Final answer:
To prove or disprove the statement, we can use a proof by contradiction. Assume that x² is irrational, but x³ is rational. By taking the cube root of both sides of the equation, we find that x is rational, which contradicts our assumption. Therefore, if x² is irrational, then x³ is also irrational.
Step-by-step explanation:
To prove or disprove that if ² is irrational, then x³ is irrational, we can use a proof by contradiction. Let's assume that x2 is irrational, but x³ is rational. That means there exist two integers a and b (where b is not equal to 0), such that x³ = a/b.
If we take the cube root of both sides of the equation, we get x = (a/b)1/3.
Since the cube root of a rational number is also rational, we can rewrite (a/b)1/3 as (m/n)1/3 where m and n are integers. So now we have x = (m/n)1/3.
But this implies that x is rational, which contradicts our assumption that ² is irrational. Therefore, our assumption is false, and we can conclude that if x² is irrational, then x³ is also irrational.