Q: Is the product of a nonzero rational number and an irrational number rational or irrational?

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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)

Yes.

Yes, always.

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

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

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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)

It is irrational.

The product will be irrational.

It is an irrational number.

Yes.

Yes, always.

Yes, always.

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

An irrational number.

Suppose a is rational (and non-zero) and x is irrational. Suppose ax is rational;write ax = b where b is rational.Then x = b/a, and x would be rational, contradiction.

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.