In addition to this, OP seems to assume that an "optimal choice" is always an option. In reality, it can be really really hard to make a good AI, so the chess AI you're playing against might actually be "doing its best", because whoever made it couldn't make it any better.

A fission reactor is actively smashing the fuel to pieces, in order to release the energy faster. There are RTGs that only use the normal decay as its energy source, but they produce far less energy.

Yes, and that's my point.

2838393^(72829) is 72,828 multiplications resulting in a number with 469,965 digits. A 64-bit processor can't directly work with numbers larger than 20 digits, which means it'll have to perform multiple operations for a single multiplication.

All in all, it's not enough that modern computers are really fast, if it just tried to brute-force the calculation it'd still take a lot of time.

There are often shortcuts that can be used to make the problem easier. For example the fact that:

>(x^(a))^(b) = x^(ab)

Using this knowledge you can break down the exponent in your example, by finding its factors:

>67 * 1087 = 72829

1087 is still a bit large, so we can use another trick:

>x^(a) * x^(b) = x^(a+b)

And for example break it down to:

>500 + 587 = 1087

Using all this we can then perform the calculation:

>(2838393^(500) * 2838393^(587))^(67)

Which is a lot less daunting. It could probably use other rules to break it down further, but this is the general concept.