document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=293344>Question 293344</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how would you solve this proof?<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 #211778 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=211778\">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%286%29-sqrt%282%29%29%2F4+=++sqrt%282-sqrt%283%29%29%2F2+ ALIGN=MIDDLE  ALT=\"+%28sqrt%286%29-sqrt%282%29%29%2F4+=++sqrt%282-sqrt%283%29%29%2F2+\"   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=+sqrt%286%29-sqrt%282%29+=++2%2Asqrt%282-sqrt%283%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt%286%29-sqrt%282%29+=++2%2Asqrt%282-sqrt%283%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by the LCD 4 to clear out the fractions.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+sqrt%286%29-sqrt%282%29+=++sqrt%284%29%2Asqrt%282-sqrt%283%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt%286%29-sqrt%282%29+=++sqrt%284%29%2Asqrt%282-sqrt%283%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%29 ALIGN=MIDDLE  ALT=\"sqrt%284%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=+sqrt%286%29-sqrt%282%29+=++sqrt%284%282-sqrt%283%29%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt%286%29-sqrt%282%29+=++sqrt%284%282-sqrt%283%29%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the roots on the right side using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29%2Asqrt%28y%29=sqrt%28x%2Ay%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29%2Asqrt%28y%29=sqrt%28x%2Ay%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=+sqrt%286%29-sqrt%282%29+=++sqrt%288-4%2Asqrt%283%29%29+ ALIGN=MIDDLE  ALT=\"+sqrt%286%29-sqrt%282%29+=++sqrt%288-4%2Asqrt%283%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute and multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288-4%2Asqrt%283%29%29+=+sqrt%286%29-sqrt%282%29+ ALIGN=MIDDLE  ALT=\"sqrt%288-4%2Asqrt%283%29%29+=+sqrt%286%29-sqrt%282%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation.\r<BR>\n" );
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document.write( "We essentially have a nested radical on the left side. In general, we have an equation in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%2Bb%2Asqrt%28c%29%29=sqrt%28d%29%2Bsqrt%28e%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%2Bb%2Asqrt%28c%29%29=sqrt%28d%29%2Bsqrt%28e%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The goal from here is to square both sides, gather like terms, and equate rational and irrational parts. So let's do these next tasks. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288-4%2Asqrt%283%29%29+=+sqrt%286%29-sqrt%282%29+ ALIGN=MIDDLE  ALT=\"sqrt%288-4%2Asqrt%283%29%29+=+sqrt%286%29-sqrt%282%29+\"   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=8-4%2Asqrt%283%29+=+%28sqrt%286%29-sqrt%282%29%29%5E2+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+%28sqrt%286%29-sqrt%282%29%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-4%2Asqrt%283%29+=+%28sqrt%286%29%29%5E2-2%2Asqrt%286%29%2Asqrt%282%29%2B%28sqrt%282%29%29%5E2+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-4%2Asqrt%283%29+=+6-2%2Asqrt%286%29%2Asqrt%282%29%2B%28sqrt%282%29%29%5E2+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+6-2%2Asqrt%286%29%2Asqrt%282%29%2B%28sqrt%282%29%29%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get 6.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-4%2Asqrt%283%29+=+6-2%2Asqrt%286%29%2Asqrt%282%29%2B2+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+6-2%2Asqrt%286%29%2Asqrt%282%29%2B2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Square <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;\" > to get 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-4%2Asqrt%283%29+=+8-2%2Asqrt%286%29%2Asqrt%282%29+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+8-2%2Asqrt%286%29%2Asqrt%282%29+\"   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=8-4%2Asqrt%283%29+=+8-2%2Asqrt%286%2A2%29+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+8-2%2Asqrt%286%2A2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the roots on the right side using the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29%2Asqrt%28y%29=sqrt%28x%2Ay%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29%2Asqrt%28y%29=sqrt%28x%2Ay%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=8-4%2Asqrt%283%29+=+8-2%2Asqrt%2812%29+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+8-2%2Asqrt%2812%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-4%2Asqrt%283%29+=+8-2%2Asqrt%284%2A3%29+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+8-2%2Asqrt%284%2A3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Factor 12 into 4*3 (take note that 4 is a perfect square)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-4%2Asqrt%283%29+=+8-2%2Asqrt%284%29%2Asqrt%283%29+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+8-2%2Asqrt%284%29%2Asqrt%283%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the root using the identity given above.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-4%2Asqrt%283%29+=+8-2%2A2%2Asqrt%283%29+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+8-2%2A2%2Asqrt%283%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the square root of 4 to get 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-4%2Asqrt%283%29+=+8-4%2Asqrt%283%29+ ALIGN=MIDDLE  ALT=\"8-4%2Asqrt%283%29+=+8-4%2Asqrt%283%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "From here, we can see that the two numbers are equal since the rational parts are equal and the irrational parts are equal. So this confirms that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28sqrt%286%29-sqrt%282%29%29%2F4+=++sqrt%282-sqrt%283%29%29%2F2+ ALIGN=MIDDLE  ALT=\"+%28sqrt%286%29-sqrt%282%29%29%2F4+=++sqrt%282-sqrt%283%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: Ideally, you'll only manipulate one side to transform it into the other. However, this is much more difficult to do with nested radicals. I find that algebraic manipulations suffice in this case. As a final check, you can use a calculator to find that both sides approximate to the value 0.258819045 (which helps verify the answer). </SPAN>\n" );
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