Today, I wanted to do something that I haven't done in a while. I wanted to prove something, but with no patterns involved. Just take a question and prove the answer.
What I want to prove is that an irrational number to an irrational power can be rational. Like, could you raise π to a power and get a whole number, or a simple fraction?
π is a very hard number to work with since there is not a clean way of deriving it (click here to see how you can derive it), so let's take a different irrational number and use that for the example. Say, the square root of 2. We proved it irrational in November of 2011 (click here for that), so we can use that for this example.
Let's take (√2)^(√2). What does that equal?
Most people, myself included, would just say that they don't know. That is a good answer. Let's stick to two possibilities.
1. It is a rational number (it can be written as a fraction, so it is a terminating or repeating decimal)
2. It is an irrational number (it cannot be written as a fraction)
If it is a rational number, we are done. We proved that an irrational number to an irrational power is rational. That would be easy!
What do we do if it's irrational? Since we know it's irrational, we can use it as the irrational number being raised to the irrational power. Let's just see what happens if we use [(√2)^(√2)] as the base and (√2) as the exponent.
[(√2)^(√2)]^(√2)
The law of exponents says that (a^b)^c = a^bc, so we can use that to our advantage.
[(√2)^(√2)]^(√2)
(√2)^(√2)(√2)
(√2)^2
2
So, we ended up with a rational number. This means that if it were irrational, we have proven the original statement true as well. For either of the two possibilities, we have a proof.
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