I have the following problem:
Prove that if ab is irrational for any arbitrary positive integer b, then a
I was thinking that if a was already an irrational number like the square root of 2. and b was equal to 3 then I would have the cubed root of 22 which is :
Is this sufficient enough to prove a statement? I don't need to use any type of laws. Just need to prove if it's true/false.
I don't think you are understanding the key here. You can say it, but you haven't PROVEN it. What you need to suppose, is that, ALL YOU KNOW about the number a is that a^b is irrational for any natural number b. Based on this and based on ONLY THIS AND NOTHING ELSE, you need to establish that a must be irrational. You can't simply point out that "it would make sense" if your intended conclusion is the case, you need to show that it is the only possible case. For example, lets say someone asked you to prove that your friend Mike always told the truth. It is insufficient to point out an example of him telling the truth.
Now I think there might be a poor choice of words in your assigned problem or lost somewhere in communication. Your original post as literally interpreted is trivial. If a^b is irrational for EVERY natural b, then a^1 is irrational, and thus a is irrational. Q.E.D.
What I really think someone was trying to say, is that assume all you know is that a^b is irrational for SOME natural number b. Perhaps it's only one natural number or perhaps many or all, but all you know is that it's true for some b. The way to see this is to see what would happen if it WEREN'T the case that a is irrational. If a is rational, then any a^b must be rational, because it is simply a rational number multiplied by itself a finite number of times. This violates what you KNOW about a, specifically that a^b is irrational or some b. Therefore, it must be false that a is rational, hence a is irrational.
Your claim to prove is: For all real a, if there exists a positive integer b such that a^b is irrational, then a is irrational. Any proof must start with, "Fix an arbitrary real a". Then you need to prove an implication, so it is natural to assume the premise: "Assume there exists a positive integer b such that a^b is irrational". Then you need to prove that a is not rational. Negative statements are proved by contradiction, i.e., you say, "Assume that a is rational". Then see Hartlw's post.