Given the relationship:
$$\pi = \left(\frac{15ab^2c^3}{2d^4}\right)^{\frac{1}{5}}$$
where a, b and c are non-zero positive constants, can it be assumed at least one of a, b or c must be a transcendental number, since $\pi$ is a transcendental number?
After re-arranging we get:
$$\frac{15}{2} = \frac{d^4\pi^5}{ab^2c^3} .$$
Now the expression on the right must be a rational number. Can I assume one (or 3?) of the constants is/are a transcendental number that somehow cancels out $\pi$ being a transcendental number, to produce a rational result?
Can we deduce anything else about the constants? Must at least one be an irrational number? Is it possible to determine which of the constants are transcendental numbers?
At least one must be transcendental because the products, quotients, and roots of algebraic numbers are also algebraic. Hence if $a,b,c,d$ all satisfied some respective integer polynomials then so would your expression.
But one is all that can be guarunteed. For example, take $a=\frac{2}{15}\pi ^5$ and $b=c=d=1$ as an example where only one variable is transcendental (or irrational for that matter).