Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprising depth of mathematical knowledge.

Here’s a question I once received:

255/256 to what power is equal to 1/2? And please don’t use a calculator.

Here’s how I answered this question without using a calculator… in fact, I answered it without writing anything down at all. I thought of the question as

.

I was fortunate that my class chose 1/2, as I had memorized (from reading and re-reading *Surely You’re Joking, Mr. Feynman!* when I was young) that . Therefore, we have

.

Next, I used the Taylor series expansion

to reduce this to

,

or

.

For my students’ problem, I had , and so

.

So all I had left was the small matter of multiplying these two numbers. I thought of this as

.

Multiplying and in my head took a minute or two:

.

Therefore, and . Therefore, I had the answer of

.

So, after a couple minutes’ thought, I gave the answer of 177. I knew this would be close, but I had no idea it would be so close to the right answer, as

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