Question

When is the product of the square root of a prime number and a nonzero rational number a rational number?

Answer 1

The square root of a prime number is always an irrational number, then the product of the square root of a prime number and a nonzero rational number is equivalent to:

"the product of an irrational number and a rational number different than zero...."

We know that the product of an irrational number and a rational number (different than zero) is ALWAYS an irrational number.

Then "When is the product of the square root of a prime number and a nonzero rational number a rational number?"

Never, the product will always be irrational.