In a word 'yes'. Battery powered portable tools are designed to work only at their specified voltage. Most portable tools will accept only the correct battery type and voltage. 

Power = volts x amps. So, for a constant power, a 48V battery would have half the amps of a 24V battery. To double the power, double either the voltage or the amps. Doubling both power and amps quadruples the power.

That's the theory, but in practice you must use the correct voltage battery the tool is designed for! An incorrect battery either physically won't fit, or won't work.

Equally you must use the right charger for your power-tool battery.

There are some chargers which can detect the chemistry and voltage of a battery when you insert it, and adjust the charge-cycle to match.
But I'd be very wary of buying such a charger if it was manufactured by an unknown company in the Far East.

Most hand-held power tool batteries are built from a set of cylindrical lithium cells with the size format known as 18650.
The enclosure which holds them also contains a protection board which should have a temperature sensor.

The chemistry of the 18650 is LiMNC (Lithium, Manganese, Nickel, Cobalt).
It's the same as deployed in the majority of EVs and is selected for its high power to weight ratio.

However, it's also the lithium cell which is most prone to burst into flames if the charger doesn't stop when the cells are full.

That's why you are increasingly seeing videos of explosions in homes where electric bikes, scooters and skate-boards are left on charge at night.

It's not the fault of the cells, despite what news reporters keep saying.
The problem usually lies with the charger, which kept going because it failed to detect the 'battery full' condition.

By way of comparison, see the topic on How to quickly top-balance EVE 280Ah cells, where @vaugi suffered an incident which caused damage to his £2000 set of new lithium cells.

There was no fire or explosion.
Those EVE 280Ah cells use Lithium Ferrous Phosphate chemistry (LiFePO4).

Yes, they're heavier per Ah of capacity.
But that's not a problem when they're going to remain static in a house.

The more important issue is that they are inherently safe, and are therefore unlikely to cause a rise in insurance premiums.

@allyfish, I thought the higher the voltage, the more powerful the tool will be. For example, a 60V  battery will be more powerful than a 24V battery. Amps, meanwhile, measure the amount of electrical current that a battery can provide. ie. The higher the amps, the longer the tool will run before it needs to be recharged.

So a 2amp 24V and 2 amp 60V battery should run for the same amount of time in their respective tools. ie. 24V and 60V lawnmower, equipped with the same amperage batteries (for the correct voltage respectively) will run for more or less the same time.

Is this correct? 

@transparent, I was going to ask about chargers next and you've beaten me to the punch, and it is as I suspected. 

What's also really interesting is that there are gardening tools that uses two 24V batteries to power 48V motors – in instances like this, I guess you can't just one 24V battery.

Posted by: @editor

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So a 2amp 24V and 2 amp 60V battery should run for the same amount of time in their respective tools. ie. 24V and 60V lawnmower, equipped with the same amperage batteries (for the correct voltage respectively) will run for more or less the same time.

Is this correct? 

Hi Mars, generally the higher the voltage the more powerful the portable tool. But a higher voltage battery can provide a longer run time than a lower voltage one. The run time of a portable tool is a function of the battery capacity, measured in Ah, the battery & motor voltage and the power demand.

Simplified example:

A 24V 5Ah battery powering a 50W motor: P=VxI, I = 50/24 = 2.08A. A 5.0Ah battery will last 5/2.08 = 2.4hrs

A 48V 5Ah battery powering a 50W motor: P=VxI, I = 50/48 = 1.04A. A 5.0Ah battery will last 5/1.04 = 4.8hrs

Li-ion batteries tend to hold about 10% minimum charge when flat, and DC brushless motors are about 80-90% efficient, so your run time is less in reality.

As this is on an open forum, can I make a response which even more simple?!

1. Each power tool has a voltage at which it must operate.
You must match the battery to that voltage.

A 24v drill requires a 24v battery. Full stop.

2.

Posted by: @editor

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Amps, meanwhile, measure the amount of electrical current that a battery can provide.

It's best looking at it the other way around.

It is the power tool which 'decides' how many Amps it will pull from the battery.
If you're drilling through a tougher material, then it needs more current to keep the drill speed from dropping off.

The battery itself is capable of delivering much more current than the tool actually requires.
You don't need to 'measure' that.