Question 132680
{{{(2)/(sqrt(2)+sqrt(3))}}} Start with the given expression



{{{((2)/(sqrt(2)+sqrt(3)))((sqrt(2)-sqrt(3))/(sqrt(2)-sqrt(3)))}}} Multiply by the fraction by {{{(sqrt(2)-sqrt(3))/(sqrt(2)-sqrt(3))}}}. Note  {{{sqrt(2)-sqrt(3)}}} is the conjugate of {{{sqrt(2)+sqrt(3)}}}.



{{{((2)(sqrt(2)-sqrt(3)))/((sqrt(2)+sqrt(3))(sqrt(2)-sqrt(3)))}}} Combine the fractions



{{{((2)(sqrt(2)-sqrt(3)))/(sqrt(2)*sqrt(2)+sqrt(2)*sqrt(3)-sqrt(3)*sqrt(2)+sqrt(3)*sqrt(3))}}} Foil the denominator



{{{((2)(sqrt(2)-sqrt(3)))/(sqrt(2)*sqrt(2)+sqrt(3)*sqrt(3))}}} Cancel like terms



{{{((2)(sqrt(2)-sqrt(3)))/(2-3)}}} Multiply



{{{((2)(sqrt(2)-sqrt(3)))/(-1)}}} Combine like terms



{{{-2(sqrt(2)-sqrt(3))}}} Divide and simplify



{{{-2sqrt(2)+2sqrt(3)}}} Distribute



So {{{(2)/(sqrt(2)+sqrt(3))}}} simplifies to {{{-2sqrt(2)+2sqrt(3)}}}




In other words, {{{(2)/(sqrt(2)+sqrt(3))=-2sqrt(2)+2sqrt(3)}}}