Does the Nikon 16-35mm work on a DX format camera?

I own a D7100 that came with a Nikkor 18-140mm lens.

Yes. You can step down from FX to DX, but you get vignetting when you try to use an FX lens on a DX camera.

Of course it will. However, before rushing out to buy it I urge you to consider the Sigma 18-35mm f/1.8 DC HSM which is currently the fastest zoom lens available.

At B&H the Nikon 16-35mm f/4 zoom is shown for $1,256.95 while the Sigma 18-35mm f/1.8 is shown for $799.00.

Note that the Sigma f/1.8 is 2-1/3 stops faster than the Nikon f/4 and when shooting indoors that's a MAJOR advantage.

Suppose you had to use ISO 1600 @ f/4 to get a 1/250 sec. Shutter speed. With the Sigma @ f/1.8 an ISO of 320 will give you the same 1/250 sec. Shutter speed.

NOTE: Unfortunately, retiredPhil is incorrect about using a DX format lens on a Nikon FX (full-frame) format camera body. The camera simply crops the FX sized sensor to the same size as a DX sensor. This is a MAJOR advantage Nikon gives its users over Canon. With Canon, their crop-sensor EF-S lenses are totally useless on their full-frame bodies (6D, 5D, 1D).

Yes it will work but you do realize that 18-35mm only adds 2mm at the wide end. 18-35mm is already within your 18-140mm lens. You could simply take some steps backward with your existing lens and approximately get the same result.