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Since 12V Dc is appox 10% of 110VAC, the current draw rating of a 110V appliance will be approx 10 times that at 12V. Your 15amp hair dryer will therefore draw 150 amps at 12VDC. Divide by the fraction of an hour of expected usage (15 minutes) equals 25% of 150 amps or approx. 37 amps.
If you have a 100 amphr rated battery, you just used 37% of it. You can do the math to extrapolate to whatever you really have for batteries and assuming you understand the concept of not drawing down batteries beyond 50% capacity, can figure out what size bank you would need just for the hair dryer.
This can get more complex but the about is a simplistic description.
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