Question 173273
{{{(2+sqrt(5))/(4+3*sqrt(5))}}} Start with the given expression.



{{{((2+sqrt(5))/(4+3*sqrt(5)))((4-3*sqrt(5))/(4-3*sqrt(5)))}}} Multiply the fraction by {{{(4-3*sqrt(5))/(4-3*sqrt(5))}}} (which is the conjugate of the denominator)



{{{((2+sqrt(5))(4-3*sqrt(5)))/((4+3*sqrt(5))(4-3*sqrt(5)))}}} Combine the fractions.



{{{((2)(4)+(2)(-3*sqrt(5))+(sqrt(5))(4)+(sqrt(5))(-3*sqrt(5)))/((4+3*sqrt(5))(4-3*sqrt(5)))}}} FOIL the numerator.



{{{((2)(4)+(2)(-3*sqrt(5))+(sqrt(5))(4)+(sqrt(5))(-3*sqrt(5)))/((4)(4)+(4)(-3*sqrt(5))+(3*sqrt(5))(4)+(3*sqrt(5))(-3*sqrt(5)))}}} FOIL the denominator.



{{{(8-6*sqrt(5)+4*sqrt(5)-3(sqrt(5))^2)/(16-3*sqrt(5)+3*sqrt(5)-9(sqrt(5))^2)}}} Multiply.



{{{(8-6*sqrt(5)+4*sqrt(5)-3(5))/(16-3*sqrt(5)+3*sqrt(5)-9(5))}}} Square {{{sqrt(5)}}} to get 5.



{{{(8-6*sqrt(5)+4*sqrt(5)-15)/(16-3*sqrt(5)+3*sqrt(5)-45)}}} Multiply.



{{{(-7-2*sqrt(5))/(-29)}}} Combine like terms.



{{{(7+2*sqrt(5))/(29)}}} Reduce




So {{{(2+sqrt(5))/(4+3*sqrt(5))=(7+2*sqrt(5))/(29)}}}.



So you are correct.