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

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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 must be a generalised rational number. Otherwise, if you select a rational number to multiply, then you will only prove it for that number.

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

Yes, it is possible only if an irrational number is multiplied with 0.

Negative. Only like x like = positive

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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.

The product of two rational numbers is always a rational number.

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

If the multiplicative inverse exists then, by definition, the product is 1 which is rational.

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)

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

The product will also be a rational number

another rational number

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.

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!

It is a rational number.

It is a rational number.