Question 191291
{{{(sqrt(3)-sqrt(4))/(sqrt(3)+sqrt(4))}}} Start with the given expression.



{{{((sqrt(3)-sqrt(4))(sqrt(3)-sqrt(4)))/((sqrt(3)+sqrt(4))(sqrt(3)-sqrt(4)))}}} Multiply both the numerator and denominator by {{{sqrt(3)-sqrt(4)}}}



Note: {{{sqrt(3)-sqrt(4)}}} is the conjugate of the denominator {{{sqrt(3)+sqrt(4)}}}. Multiplying these two expressions yields a rational expression



{{{((sqrt(3)-sqrt(4))(sqrt(3)-sqrt(4)))/((sqrt(3))^2-(sqrt(4))^2)}}} FOIL the denominator (use the difference of squares formula)




{{{((sqrt(3))^2-2*sqrt(4)*sqrt(3)+(sqrt(4))^2)/((sqrt(3))^2-(sqrt(4))^2)}}} FOIL the numerator (use the perfect square formula)



{{{(3-2*sqrt(4)*sqrt(3)+4)/(3-4)}}} Square each term



{{{(3-2*sqrt(12)+4)/(3-4)}}} Combine and multiply the roots.



{{{(7-2*sqrt(12))/(-1)}}} Combine like terms.



{{{(7-2*2*sqrt(3))/(-1)}}} Simplify {{{sqrt(12)}}} to get {{{2*sqrt(3)}}}



{{{(7-4*sqrt(3))/(-1)}}} Multiply



{{{-7+4*sqrt(3)}}} Reduce



So {{{(sqrt(3)-sqrt(4))/(sqrt(3)+sqrt(4))=-7+4*sqrt(3)}}}