Why is exponent 0 always 1

The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Why is everything except 0 to the power of 0 always 1? Asked 3 years, 9 months ago. Active 3 years, 8 months ago. Viewed times.

SilentMath 2, 8 8 silver badges 25 25 bronze badges. Mecke M. Mecke 21 2 2 bronze badges. But zero is a very special number in addition: it's called the additive identity, because it's the only number which you can add to any other number and leave the other number the same.

So, by this reasoning, it makes sense that if adding no numbers at all gives back the additive identity, multiplying no numbers at all should give the multiplicative identity. Now, what's the multiplicative identity?

Well, it's the only number which can be multiplied by any other number without changing that other number. So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1.

It's exciting to me that you asked this question. The fact is these rules are presented as somewhat arbitrary, but there is always, always well almost always good reason for them.

Keep it up! If it ever sounds arbitrary then hound your teacher. If your teacher can't give you compelling reasons why something is true, hound us or hound Google. Okay, enough, onto your question:. Mathematics was initially developed to describe relationships between everyday quantities generally whole numbers so the best way to think about powers like a b 'a' raised to the 'b' power is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'.

For example, 2 3 is 8. There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2: 1,1,1 1,1,2 1,2,1 2,1,1 2,1,2 2,2,2 1,2,2 2,2,1. So what does 3 0 represent? It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them!

How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table? Just one I know this sounds a little fishy since we started with a rule I could have just made up which is why I gave the other reason first , but these formulas are all consistent and there is never any magic step, I promise!

One rule for exponents is that exponents add when you have the same base. Since 1 is the multiplication identity, the product is never changed by multiplying by 1. When we multiply numbers, we can start with a 1 identity and then multiply it by the rest of the numbers, just like when we add or subtract numbers, we can start with 0 identity , and then move to the right addition or left subtraction on the number line.

Now consider that an exponent power is merely the number of times you need to multiply the base the five ; and don't forget to also multiply by one 1.

Now, move the 5 in the denominator to the numerator. Now, you have a product of a base raised to different powers.