11. Prove that for any prime positive integer

Solution:

Consider

   

as a rational number

Assume

   

 where a and b are integers and

   

By squaring on both sides

   

   

                                                        

p and b are integers

   

will also be an integer

But we know that

   

is a rational number so our supposition is wrong

Therefore,

   

is an irrational number.