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Q: Is the product of a rational number and irrational number always rational?

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The product of a rational and irrational number can be rational if the rational is 0. Otherwise it is always irrational.

Such a product is always irrational - unless the rational number happens to be zero.

No. 0 is a rational number and the product of 0 and any irrational number will be 0, a rational. Otherwise, though, the product will always be irrational.

It is always irrational.

Provided that the rational number is not 0, the product is irrational.

Not if the rational number is zero. In all other cases, the product is irrational.

Actually the product of a nonzero rational number and another rational number will always be rational.The product of a nonzero rational number and an IRrational number will always be irrational. (You have to include the "nonzero" caveat because zero times an irrational number is zero, which is rational)

The product of 0 and an irrational is 0 (a rational), the product of a non-zero rational and any irrational is always irrational.

No, but the only exception is if the rational number is zero.

It is always rational.

The product of an irrational number and a rational number, both nonzero, is always irrational

Yes, always.

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