questions on rationalization

in most of these questions, the square of the rational part is greater than the square of the irrational part by unity.
in such a case if x = a + √b, then 1/x = a - √b
now use the formula , X² + 1/X² = (X + 1/X)² – 2

 you will notice that the answer is simply = 4a² – 2

 

eg:

X = 6 + √35

1/X = 6 - √35

X² + 1/X² = 142