A quantity which does not equal zero is said to be nonzero.
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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)
Any nonzero number raised to the power of zero is equal to one (1).By definition.
Yes.
Yes, always.
It is always irrational.