Final answer:
An irrational number raised to an irrational power can indeed be rational, as demonstrated by the example where sqrt(2) raised to the power of sqrt(2) raised to sqrt(2) equals 2.
Step-by-step explanation:
Yes, an irrational number raised to an irrational power can sometimes be rational. This is a surprising fact to many, but it can occur under special circumstances. For example, consider the irrational number sqrt(2), which, when raised to the power of sqrt(2), remains irrational. However, if we raise sqrt(2) to the power of sqrt(2) raised to sqrt(2), which is itself irrational, we end up with (sqrt(2))^(sqrt(2)^sqrt(2)) = 2, which is rational. This phenomenon can be thought of in terms of properties of exponents, wherein we can manipulate exponents raised to other exponents. In particular, we can use the property that raising a power to another power multiplies the exponents: (a^b)^c = a^(b*c).