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How do you show proof by contradiction That 3 to the power of x equals 2 Is Irrational?
Wiki User
∙ 7y ago
Using "^" for powers.
3^x = 2. Assume that x is a rational number, and call it p/q. So:3^(p/q) = 2
Raise both parts to the power "q", and you get:
(3^(p/q))^q = 2^q
or:
3^p = 2^q
You must now raise 3 to an integer power "p", and 2 to an integer power "q", and get the same result. This is clearly impossible, since 3 and 2 have no common factors.
Put it another way: 3 raised to any power will always be ODD, 2 raised to any power will always be EVEN. The two therefore can't be equal.
This contradicts the initial assumption, that p and q are integers, i.e., that the solution is an integer number, p/q.
Wiki User
∙ 7y ago
Wiki User
∙ 7y ago3^x = 2 is an equation. It is neither rational nor irrational.
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