Why is anything raised to the 0 power 1

This is an excellent question! There are lots of different ways to think about it, but here's one: let's go back and think about what a power means. So when we raise a number to the zeroth power, that means we multiply the number by itself zero times - but that means we're not multiplying anything at all!

What does that mean? Well, let's go even farther back to the simplest case: addition. What happens when we add no numbers at all? Well, we'd expect to get zero, because we're not adding anything at all.

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. Math 8th grade Numbers and operations Exponents with negative bases. Exponents with negative bases. Practice: Exponents with integer bases. Practice: Exponents with negative fractional bases. Sign of expressions challenge problems. Practice: Signs of expressions challenge. Next lesson. Current timeTotal duration Google Classroom Facebook Twitter. Video transcript What I want to do in this video is think about exponents in a slightly different way that will be useful for different contexts and also go through a lot more examples.

So in the last video, we saw that taking something to an exponent means multiplying that number that many times. So if I had the number negative 2 and I want to raise it to the third power, this literally means taking three negative 2's, so negative 2, negative 2, and negative 2, and then multiplying them.

So what's this going to be? Well, let's see. Negative 2 times negative 2 is positive 4, and then positive 4 times negative 2 is negative 8. So this would be equal to negative 8. Now, another way of thinking about exponents, instead of saying you're just taking three negative 2's and multiplying them, and this is a completely reasonable way of viewing it, you could also view it as this is a number of times you're going to multiply this number times 1.

So you could completely view this as being equal to-- so you're going to start with a 1, and you're going to multiply 1 times negative 2 three times. So this is times negative 2 times negative 2 times negative 2. So clearly these are the same number. Here we just took this, and we're just multiplying it by 1, so you're still going to get negative 8.

And this might be a slightly more useful idea to get an intuition for exponents, especially when you start taking things to the 1 or 0 power. So because of these problems, zero to zeroth power is usually said to be indeterminate. However, if zero to zeroth power needs to be defined to have some value, 1 is the most logical definition for its value. This can be "handy" if you need some result to work in all cases such as the binomial theorem.

See also What is 0 to the 0 power? How is that proved? What is the difference between power and the exponent? Varthan The exponent is the little elevated number. In this case, 3 is the exponent, and 2 3 the entire expression is a power. Math Lessons menu.

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