This explanation of exponentiation avoids defining it as repeated multiplication:

Let me introduce the Expand-o-tron 3000.

Yes, this device

lookslike a shoddy microwave — but instead of heating food, it grows numbers. Put a number in and a new one comes out. Here’s how:

- Start with 1.0
- Set the
growthto the desired change after one second (2x, 3x, 10.3x)- Set the
timeto the number of seconds- Push the button

And shazam! The bell rings and we pull out our shiny new number. Suppose we want to change 1.0 into 9:

- Put 1.0 in the expand-o-tron
- Set the change for “3x” growth, and the time for 2 seconds
- Push the button

The number starts transforming as soon as we begin: We see 1.0, 1.1, 1.2… and just as finish the first second, we’re at 3.0. But it keeps going: 3.1, 3.5, 4.0, 6.0, 7.5. As just as we finish the 2nd second we’re at 9.0. Behold our shiny new number!

Mathematically, the expand-o-tron (exponent function) does this:

or

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