simplify 1/a+root 2 + 1/root 2=root 3+ 2/root3 +root 5

Question

it is related to number system

Answer 1

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.