Let us take an irrational number ‘k’ and a non-zero rational number ‘’ where (a,b≠ 0).
Assume that the product of an irrational number and non-zero rational number is an rational number.
where c/d is another rational number.
We can say that (bc) and (ad) are also integers with (ad ≠0).
So, bc/ad is a fraction with integer values in the numerator and denominator (denominator not zero) making it a rational number.
This is a contradiction to the fact that ‘k’ as an irrational number.
So, the assumption is wrong.
Therefore, the product of any irrational number and non-zero rational number is an irrational number.
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