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I recently bought a new table saw. The nameplate says it uses 8 amps at 240 volts.  I was planning on running it on my 2500 watt generator. Will is work, or will I damage my generator?

Well, based on the numbers, your saw will draw just under 2000 watts when it is in operation. However, it will draw several times more than that for a split second when it is initially turned on. This is what has the greatest danger of burning out your generator and also the table saw motor, due to current starvation on power-up.
You should check with the motor manufacturer to find out what the surge or inrush current rating is. Multiply the inrush current by 240 volts, and then add 25% to that number. That will give you the wattage rating your generator needs to be able to provide (at 240 volts). One other note: many generators today list two ratings- the normal wattage output, and the maximum surge wattage. This surge wattage is designed to help when starting motors. If your generator has a surge rating listed on it, then it may be up to the task of running your saw.

I want to install an outside outlet about 150 feet away from my garage sub panel in order to operate a 1500 watt water trough heater. The panel currently is full with breakers in every position. Can I run this span with 14/2 wire? And does it have to go to the panel, or can I feed it from the closest outlet?

For that kind of run with a 1500 watt load on it, you may want to consider running the circuit in 12/2 to allow for voltage loss. Since the Code would define the trough heater as a continuously operating appliance, the power requirements need to be upsized to 125% of the rated load. As such, it should be on a dedicated 20 amp circuit. You may be able to use a tandem circuit breaker in the garage panel to accommodate the new circuit.

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