Question 152697
{{{(sqrt(7) - sqrt(5)) / (sqrt(5) + 3*sqrt(7))}}} Start with the given expression.




{{{((sqrt(7) - sqrt(5)) / (sqrt(5) + 3*sqrt(7)))((sqrt(5) - 3*sqrt(7))/(sqrt(5) - 3*sqrt(7)))}}} Multiply both numerator and denominator by {{{sqrt(5) - 3*sqrt(7)}}}



{{{((sqrt(7) - sqrt(5))(sqrt(5) - 3*sqrt(7))) / ((sqrt(5) + 3*sqrt(7))(sqrt(5) - 3*sqrt(7)))}}} Combine the fractions




{{{(sqrt(7)sqrt(5)- 3*sqrt(7)sqrt(7) - sqrt(5)sqrt(5)+3*sqrt(5)sqrt(7)) / (sqrt(5)sqrt(5)-3*sqrt(5)sqrt(7)+3*sqrt(5)sqrt(7)-3*3*sqrt(7)sqrt(7))}}} FOIL




{{{(sqrt(35)- 3*7 - 5+3*sqrt(35)) / (5-3*sqrt(35)+3*sqrt(35)-9*7)}}} Combine and multiply the roots.




{{{(-26+4*sqrt(35)) / (-58)}}} Combine like terms



{{{(13-2*sqrt(35)) / (29)}}} Reduce



So {{{(sqrt(7) - sqrt(5)) / (sqrt(5) + 3*sqrt(7))=(13-2*sqrt(35)) / (29)}}}