No.
Sqrt (2) is irrational.
Square it, or raise it to any even power, and it becomes rational.
The set is not closed under exponentiation.
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Hennd
No. It is not even closed. sqrt(3)*sqrt(3) = 3 - which is rational.
It cannot be closed under the four basic operations (addition, subtraction, multiplication, division) because it is indeed possible to come up with two negative irrational numbers such that their sum/difference/product/quotient is a rational number, indicating that the set is not closed. You will have to think of a different operation.
The set of even numbers is closed under addition, the set of odd numbers is not.
The numbers are not closed under addition because whole numbers, even integers, and natural numbers are closed.