Calculate magnification from image and object size.
Linear magnification is simply the ratio of image size to object size, and because magnification also equals the ratio of image distance to object distance, knowing the object distance lets you locate where the image must form. Area magnification is the square of the linear figure, which is why an 8x linear magnification spreads the same light over 64 times the area and makes high magnification progressively dimmer.
Magnification from sizes
m = image size / object size; area magnification = m^2; image distance = m x object distance
Because brightness falls with it. The same collected light spread over m squared times the area is m squared times dimmer, which is the fundamental reason high-magnification microscopy needs strong illumination and larger numerical apertures.
A compound microscope multiplies the objective and eyepiece magnifications — a 40x objective with a 10x eyepiece gives 400x. Enter the final image and original object sizes and you recover that total.
No. Beyond roughly 1000x for light microscopy you get empty magnification: the image is larger but no more detailed, because resolution is limited by the wavelength of light and the numerical aperture, not by magnification.