Q: What is the product of a nonzero rational number and an irrational number?

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It is an irrational number.

Yes, always.

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.

The question cannot be answered because it is based on a false premise.The product of a (not an!) rational number and an irrational number need not be irrational. For eample, the product ofthe rational number, 0, and the irrational number, pi, is 0. The product is rational, not 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 always irrational.

It is 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.