If an iPod touch 5th Gen is completely powered off, can apps still run down the battery?

Hello,

I own an iPod touch 5th generaion that is less than 2 months old, it worked great until about a week ago then the battery started depleting rapidly, usually fully within a 12 hour period even if I did not use the unit.

I started trying to figure out if there is anyting I cold do and learned a lot about apps left open draining the battery, etc. but making sure they were all closed did not help.

I then restored the unit to factory default, but this does not help.

The last thing I tried was to turn the unit completed off at night using the power switch by holding it until the unit turns off.

My question is, once I do this can any app use any power at all?

The reason I ask is because I had spoken with 3 Apple technicians and all 3 of them do not understand or can answer the question. They keep asking me if I know how to look in the taskbar to turn off the apps which once the power is off is not an available option.

I will greatly appreciate your response.

Mike.