Let a = sqrt(4/3) - sqrt(3/4)a=√43−√34
Let b = sqrt(4/3) + sqrt(3/4)b=√43+√34
ab = 4/3 - 3/4ab=43−34
ab = 7/12ab=712
a+b = 2sqrt(4/3)a+b=2√43
a+b = 4/sqrt3a+b=4√3
Now as simultaneous equations:
a(4/sqrt3 -a) = 7/12a(4√3−a)=712
a^2 - (4a)/sqrt3 + 7/12 = 0a2−4a√3+712=0
Quadratic equation:
a = (4/sqrt3 +- sqrt((4/sqrt3)^2 - 4 xx 7/12))/(2)a=4√3±√(4√3)2−4×7122
a = (4/sqrt3 +-sqrt(16/3 - 7/3))/(2)a=4√3±√163−732
a = (4/sqrt(3) +- 3/sqrt3)/2a=4√3±3√32
a = (4/sqrt(3) - 3/sqrt3)/2a=4√3−3√32
a = 1/(2sqrt3)a=12√3
Therefore sqrt(4/3) - sqrt(3/4) = 1/(2sqrt3)√43−√34=12√3
and we also get sqrt(4/3) + sqrt(3/4) = 7/(2sqrt3)√43+√34=72√3