No.[5 + sqrt(3)] + [1 - sqrt(3)] = 6 is rational
[5 + sqrt(3)] - [1 + sqrt(3)] = 4 is rational
[5 + sqrt(3)] * [5 - sqrt(3)] = 22 is rational
sqrt(8)/sqrt(2) = +/- 2 is rational.
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No. You can well multiply two irrational numbers and get a result that is not an irrational number.
No. sqrt(8) is irrational sqrt(2) is irrational but sqrt(8) /sqtr(2) = sqrt(4) = ±2 is not irrational.;
Yes.
Yes. The empty set is closed under the two operations.
No, it is not. Root2 and root 8 are each irrational. Root8 / root2 =2. 2 is not a member of the set.