Pi, e, and certain other real numbers are called transcendental numbers. In some branch of mathematics, there is a theorem which proves that such numbers belong in a class by themselves.

They cannot be expressed using a finite number of simple arithmetic operations without using transcendental numbers (roots and powers allowed). They cannot be the roots of a polynomial with rational coefficients.

On the basis of the above, I believe that your search for a so called rationalizing factor is doomed.

Oddly bit of related information.

e^i*Pi = -1

Where i is SquareRoot(-1), and e is 2.71828...

BTW: I am pretty sure that numbers which can be roots of a polynomial are called algebraic numbers. The algebraic numbers are a bit stranger than you might expect. For 2nd, 3rd, & 4th order polynomials, there are direct formulae or algoithms which result in determing the roots. Except for special cases, the roots of higher order polynomials cannot be determined by any direct formulae or algorithms. You can use numerical methods to get accurate approximations, but there is no formula or expression which precisely describes the roots using a finite number of arithmetic operations. There is nothing remotely similar to the formula for the roots of a quadratic.