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### Course: Arithmetic > Unit 17

Lesson 5: Intro to exponents# The zeroth power

Sal Khan considers two different ways to think about why a number raised to the zero power equals one: 1) if 2^3 = 1x2x2x2, then 2^0 = 1 times zero twos, which equals 1. 2) By following a pattern of decreasing an exponent by one by dividing by the base, we find that when we get to the 0 power, we end up dividing the base by itself, resulting in 1.

## Want to join the conversation?

- I watched this video and it explained a lot because I didn't understand this before, but at the end of the video I realized that if you used the first method, that 0 to the power of 0 = 1. But 0 to the power of 1 = 0! Can someone please explain why that is?(48 votes)
- 0 to the 0th power is debated within the mathematical community. Some may say that it is 1, some may say that it is 0. Most people would just agree that it is indeterminate.(55 votes)

- I know that n/0 is undefined , because you can add 0 as many as times you want but you can not get n . This means that division by zero is not defined as written in my textbook . But what if we have 0 / 0 . Isn't it possible. Because you can add zero as many as times and you will always get zero . So here(in 0 /0) we are getting an answer which can be any number. So in this division by zero is defined and it is correct . And answer to this division problem is possible.

Am I right?

Please tell(18 votes)- Good question! The fact that 0/0 can be any number does not make 0/0 defined, because it is not a definite number. The best thing to call 0/0 is indeterminate (which means inconclusive).

If you study calculus later, you will frequently encounter indeterminate situations in limit problems.(14 votes)

- I know he said at4:23to "ponder Zero to the Zeroth power", but I am curious. What other science is behind it? I have been out of 6th grade for 4 years now, but I never really got an answer when I asked around. Not even Google is really helpful. Which theory is more supported? Personal opinions? Any answer is appreciated.(10 votes)
- Most people say that it is indeterminate. You can get 0 or 1 if you use different solutions, but 0 is not equal to 1, so that is why it is indeterminate.(12 votes)

- how do you times decimals with exponets(11 votes)
- If you mean having a decimal value being applied an exponent, the best way to solve for that is my following method: pretend the base value was a whole number, find the whole number value, then move the decimal "invisible decimal point" at the very end towards the left by the number of times, as if you were working with a negative exponent on scientific notation. For example, let's say you have *(1.2)^2* [1.2 to the second power]. You first take the 2nd power of 12, which equals 144. After the value (without any decimal places) was found, you then move the decimal two places toward the left so you get 1.44; this works because
**one-tenth**is basically**1/10**. However, due to the exponent, we have two "**one-tenths**" being multiplied this time; 12 squared gives us 12x12 and (0.1) squared gives us (0.1)x(0.1)=0.01

On the other hand, if you are talking about*an exponent that is a decimal value (not a whole number)*, what is called "radicals" or "roots" are likely needed. Change the exponent into an improper fraction (if possible) to make things easier to read. Then move the value of the denominator as the base of a root, while keeping the numerator a whole number exponent. For example, let's say you have 9^(2.5) [This reads "nine to the power of 2 point 5"] But the final value is in fact 243, not very difficult to work with and only has three digits. Why is that? First, let us see what happens when we change the decimal into a mixed number, then convert it into its equivalent improper fraction.

2.5 = 2 and 1/2; it's equivalent improper fraction is 5/2. You may not have learned of inverses, negative exponents, or even square roots by 6th grade, but the way things work is this:

Let's consider the numerator as a whole number itself, then look at the "**proper fraction**" version we will now get. Before we multiplied by a 5, we had 1/2. Its inverse is a multiplying of a 2. So taking the 2nd power would mean we had 9 squared; but because the value is the inverse, we now have to take the square root of the 9, which equals 3.

Afterward, you take the 3 and apply the 5th power to it, which means you have 3x3x3x3x3 and that equals 243.

P.S. (sorry about making a post so late; but even so I hope this explains a little about how decimal values and exponents work). Things may not always work out so easily as I have shown here (sometimes giving what is called irrational values). But I hope these explanations and examples are simple enough that the individual sections and steps may be understood to some extent.(1 vote)

- What is zero to the zeroth power?(7 votes)
- Look at previous responses, the number one voted gives a good answer.(8 votes)

- are there exponents like -1(5 votes)
- Yes, there are negative exponents, since it is hard to explain in words (for me) I'm just going to show an example,

5^3=125

5^2=25

5^1=5

5^0=1

5^-1= 1/5 = 0.2

5^-2= .2/5 = 0.04(12 votes)

- Since 10^0, is 1. Does this mean it is 10^1? If I am multiplying 10^0 x 10^5, we add the exponents but am I adding zero, or one? If 10^0 is equal to 10^1, then the product should be 10^6. Is this right?(7 votes)
- Yes, 10^0 = 1, but this is not the same as 10^1, which would equal 10.

