In spherical coordinates, we have
$x = \rho \sin \phi \cos \theta,$
$y = \rho \sin \phi \sin \theta,$
$z = \rho \cos \phi.$
We are given that the rectangular coordinates are $(-5, -7, 4)$ and the spherical coordinates are $(\rho, \theta, \phi)$.
So we have
$-5 = \rho \sin \phi \cos \theta,$
$-7 = \rho \sin \phi \sin \theta,$
$4 = \rho \cos \phi.$
To find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi)$, we can use the same equations but with $-\phi$ instead of $\phi$.
So we have
$x' = \rho \sin (-\phi) \cos \theta = -\rho \sin \phi \cos \theta,$
$y' = \rho \sin (-\phi) \sin \theta = -\rho \sin \phi \sin \theta,$
$z' = \rho \cos (-\phi) = \rho \cos \phi.$
Therefore, the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi)$ are $(-\rho \sin \phi \cos \theta, -\rho \sin \phi \sin \theta, \rho \cos \phi)$.
Substituting in the given values, we have $(-\rho \sin \phi \cos \theta, -\rho \sin \phi \sin \theta, \rho \cos \phi) = (-(-5), -(-7), 4) = (5, 7, 4)$.
So the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi)$ are $\boxed{(5, 7, 4)}$.The answer is: (5,7,4)
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Given a point with rectangular coordinates $(-5, -7, 4)$ and spherical coordinates $(\rho, \theta, \phi)$, determine the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, -\phi)$.

### Answer:
