Prove that the following are irrationals:

(i) Let  is rational.
Therefore,  we can find two integers p &  q where, q ≠0  such that
is rational as p and q are integers.
Therefore,  is rational which contradicts to the fact that  is irrational.
Hence, our assumption is false and  is irrational.

(ii)Let  is rational.
Therefore, we can find two integers p & q where, q≠0  such that
 for some integers p  and q
  is rational as p and q are integers.
Therefore,  should be rational.
This contradicts the fact that  is irrational.
Therefore our assumption that  is rational is false.
Hence,  is irrational.

(iii)Let  be rational.
Therefore, we can find two integers p & q where  q≠0, such that

 
Since p and q are integers,  is also rational
 Hence,  should be rational, this contradicts the fact that  is irrational.
So, our assumption is false and hence,  is irrational.