Ever wondered roughly how big a number like, say, 264 is? Here a quick rule of thumb:
Take the exponent, divide by 10 and multiply by 3. That is the number of zeroes after the first digits.
E.g. with 264, the exponent is 64. And 64/10×3 is 19.2. So 264 is roughly some digit followed by 19 zeros. Actually it is more like 1.844×1019 – which is pretty good for rule of thumb.
Why is this so? Well, 210 is 1024, and 103 is 1000. That’s only 2.4% off, which is more than acceptable for a rule of thumb.
Unfortunately, while the offset initially starts out quite small, it will compound with interests interest. For 220 it is off by almost 4.9%, which is slightly more than double the error. At 21000 it is off by more than a factor of 10, ie. an extra digit (but then, we are talking about roughly 1.072×10301, and who will notice the extra zero at the end, when there already are 300 of them…? 😉 ).
If you want to be a bit more precise, you can add an extra digit for each thousands of exponent of two, though this will of course also fail to account for the compound interest accrued as the exponent increases. So you might just as well stick to the original rule of thumb.