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

I’d think about it like this: Imagine 2 expand-o-trons going at the same time, side by side:

* The first takes x = 3 seconds to turn 1 into a^3
* The second takes x = 3 seconds to turn 1 into b^3

Ok. Now, when we write (a^3) * (b^3) I think “I want to apply 2 separate growth processes, one after another”.

What if we wanted to apply them at the same time? What would the equivalent be? Well, every second you’d have to grow by a AND b, i.e a*b. (Growing by 2x and 5x simultaneously would be growing by 10x).

So, the result is (ab)^3 => each second you grow “ab” which takes each individual growth rate into account. Hope this helps!

Do you have any insightful ways of thinking about why (a^x).(b^x)=(ab)^x ? I can see how to extend from natural numbers etc but can’t think of anything that makes sense on a deeper level. I thought about picturing the whole surface z=x^y (which I think is helpful for example for linking the generally rather separated school topics of indices and surds) but seemed to get caught up in rather a lot of dimensions if trying then to take a product between z=x^y (for a^x) and z1=x1^y1 (for b^x). So basically confusion… Hope what I’m trying to say makes sense…

Thank you (and thank you also for all the fantastic material on this site!)

Tom

original * growth^duration = new

so if we have 0^0 we get

original * 0^0 = new
original * 1 = new
original = new

I.e., we didn’t change at all. The scale factor is multiplied into the original amount, so 0^0 = 1 is another way of saying “we didn’t change at all”.

For something like 3^2, we write

original * 3^2 = new
original * 9 = new

which is to say, our new value will be 9x our original one. Normally we don’t think about the original amount and write “3^2 = 9″ but then it becomes harder to see why 3^0 = 1. So in my head, I even see 3^2 as “1 * 3^2″, i.e. we have a starting point and begin to change it.