The ohms of that the speaker is rated at is produced from the lm386 amp IC on the board, as long as that IC has sufficient power (4v -12v) from the circuit it will output correctly. The main issue is missed here, what you really need to know is how many amps/hour (or milliamps/hour 'mAH' 1/1000 of an amp/hour) the device in question uses or draws.
An AA battery is 1.5 volts DC and can push about <font color="red">1120 mAH
A 9V battery is 9 volts DC and can push about <font color="red">565 mAH
Now link 6 AA batteries + to - (in series) and you get 9 volts with 1120 mAH of push.
That would be like this <font color="red">+[ ]-~+[ ]-~+[ ]-~+[ ]-~+[ ]-~+[ ]<font color="red">- you connect the device at the red ends.
The black Rat Shack amp uses the AA batteries in the 9v/1120 mah (in series) configuration, so you could sub in a 9v battery but expect it to burn out in about half the time or less depending on the pull of the device. Although they both output 9v the 9v battery has about half the push/hour (565 mAH) behind it compared to the series of AA batteries (1120 mAH)
Confused yet 
Also remember Rat Shack had two amps a white one and a black one. White uses 9v by default and the black used 6 AA by default, the circuit in both is VERY simular.
