Show that 2^1/3, 5^1/7, and 13^1/4 do not represent rational numbers.© BrainMass Inc. brainmass.com April 3, 2020, 2:55 pm ad1c9bdddf
We will prove this by contradiction.
The idea of a proof by contradiction is to:
First, we assume that the opposite of what we wish to prove is true.
Then, we show that the logical consequences of the assumption include a contradiction.
Finally, we conclude that the assumption must have been false
Recall, A number r is rational if it can be written as a fraction r = p/q where both p and q are integers.
Proof: For 2^1/3
(If 2^1/3 is rational, it should be representable as a fraction r = p/q)
Assume that a rational number r exists such that r^3 = 2. (or, r = 2^1/3)
As we have already assumed that, r is rational, in the representation r=p/q assume p and q are mutually prime, i.e. have no common divisors. The fraction p/q in this case is called irreducible. In other words, a/b is ...
It is shown that 2^1/3, 5^1/7, and 13^1/4 do not represent rational numbers.