Question 393898
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{{{ (x)/(4+ sqrt(2)) - (-1+ 3sqrt(2)) }}}

{{{ (x/(4+ sqrt(2))) - (-1+ 3sqrt(2)) }}}

Write the second term over 1

{{{ (x/(4+ sqrt(2))) - ((-1+ 3sqrt(2))/1) }}}

Get an LCD of {{{4+sqrt(2)}}}

{{{ (x/(4+ sqrt(2))) - ((-1+ 3sqrt(2))/1)((4+sqrt(2))/(4+sqrt(2))) }}}

{{{ (x/(4+ sqrt(2))) - ((-4-sqrt(2)+12sqrt(2)+3*2)/(4+sqrt(2))) }}}

{{{ (x/(4+ sqrt(2))) - ((-4+11sqrt(2)+6)/(4+sqrt(2))) }}}

{{{ (x/(4+ sqrt(2))) - ((2+11sqrt(2))/(4+sqrt(2))) }}}

{{{ (x-(2+11sqrt(2)))/(4+ sqrt(2)) }}}

{{{ (x-2-11sqrt(2))/(4+ sqrt(2)) }}}

Now rationalize the denominator:

{{{ ((x-2-11sqrt(2))/(4+ sqrt(2)))((4-sqrt(2))/(4-sqrt(2))) }}}

Group the {{{(x-2)}}} so you can use FOIL on top as well as the bottom:

{{{ (((x-2)-11sqrt(2))/(4+ sqrt(2)))((4-sqrt(2))/(4-sqrt(2))) }}}

{{{((x-2)*4-(x-2)sqrt(2) - 44sqrt(2) + 11*2)/(16-4sqrt(2)+4sqrt(2)-2)}}}

{{{(4x - 8 - x*sqrt(2) + 2sqrt(2) - 44sqrt(2) + 22)/(16-2)}}}

{{{(4x-x*sqrt(2)+14-42sqrt(2))/14}}}

You can leave it like the above or maybe

{{{(4x+14-x*sqrt(2)-42sqrt(2))/14}}}

and/or then maybe minimize the number of square roots 

{{{(4x+14-sqrt(2)(x+42))/14}}}

Edwin</pre>