Not necessarily. The cube roots of 4, 6 and 9 are all irrational (and different). But their product is 6, not just rational, but an integer.
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Any irrational number does.
You can't get an irrational number by applying a finite number of additions, subtractions, multiplications and divisions to a set of rational numbers.
Yes, unless the rational number is 0.
The real numbers are divided into rational numbers and irrational numbers.
The product will also be irrational.