Consider $x^e \equiv a\pmod n$, given $n$, $a$, and $e>2$, with $n$ being a composite integer and unknown $x$.
Can a hypothetical function $f(a)=x$, an $eth$ root extractor, be used / adapted to factor $n$, in the same way that the multiplicative order of $x \space modulo\space n$ can be used to factor $n$ like in the classical part of Shor's algorithm?
The RSA problem, which you describe, is not known to be equivalent to factoring and there is evidence both ways. In [BV] it is shown that this barrier might be inherent: using a black-box separation technique called meta-reductions, they show that certain restricted class of reductions are not possible. On the other hand, it was shown later in [AM] that in the generic ring model (see [JS]), these problems are equivalent. That is, any speed-up in breaking RSA has to exploit the representation of $\mathbb{Z}_N^*$.
You can read about more related works in §1.3 in [AM].
[BV]: Boneh and Venkateshan, Breaking RSA may not be equivalent to factoring, Eurocrypt'98
[AM]: Aggarwal and Maurer, Breaking RSA Generically Is Equivalent to Factoring, Eurocrypt'09
[JS]: Jager and Schwenk, On the Analysis of Cryptographic Assumptions in theGeneric Ring Model, Asiacrypt'09
