Exponents of Negative Numbers

This is an add-on in the series The Order of Operations Explained.
For the other articles in this series, click here to visit the introduction.

Evaluating exponents of negative numbers, or raising a negative number to a power, can get weird. And it has to do with the order of operations.

Exponents come way before subtraction (PEMDAS). So something like has to take this into consideration.

Even though from left to right it seems the negative is on the 3, it’s really the tiny 2 that’s supposed to come first.

All that’s fine and dandy. But how about a nicer way to think about it?

Exponents are copying instructions. Parenthesis are paperclips.

Imagine you’re heading to the copy center with two pieces of paper – one paper has only a “-” and the other a number (like 3).

If you have the pages clipped together with the stickie note reading “Please make 2 copies,” it looks like this:

If you have the pages separate and the stickie is only on the page with the 3, it looks like this:

What will you get from the copy center?

Now think of what the copier-dude will do with your first set. He’s going to make two copies of the whole thing. He’ll likely give you two stapled sets.

On the second set, he’ll make two copies of the “3” page and put the “-” page to the side. He’ll think, “Hmm… I wonder what I’m supposed to do with this? Oh, well. I’ll just put it back on the stack after I’m done.”

But there’s an even better way.

If we would put parenthesis where we mean things to happen first, we could avoid all this.

Using and would solve this whole issue, right?

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