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Q: Is a rational number multiplied a rational number rational?

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Only if the rational number is 0.

Every rational number does.

Any number will be a rational number when multiplied.0 multiplied by any real number is rational and so it will produce a rational number when multiplied.If x is any non-zero number (rational or not), then since it is non-zero, 1/x is defined and x*(1/x) = 1 which is rational. So any non-zero number will produce a rational number when multiplied.Thus any number will produce a rational number when multiplied.

Every rational number does.

Yes, unless the rational number is 0.

It is the square of the rational number.

Any irrational number multiplied by 0.5 will remain irrational. Any rational number multiplied by 0.5 will remain rational.

If an irrational number is added to, (or multiplied by) a rational number, the result will always be an irrational number.

Any and every rational number.

Any already rational number

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.

rational * irrational = irrational.

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