The sum or product of a non-zero rational number and an irrational number is always

Both the sum and the product are irrational. If adding a rational number to an irrational number gave a rational result, subtracting the rational number would make the irrational number rational—a contradiction. If multiplying a nonzero rational number by an irrational number gave a rational result, dividing by the nonzero rational number would make the irrational number rational, also a contradiction. So the correct choice is “irrational.”

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