Question 1209233
<pre>

{{{1/(sqrt(2) + 2 + sqrt(3)) + 1/(sqrt(2) - 2 - sqrt(3))}}}

We combine the fractions over the LCD, using {{{A/B+C/D=(AD+BC)/(BD)}}}

{{{ (( sqrt(2) - 2 - sqrt(3)^"" )+ ( sqrt(2) + 2^"" + sqrt(3) ))/((sqrt(2) + 2^"" + sqrt(3))(sqrt(2) - 2^"" - sqrt(3)))}}}

{{{ ( sqrt(2) - 2 - sqrt(3)^"" +  sqrt(2) + 2^"" + sqrt(3) )/(((sqrt(2) + 2^"") + sqrt(3)^"")((sqrt(2) - 2^"") - sqrt(3)^""))}}}

{{{ (2sqrt(2) - 2 - sqrt(3)^""  + 2^"" + sqrt(3) )/(((sqrt(2) + 2^"") + sqrt(3)^"")((sqrt(2) - 2^"") - sqrt(3)^""))}}}

Combine the terms on top, and FOIL out the bottom:

{{{2sqrt(2)^""/( (2-4^"")-(sqrt(6)+2sqrt(3))+(sqrt(6)-2sqrt(3)^"")-3   )}}}

{{{2sqrt(2)^""/( 2-4^""-sqrt(6)-2sqrt(3)+sqrt(6)-2sqrt(3)^""-3   )}}}

{{{2sqrt(2)^""/(-5-4sqrt(3))}}}

Rationalize the denominator:

{{{2sqrt(2)^""/(-5^""-4sqrt(3))}}}{{{""*""}}}{{{(-5+4sqrt(3)^"")/(-5^""+4sqrt(3))}}}

{{{(-10sqrt(2)+8sqrt(6))/(25-16*3)}}}

{{{(-10sqrt(2)+8sqrt(6))/(25-48)}}}

{{{(-10sqrt(2)+8sqrt(6))/(-23)}}}

Multiply top and bottom by -1

{{{(10sqrt(2)-8sqrt(6))/(23)}}}

Edwin</pre>