For this kind of a question you should multiply the nominator and the denominator with the demonator but with the minus sign. (forgive me for my bad english :). What i mean is, this is what you should do;
((sqrt7)-(sqrt2))/((sqrt7)+(sqrt2))(√7)−(√2)(√7)+(√2) since there is ((sqrt7)+(sqrt2))((√7)+(√2)) in the denomintor you should multiply the whole equation with ((sqrt7)-(sqrt2))((√7)−(√2))
The reason we do this to reach this result in the denominator part;
((sqrt7)+(sqrt2))((√7)+(√2)) . ((sqrt7)-(sqrt2))((√7)−(√2)) = 7 - 2 7−2 = 55
I came to that from this formula;
(x+y) . (x-y) = x^2 - y^2 (x+y).(x−y)=x2−y2 => ⇒ This is a formula that should be memorized in order to solve this kind of question.
Also you have to know this too;
(x-y)^2 = x^2 - 2xy + y^2 (x−y)2=x2−2xy+y2
So if i solve the entire question the result will be;
((sqrt7)-(sqrt2))/((sqrt7)+(sqrt2))(√7)−(√2)(√7)+(√2) = ((sqrt7)-(sqrt2))/((sqrt7)+(sqrt2)) * ((sqrt7)-(sqrt2))/((sqrt7)-(sqrt2))(√7)−(√2)(√7)+(√2)⋅(√7)−(√2)(√7)−(√2) = ((sqrt7)-(sqrt2))^2/(7-2)((√7)−(√2))27−2 = (7-2sqrt14 + 2)/ 57−2√14+25 = (9- 2sqrt14)/59−2√145 =>⇒ this is the simplest version i can write. I hope it helps.
