Final answer:
To prove or disprove that if a and b are rational numbers, then aᵇ is also rational, we consider both cases. If aᵇ is rational, then a is rational. If aᵇ is irrational, it does not necessarily mean that a is irrational.
Step-by-step explanation:
To prove or disprove that if a and b are rational numbers, then aᵇ is also rational, we need to consider both cases.
Case 1: aᵇ is rational:
If we assume that aᵇ is rational, then there exist two integers p and q (where q is not equal to 0) such that aᵇ = p/q. We can rewrite this equation as a = (p/q)^(1/b). Since p/q and 1/b are rational numbers, raising p/q to the power of 1/b will still result in a rational number. Therefore, in this case, a is also rational.
Case 2: aᵇ is irrational:
If we assume that aᵇ is irrational, then by definition, it cannot be expressed as a ratio of two integers. However, this does not necessarily mean that a is irrational. There might be cases where a is a rational number even if aᵇ is irrational. Therefore, we cannot conclude that a is irrational.
Based on these two cases, we can conclude that if a and b are rational numbers, then aᵇ can be rational or irrational. It depends on the specific values of a and b.