Turn a power of $2$ into a multiple of $7$, maybe adding a $2$ but not a $7$ On St. Patrick's Day, the Lucky Charms leprechaun wagered me a bottle of Glenfiddich I couldn't solve this math problem before midnight:

There is a power of $2$ that can be turned into a multiple of $7$ with a simple rotation of a representation. Furthermore, that same power of $2$ can be turned into that same multiple of $7$ by adding a $+2$ to another representation. And if you really want to use a $7$ to get there, that can also be done, but you need a couple more things besides the $+2$ to accomplish the transformation.

I immediately said $1024$, since $2401 = 7^4$, but he said that's wrong, because that requires a rotation and a swap. "Think easier, lad," he said.
Midnight came and went and I didn't figure it out. But I still want to know the answer. I know it's something so easy I will feel stupid once the answer is revealed to me. The answer doesn't require heavy duty number crunching. I've been looking at the powers of $2$ up to $2^{64}$ but I can't figure it out. This riddle has me stumped.

It's still possible to make $1024$ work. Do $4102 = 7 \times 586$. Now, $1024 = 4^{9 - (3 + 1)}$ and $4102 = 14 \times 293$. What I did there was rearrange the digits and move a $2$ in there, and replace the various operators with a single $\times$ sign. I doubt this would've won you the Glenfiddich, however.
