Let's first rationalize the denominators of the equations.
1√2+√3=1√2+√3×1=1√2+√3×√2−√3√2−√3=√2−√3(√2+√3)×(√2−√3)=√2−√3√22−√32=√2−√32−3=−(√2−√3)=√3−√2
1√2−√3=1√2−√3×1=1√2−√3×√2+√3√2+√3=√2+√3(√2+√3)(√2−√3)=√2+√3√22−√32=√2+√32−3=−(√2+√3)=−√2−√3
Now we can see that the origional equation is just (sqrt(3) - sqrt(2)) + (-sqrt(2) - sqrt(3)). This simplifies to -2sqrt(2)