Question

question 8
are irrational numbers closed under multiplication? justify your reasoning.
no, the product of any two irrational numbers is not always an irrational number.
yes, the product of any two irrational numbers is always an irrational number.
yes, the product of any two irrational numbers is not always an irrational number.
no, the product of any two irrational numbers is always an irrational number.

question 10
the product of a rational number and an irrational number...
is sometimes rational and sometimes irrational
is always a rational number.
is never rational or irrational.
is always an irrational number.

Explanation:

Question 9 Step1: Define closure under multiplication

A set is closed under multiplication if the product of any two elements in the set is also in the set.

Question 9 Step2: Provide counterexample

Take $\sqrt{2}$ (irrational) and $\sqrt{2}$ (irrational). Their product is $\sqrt{2} \times \sqrt{2} = 2$, which is rational. This shows the product of two irrationals is not always irrational.

Question 10 Step1: Case 1: Rational is 0

Let rational number be $0$ and irrational number be $\sqrt{3}$. The product is $0 \times \sqrt{3} = 0$, which is rational.

Question 10 Step2: Case 2: Rational is non-zero

Let rational number be $2$ and irrational number be $\sqrt{3}$. The product is $2 \times \sqrt{3} = 2\sqrt{3}$, which is irrational.

Question 10 Step3: Conclude result

Since both rational and irrational products are possible, the product is sometimes rational and sometimes irrational.

Answer:

Question 9: No, the product of any two irrational numbers is not always an irrational number.
Question 10: is sometimes rational and sometimes irrational.