Question

(1÷1+√7)+(1÷√2+√3)+(2÷√3+√5) simplify

Answer 1

This is a rationalisation problem,

Multiply the numerator and denominator by a value that makes the denominator vanish,
$a^{2}-b^{2} =( a+b ) ( a-b )$

Thus we try to convert the denominator in the form $a^{2}-b^{2}$

Multiply the numerator and denominator of the first part  1/(1 + √7) by (1-√7) 

We get,

1-√7 / -6....(1)

Multiply the numerator and denominator of the second part  1/(√2 + √3) by (√2-√3) 

We get,

-(√2-√3)...(2)

Multiply the numerator and denominator of the third part  2/(√3 + √5) by (√3 + √5) 

We get,

-(√3 + √5)...(3)

Adding  (1),(2) & (3),

√7-1/6 - √2 +√5