Hazelnut Powers of ten my dude. Think of it this way: assume the numbers at 6:58 are right and there are 1.5e29 particles per human (using "e" here as short for "times ten to the power of), 3.3e80 particles in the universe, so enough particles for 2.25e51 humans. There are 7.5 billion humans now which is 7.5e9.

You can pretty much round to the nearest power of ten without affecting the answer, so let's ask how long would take to go from 1e10 (ten billion) humans to 1e52 (10 sexdecillion per https://en.wikipedia.org/wiki/Names_of_large_numbers) humans at 1% growth per year?

The direct way to solve this is with a logarithm as in 7:54, but that doesn't help the intuition - 8,604 years seems way too short a time!

Instead, to help our intuition, let's ask how many years of 1% growth does it take to multiply our population by ten -- to add one more number to the right of the "e"? This is independent of original population size. It takes as long to go from 3,000 to 30,000 humans (3e4 => 3e5) as to go from ten billion to a hundred billion (1e9 => 1e10) if you're always expanding at the same % rate.

Well, if you open a spreadsheet, start a cell at 1,000, and multiply each cell below by 1.01 (for 1% growth), you'll see it takes 231 cells to get to 10,000. So at 1% growth it takes 231 years to grow a population by 10x. That... actually feels pretty intuitive, right? Especially when you look at the numbers in the spreadsheet, or plot it in a line graph.

So now how long does it take to grow the population from 1e10 to 1e52? We have to raise the exponent by 42, that is, multiply the population by ten (increase the exponent by 1) forty-two times over. So that will take 231 years x 42 times... which is 9702 years!

(The difference between that and the 8,604 is partly because I rounded the population but mostly because the video says a 1.1% growth rate, not 1% -- a small difference in the growth rate makes centuries of difference.)

But wait, haven't humans been around for half a million years or more? Sure have, but obviously we have not growing at 1% a year. That is a very recent and totally unsustainable phenomenon.

On a side note this kind of exponential growth, on a MUCH shorter time scale, is what has a lot of smart people worried about computers undergoing an intelligence explosion or "singularity".