First we spent a week on Exponent Boot Camp, in which I review WHAT the exponent properties are, and also WHY they are, including negative exponents, rational exponents, and rational bases with positive and negative exponents. That covered the first part - evaluating expressions with all kinds of bases and exponents.
On this day, however, it was all about going the other way, developing those exponent lenses. After warming up with a few evaluation examples, I put 16 on the board and asked how can we write this number as a power? The answers I got were as expected:
Anyway, so I showed them this:
We spent a few minutes working out each of these, just to re-convince everyone that these were in fact all equal to 16.
Back to 16, and I asked again, "That's it now, right? No other possibilities?"
My students know me well enough to know that answer to that. So we moved on to rational exponents:
I got a few really great answers added here, like 65536^(1/4) and 1048576^(1/5).
Me: So that's it right?
Students: Nope, that's never it is it?
So now that we all knew that there were in fact infinitely many ways to express a number as a power, I asked everyone to write a power on the board that equals 81, specifying that you can't use one that's already there. That went really well. Definitely you should do this again next year Future Audrey.