SOLUTION: Prove that 7+3root2 is not a rational number.

Let assume that  is a rational number.   (1)

Then   = , where m and n are integer numbers.

Then  =  =  is the rational number.

But {{sqrt(2)}} is an irrational number  (well known fact.
                                          For the proof see the lesson Proving irrationality of some real numbers in this site). 

Contradiction.

You got the contradiction.

Hence, the original assumption (1) is wrong.

It implies that    is an irrational number.

The proof is completed.