Exponentiation is not repeated multiplication

This explanation of exponentiation avoids defining it as repeated multiplication:

Let me introduce the Expand-o-tron 3000.

Yes, this device looks like 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 growth to the desired change after one second (2x, 3x, 10.3x)
  • Set the time to 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:

\displaystyle{original \cdot growth^{duration} = new}

or

 

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