nth root of n equal to the cube root of 3

[amigocabal]

There are two positive real numbers n such that its nth root is equal to the cube root of 3. One, of course, is 3. The other is approximately 2.478 to the nearest thousandth.

Here is a short proof that this other number of transcendental.

Theorem 1, If the nth root of n is equal to the cube root of 3, n is either 3 or irrational.

First, we assume that n is rational. Then there are two positive coprime integers, p and q, such that p>q>1. (As established above, the positive real numbers n whose nth root equals the cube root of 3 are 3 and a real number approximately 2.478.) Thus, the (p/q)th root of (p/q) is equal to the cube root of 3.

We cube both sides, raise both sides to the *(p/q)th power, and then take both sides to the q power. On side we have p/q raised to the 3qth power. because 3q is an integer, and p and q are coprime with q>1, the left side is a rational non-integer. The right side is 3 to the p power. p is an integer, so the right side is an integer. This is a contradiction, and therefore, n is irrational.
Theorem 2. If n does not equal 3, and the nth root of n is equal to the cube root of 3, n is transcendental

We then assume n is algebraic. We raise both sides to the nth power. The left side is n, which we assume to be algebraic. The right side is the cube root of 3 (an algebraic number not equal to zero or one) raised to the nth power. n is algebraic and irrational, thus implying through the Gelfond-Schenider theorem that the right side is transcendental. This is a contradiction, and therefore we must prove that the other positive real number n whose nth root equals the cube root of 3 must be transcendental.

[madd0ct0r]

I'm having trouble with "the (p/q)th root of (p/q) is equal to the cube root of 3."

[amigocabal]

I'm having trouble with "the (p/q)th root of (p/q) is equal to the cube root of 3."

There are two positive real numbers n such that the nth root of n equals the cube root of 3. 3, of course, is one of those numbers, and the other n is about 2.478.

I proved that the other n is irrational, first by assuming there are coprime positive integers p and q with n=p/q and p>q>1, and then arriving at a contradiction.