Question 116791


{{{(2+sqrt(3))/(sqrt(3)+sqrt(5))}}} Start with the given expression



{{{((2+sqrt(3))/(sqrt(3)+sqrt(5)))((sqrt(3)-sqrt(5))/(sqrt(3)-sqrt(5)))}}} Multiply by the fraction by {{{(sqrt(3)-sqrt(5))/(sqrt(3)-sqrt(5))}}}. Note  {{{sqrt(3)-sqrt(5)}}} is the conjugate of {{{sqrt(3)+sqrt(5)}}}.



{{{((2+sqrt(3))(sqrt(3)-sqrt(5)))/((sqrt(3)+sqrt(5))(sqrt(3)-sqrt(5)))}}} Combine the fractions



{{{((2+sqrt(3))(sqrt(3)-sqrt(5)))/(sqrt(3)*sqrt(3)+sqrt(3)*sqrt(5)-sqrt(5)*sqrt(3)+sqrt(5)*sqrt(5))}}} Foil the denominator



{{{((2+sqrt(3))(sqrt(3)-sqrt(5)))/(sqrt(3)*sqrt(3)+sqrt(5)*sqrt(5))}}} Cancel like terms



{{{((2+sqrt(3))(sqrt(3)-sqrt(5)))/(3-5)}}} Multiply



{{{((2+sqrt(3))(sqrt(3)-sqrt(5)))/(-2)}}} Combine like terms




{{{((2)(sqrt(3))+(2)(-sqrt(5))+(sqrt(3))(sqrt(3))+(sqrt(3))(-sqrt(5)))/(-2)}}} Foil the numerator




{{{(2sqrt(3)-2sqrt(5)+3-sqrt(3)sqrt(5))/(-2)}}} Multiply