(a)This can be proved by showing a contradiction. If p is a rational number and q is an irrational number and suppose , the product of p and q is another rational number r. Then,

pq=r

therefore q= r/p

Since both r and p are rational numbers r/p will be a rational number. r/p cannot be an irrational number

But q is an irrational number which can not be a rational number also.

Hence, it is a contradiction. The assumption that product of p and q is rational is not valid. This means that the product of a rational and a irrational number must be irrational.

(b) if m>n, then m² > n², both m and n both being either positive or negative integers. Hence n² cannot be > m³.

However, if m is a negative integer, then irrespective of n being < or > m, n² > m³.

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