1/(a+√2)+1/√2=√3+2/√3+√5.
Let x=1/(a+√2), so:
x+1/√2=x+√2/2=√3+2/√3+√5=√3+2√3/3+√5=5√3/3+√5=(5√3+3√5)/3.
x=(5√3+3√5)/3-√2/2=(10√3-3√2+6√5)/6.
a+√2=6/(10√3-3√2+6√5), a=6/(10√3-3√2+6√5)-√2.
The calculation can be taken further, but the result looks more complicated.