3⁰ = 1. So does 7⁰, and 100⁰. It seems strange that a power of zero gives 1, but there's a simple reason.
Divide a number by itself
3³ is 27. Divide it by itself: 27 ÷ 27 = 1.
Now write the same sum with powers
27 ÷ 27 is the same as 3³ ÷ 3³. When you divide powers of the same number, you take away the powers: 3 − 3 = 0. So the answer is 3⁰.
It's the same sum, so it must have the same answer. That means 3⁰ has to be 1.
You can see it in the pattern too
81, 27, 9, 3: that's 3⁴, 3³, 3², 3¹. Each time the power goes down by one, you divide by 3. So the next one is 3 ÷ 3 = 1. That's 3⁰.
Any number except zero
This works for any number except zero, because both methods divide by the number, and you can't divide by zero.
So what's 0⁰?
Show the answer
It's debated! Most of maths, and your calculator, says 1. Some mathematicians leave it undefined, because the "power 0 gives 1" pattern says 1 while the "0 to any power is 0" pattern says 0. GCSE won't ask you.
Save this for your next test
- A number divided by itself is 1.
- Dividing powers means subtracting them.
- So a³ ÷ a³ = a⁰ = 1.
- It works for every number except 0.
Read the video transcript
- Have you ever wondered why 3 to the power 0 is 1?
- In fact, any number to the power 0 is 1. Here's why.
- 3 to the power 3 is 27.
- Divide it by itself: 27 divided by 27 is 1.
- Now write that same sum with powers: 3 to the 3, divided by 3 to the 3.
- When you divide powers, you take away the powers. 3 minus 3 is 0.
- So the answer is 3 to the power 0.
- Same sum, same answer. So 3 to the power 0 has to be 1.
- You can see it in the pattern too. 81, 27, 9, 3. Each power down, you divide by 3.
- So the next one is 1. That's 3 to the power 0.
- It works for any number but zero. Hold that thought.
- So, what's 0 to the power 0? Tell me in the comments.
- Want maths explained like this? Book a free trial lesson at bloomeducationonline.com.
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