Myriora Maths › Maths foundations › Chapter 7: Order of operations and powers

Power rules for the curious

Ages 9–11Chapter 7 · Lesson 5

In this lesson

When you multiply powers of the same number, add the powers.
When you divide, take them away.
It works because you're counting how many times the number is multiplied.
And the numbers underneath must be the same.

Your turn

What is three to the power of six, divided by three to the power of four?

Show the answer

When you divide, you take away the powers. Six take away four is two. So the answer is three squared, which is nine!

The common mistake

Here's a question that catches lots of people. What is two cubed, times two to the power of four? Some people multiply the powers, and say two to the power of twelve. But count the twos: there are only seven of them!

Try it at home

Write 2 cubed × 2 squared as a row of twos on paper. Count them, then check the shortcut: add the powers.

Read the whole lesson

Hello, and welcome to Myriora Maths! This one is for the curious. What is two cubed, times two to the power of four? There's a lovely shortcut, and you can see for yourself why it works. Let's find it!

Let's write out all the twos. Two cubed is three twos, multiplied. Two to the power of four is four twos. Put them together, and that's seven twos multiplied: two to the power of seven!

So when you multiply powers of the same number, you add the powers. Three add four is seven. It works because you're just counting the twos!

Dividing works the other way round. Here's two to the power of five, divided by two squared. Write five twos on top, and two twos underneath. Each two on the bottom cancels a two on the top, leaving three twos: two cubed. So when you divide, you take away the powers!

But be careful: the shortcut only works when the numbers underneath are the same. Two cubed times three squared can't be joined up, because twos and threes are different. You just work them out: eight times nine is seventy-two.

Let's try one together. Five squared, times five cubed. What is that as one power of five? Say it out loud!

Add the powers: two add three is five. So it's five to the power of five!

Your turn! What is three to the power of six, divided by three to the power of four? Pick the right card. You've got twenty seconds, or pause the video. Go!

Time's up! When you divide, you take away the powers. Six take away four is two. So the answer is three squared, which is nine!

Quick review. When you multiply powers of the same number, add the powers. When you divide, take them away. It works because you're counting how many times the number is multiplied. And the numbers underneath must be the same. See you next time!