Written by Colin+ in ask uncle colin.

Dear Uncle Colin,

I’m ok with my basic power laws, but I don’t understand why $x^0$ is always 1, and I get mixed up when it’s a fraction or a negative power. Can you help?

Running Out Of Time

Hi, ROOT, and thanks for your message!

If it’s any consolation, you’re not alone — this is something that I struggled with when I was learning, and it’s a really common problem with my GCSE and A-level students.

My first and biggest bit of advice, when you see a power you’re unfamiliar with: **slow down and think**. If I had a list of Important Mathematical Skills, somewhere near the top would be “recognising where something is tricky” or “knowing your weaknesses”. This doesn’t just apply to awkward powers — every mathematician has different strengths and weaknesses — but the advice is good every time. If you think “this is something I’ve messed up frequently in the past, I’d better take extra care”, you will save yourself a lot of headache in the end.

Right, let’s get on to zero and negative powers. While you’re learning these, I recommend writing out a **list of powers** to remind yourself of the pattern, which turns out to be completely logical — once you look at it in exactly the right way.

Let’s work with 9 as our base — although there’s nothing special about 9^{1}, you could pick any number of variable you wanted here.

Start by writing $9^3 = 9 \times 9 \times 9 = 729$ on one line.

On the next, write $9^2 = 9 \times 9 = 81$ and notice that you’ve *divided by the base* to drop the power by 1.

On the next, write $9^1 = 9$ — and again, you’ve divided by the base to drop the power.

So, logically, what must $9^0$ be? Well, you divide by the base again and get $9^0=1$ on the next line.

And how about $9^{-1}$? Guess what, you divide by the base again. $9^{-1} = \frac{1}{9}$, and $9^{-2} = \frac{1}{81}$, and so on.

You can also see from here, using the power laws you’re happy with, that $9^{-2} \times 9^{2} = 9^0$, which is exactly the same as $\frac{1}{81} \times 81 = 1$.

Fractions are a bit harder to see, but they do still make sense: it’s pretty clear from your list of powers that $9^{\frac{1}{2}}$ is between 1 and 9, but where exactly? Again, your power laws can help here: you know that $9^{\frac{1}{2}} \times 9^{\frac{1}{2}} = 9^1$ because $\frac{1}{2} + \frac{1}{2} = 1$.

This means that $9^{\frac{1}{2}} is the number you multiply by itself to get 9 – or the square root of 9, which is 3.

Generally, the number on the bottom of a fraction in the power corresponds to a *root*^{2}.

Hope that helps!

– Uncle Colin