Question

Simplify the expression 3 1/3/3 1/3. Explain how the expression and its simplified form show that the set of irrational numbers is not closed under division

Answer 1

The expression given is ambiguous according to the standard rules of math (PEMDAS) due to the lack of appropriate parentheses and probably the exponentiation symbol ^.

From the context, I assume it to mean
(3^(1/3)) / (3^(1/3)) = 1

Since 3^(1/3) is the cube-root of 3, and is thus irrational.
However, the quotient is exactly 1, which is a rational number.

The example is therefore a counter-example for the statement that "set of irrational numbers is closed under division", or stated differently,
"the set of irrational numbers is not closed under division"