# Why does n to the power of 0 = 1?

There are numerous ways to show that n0 = 1 where n is not equal to 0 (for 0 to the power of 0, see my article which can be found on the homepage). I will present a list of them. I know this is gives some people confusion because something to the power of another is sometimes defined as multiplying a number of times but this does not help with numbers like 23.2 which has a result, or 90.5 = √9 = 3. I would say that when the exponent is not a number such as 3, 5, 2 etc. we are extending the definition of repeated multiplications.
Here are some ways to show that n0 = 1:

## 1 × repeated multiplications

There is one way I know where you can still have repeated multiplications. Any number will work with this, I have just decided to pick the number 9:
93 = 9 × 9 × 9 which is three multiplications. So far so good.
I can multiply by 1 without changing the result:
93 = 1 ×9 × 9 × 9 now I am going to reduce the exponent by one:

93 = 1×9 × 9 × 9 (three nines)
92 = 1×9 × 9 (two nines)
91 = 1×9 (one nine)
90 = 1 (zero nines)

## A pattern in a table

You can see a pattern if I pick a number say, 2 and you can see the result keeps halving:
2n = Result
23 8
22 4
21 2
20 1
2-1 0.5

Try it with any number (exept 0 to the power of 0) and you will see the same pattern

## Subtracting exponents

A number divided by itself is equal to 1, whatever it looks like:
 34 = 1 but we know that na = na-b 34 nb

therefore
 34 = 34-4 = 30 = 1 34
If you are experimenting and you want to replace the number 3 with a negative number, put brackets around it e.g. (-4)5 otherwise the computer or calculator will raise the number to the power of another, then make the result negative

## Multiplying with 1

What number would I put in to this to satisfy the equation?
54 × ? = 54

well multiplying by 1 will satisfy it:
54 × 1 = 54

No problem. What happens if I replace it with something that is equivalent to 1?
54 × 50 = 54

Again, adding the indices will show you that I haven't broken any mathematical laws:
54 × 50 = 54+0 = 54