That happened to me today with fractions as exponents.
I won't bother to mention how I used to teach it. It was bad. Very bad.
Today, I started by showing them this, and
Then I asked them HOW they knew it was x squared. Somebody pointed out that three x-squareds multiply to equal x^6. Someone else said that you divide 6 by 3.
I made a big deal about writing x^(6/3) before writing the answer as x squared. And we did several more of these including some square roots, so they could see that you divide by two.
Then I showed them this, and gave some time to think about it and write down what they thought the answer should be.
Almost every single student wrote down x^(2/3)! And there were angels singing.
Then we worked on going backwards, which was no biggie at all. Given x^(1/2), students could easily rewrite as square root of x and so on.
And then I didn't know what to do, because it used to take me a full class period to teach that. And some students would still be sitting there going, "Huh?". But today they just got it in, like, five minutes.
I realize that there is nothing earth-shattering about this method. The thing is, I've taught it this way before. Only I didn't lead with this part. I ended with this part. All I did today was change the order.
Oh, I love these moments of finding the tiniest little change that makes a huge difference.