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does higher Ah on a battery mean longer battery life? Answered

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In general, yes, assuming the same voltage. Ah is "amp hours" -- the product of how much current the battery can produce over how long a time period. Since watts are volts times amps, this is effectively a measurement of how much work you can get out of the battery. It ignores a bunch of issues such as exactly what the discharge curve is and what the battery's internal resistance might be, but it's a good rough indication.