Question

Prove that the following are irrational.

`6+sqrt2`

Answer 1

`6+sqrt2`

∴ We can find two co-prime integers a and b such that `6 + sqrt2 = a/b`, Where b ≠ 0

∴ `a/b - 6 = sqrt2`

or `sqrt2 = (a/b - 6)`

= `(a - 6b)/b`

From (1), `sqrt2` is a rational number, which contradicts the fact that `sqrt2` is an irrational number.

∴ Our supposition is wrong. 

= `6 sqrt2` is an irrational number.