So 10^0 x 10^5 = 10^5, not 10^6.(6 votes)

- hello people i have a question what's -1^0?(7 votes)
- When evaluating the expression -1^0, it is important to consider the order of operations. In mathematics, exponentiation takes precedence over negation. Therefore, the exponentiation is performed before applying the negative sign.

Applying the exponent of 0 to -1 gives us:

-1^0 = (-1)^0

Any non-zero number raised to the power of 0 is defined as 1. Therefore:

(-1)^0 = 1

So, the value of -1^0 is 1, not -1.(6 votes)

- This is a very good question! Consider the following two rules:

1) Zero to any positive power is 0.

2) Any nonzero number to the zero power is 1.

If we try to extend both rules to define 0^0, we get different answers. It is unclear if 0^0 should be 0, 1, or something else. Because of this, it is best to call 0^0 indeterminate (though 0^0 is often interpreted as 1).(9 votes) - i watched the video and i used the both ways but why when i searched on google i got 1 and why do we add 1 in the first way?(3 votes)
- Good question. Here's what I found on Google: "In short, the multiplicative identity is the number 1, because for any other number x, 1*x = x. 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." Hope this helped! -Johnny Unidas(5 votes)

## Video transcript

- If we think about something
like two to the third power, we could view this as taking three twos and multiplying them together,
so two times two times two, or equivalently, we could
say this is the same thing as taking a one and then
multiplying it by two three times, so actually, let's just
go with this definition right over here, and this, of course, is going to be equal to eight. Now what would, based on
this definition I just did, what would two to the second power be? Well, this would be one times two twice, so one times two times two, which, of course, would be equal to four. What would two to the first power be? Well, that would be one, and we would multiply it by one two, one times two, which, of course, is equal to two. Now let's ask ourself
an interesting question. Based on this definition
of what an exponent is, what would two to the zeroth power be? I encourage you to just think
about that a little bit. If you were the mathematics community, how would you define two to the zero power so it is consistent with
everything that we just saw. Well, the way we just talked
about it, we just said exponentiation is you start
with a one and you multiply it by the base zero times, so
we're not gonna multiply it by any two, so we're just
gonna be left with a one. So does this make sense that two to the zero power is equal to one? Well, let's think about it another way, and let's do a different base. That was with two, but
let's say we have three and I could say three to
the fourth power, that's three times three times three times three which is going to be equal to 81, and let me just write down that this is going to be equal to 81. If I said three to the third power, that's three times three
times three which is 27. Three to the second
power is equal to nine. Three to the first
power is equal to three. Do you notice a pattern every time we decrease the exponent here by one? We want three to the fourth, and now we go three to the third. What happened to the value? Well, going from 81 to 27,
we divided by three, and that makes sense,
because we're multiplying by one less three, so we divide by three to go from 81 to 27, we
divide by three again if our exponent goes down by one, and we divide by three again
when we go from nine to three, divide by three, so based
on this, what do you think three to the zero power should be? Well, the pattern is every time we decrease our exponent by
one, we divide by the base, and so we should divide by three again would be the logic if
we follow that pattern, and so three divided by
three would get us one again. So I know it might seem a little
bit counter intuitive that something to the zeroth power
is going to be equal to one, but this is how the mathematics
community has defined it because it actually makes a lot of sense. Either if you view an
exponent as taking a one and multiplying it by the base
the exponent number of times, so I'm gonna multiply
one by two three times, or if you just follow
this pattern, every time you decrease the exponent by one, you're going to be dividing by the base. Either of those would get
you to the conclusion that two to the zero power is one,
or three to the zero power is one, or frankly, any number
to the zero power is one. So if I have any number,
let's say I have some number a to the zero power, this is
going to be equal to one. Now I have an interesting
question for you, and let's just say this is the case when a does not equal zero. I'll leave you a little bit of a puzzle for you to think about. What do you think is
zero to the zeroth power? What should zero to the zero power be? And what's interesting
about zero to the zero power is you'll get a different
answer if you use this technique versus if you use this
technique right over here. This technique would actually
get you to being one, while this technique would
have you divide by zero, which we don't know how to do. Anyway, I'll leave you there to ponder the mysteries of zero to the zeroth power.