Chapter 4.8, Problem 5ES

Let S be the statement: The cube root of every irrational number is irrational. This statement is true, but the following “proof” is incorrect. Explain the mistake.
“Proof (by contradiction): Suppose not.
Suppose the cube root of every irrational number is rational. But $2\sqrt{2}$ is irrational because it is a product of a rational and an irrational number, and the cube root of $2\sqrt{2}$ is $\sqrt{2}$ which is irrational. This is a contradiction, and hence it is not true that the cube root of every irrational number is rational. Thus the statement to be proved is true.”