+21%29%2F12 ALIGN=MIDDLE  ALT=\"%28-7sqrt%282%29+%2B+21%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );