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(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.

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<span lang="EN-USRS Aggarwal - Mathematics