I have two extension cords both are 50 feet long. One is 12 gauge and the other is 14 gauge. I want to power an electric saw.

I want the least voltage drop over the combined 100 feet so I keep the current draw by the saw motor as low as possible minimizing heat damage. Which should I plug into the 120 volt receptical first? Why?

I don't think it matters. The resistance will be the sum of the two extension cords regardless of which one comes first and the distance is still the same. But I could be wrong (it wouldn't be the first time.)

Yeah....it's still, essentially two resistors in series. Don't think it really matters which "gets it" first.

If it makes you feel better, though, I'd go with the #12 first and then hook in the #14.

Of course I betcha the wiring in the wall is #14....

Would there really be THAT much voltage drop over that much wire?

It's all going to be one series circuit whichever way you do it:

#1. Hot -> 50' #12 -> 50' #14 -> Tool -> 50' #14 -> 50' #12 -> Neutral.

#2. Hot -> 50' #14 -> 50' #12 -> Tool -> 50' #12 -> 50' #14 -> Neutral.

Either way you have 100 ft. of #14 plus 100 ft. of #12 conductor in the circuit, so the resistance of the extension cords will be the same.

Nope it does not matter.
That said, even with a 16A load the drop is only right around 4 volts.

Have you thought of installing an outlet closer?

Two resistors in series: R(total)=R1+R2. No difference with placement. If it's causing you to lose sleep, spend the $20 on another 50' #12 extension.

Think about inrush current, though. Get a 100' 10/3 cord. just a thought.