Question

Show that the following numbers are irrational.

`6 + sqrt(2)`

Answer 1

Let us assume that `6+sqrt2 ` is rational. Then, there exist positive co-primes a and b such that

`6+sqrt2=a/b`

`sqrt2=a/b-6`

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

Here we see that `sqrt2` is a rational number which is a contradiction as we know that `sqrt2` is an irrational number

Hence `6+sqrt2` is irrational.