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### Course: Class 8 (Foundation) > Unit 5

Lesson 1: Properties of exponents# Exponent properties with products

To simplify expressions with exponents, there are a few properties that may help. One is that when two numbers with the same base are multiplied, the exponents can be added. Another is that when a number with an exponent is raised to another exponent, the exponents can be multiplied. Created by Sal Khan and CK-12 Foundation.

## Want to join the conversation?

- Anything to zero power is one? Does that include zero to the zero power? - thanks(219 votes)
- There is still debate, here is a link. http://www.math.hmc.edu/funfacts/ffiles/10005.3-5.shtml(93 votes)

- at10:41where did the -1 go ?(34 votes)
- (-1)^2 = +1

The set then becomes:

(+1 * x^4 * y^2)

Because you're multiplying by positive 1, the answer will not change. For this reason, the 1 is dropped entirely.(34 votes)

- Why is n^0 = 1? Since n^1 = n(n times itself once) and n^2 = n * n(n times itself twice), shouldn't n^0 = 0(n times itself 0, or no times)? I understand Sal's point of view on this, but I think that my explanation is right. Why doesn't mine work?(26 votes)
- You don't subtract n each time, you divide by n. Using that pattern:

n^2 = n * n *1, (divide by n) n^1 = n * 1, (divide by n) and n^0 = 1.

I had the same problem too for a while. If it still doesn't make sense, just do what I did for a long time before I understood:

Remember that mathematicians are slightly crazy. I only understand because I became near-mathematician-level crazy. This is not an insult. It means that you are one step closer to becoming as smart as Einstein (because you are harder to understand and only a few people could understand Einstein).(25 votes)

- I see in the comments that this has been covered. However, don't think the explanation is satisfactory as the answer does not fit within the rules set for ALL other exponents.

So if x to the 3rd is x*x*x, x to the 2nd is x*x, and x to the first is just x. Then why would x to the zero not be 0? What is the justification to make zero mean something other than a null or nothing?

Example, take any of the above I mentioned. I ask you to write the expression x to the 3rd. Then you would write down x three times, same with the second and first.

So if you are writing down x however many times the exponent shows and if it shows zero times, then you write down nothing.

I have found nothing that justifies the breaking of the rules used for all other exponents just for a null value.

If math is logical, then what makes a zero exponent separate rule logical. This is what I could not find. Sorry for the long version for the question, but had to share my thinking.(8 votes)- First, don't apologize for the long question. It's well thought out, cohesive, and my pleasure to answer.

We can determine the zero property of exponentiation by looking at ratios of exponents. If we had 2^5 / 2^4, we could write this as:

(2 * 2 * 2 * 2 * 2) / (2 * 2 * 2 * 2)

Using what we know about division, four 2's cancel out from the numerator and denominator, leaving us with just two. 2 is the same as 2^1, and the difference in the exponents that we started with was 1. Now, if we make the exponents the same, every 2 would cancel out, leaving us with 1, as x / x = 1.

We could look at taking an x away as dividing by x. If we write x three times (x*x*x), and then want to write x squared, it would be x*x*x/x, which simplifies to x*x. If we want to go from x^1 to x^0, we divide by x, just like before. x/x = 1, regardless of what x is.

Does this help with your question? It's not really a completely arbitrary rule made so that the rest of math would make sense, it works out that way. With the higher and higher abstractions of exponents and further, it becomes difficult to intuitively justify some of the rules surrounding them, especially with 0, as they aren't as often seen directly in nature.(20 votes)

- 0 to the 0 power is undefined

anything else to the 0 power is 1.(11 votes)*In this calculation, n will = the number that is being "powered"***Examples**

n^1 = 1*n

n^2 = 1*n*n

n^3 = 1*n*n*n

n^... = 1*n*n*n*n...

and so on...*Therefore if you were to do if n^0, it would = 1, as 1 is always at the beginning.*

Although, the proof of why 1 is there is not defined, many people think it is a helpful way of understanding exponents.

n^0 = 1

Because of this, 0^0 = 1, as the above equations show us.

In other words, what is 0^0? Answer: Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values.

Since n^0 is 1 for all numbers n other than 0, it would be logical to define that 0^0 = 1.

Although, these may be the only proven way of telling us 0^0 = 1, 0^0 is usually undefined, there are many debates about the topic of what 0^0 equals.`Link: https://www.homeschoolmath.net/teaching/zero-exponent-proof.php#:~:text=In%20other%20words%2C%20what%20is,define%20that%2000%20%3D%201.`

*Hope this helps!*(2 votes)

- Is anyone else here during quarantine school? Also, I know lots of people asked about 0^0 but I didn't quite get it. Can someone explain? -thanks(7 votes)
- Anything raised to the power of 0 is 1, that includes 0.(6 votes)

- Why do the videos have to be so long(7 votes)
- In the last video, Sal said 0^0 is undefined. The Khan Academy calculator says it is 1. Which is correct? IM confused.(7 votes)
- The debate still goes on as to whether 0^0 is undefined/indeterminate or simply 1. Many different fields of math define 0^0 as 1 but others say that it is indeterminate.(7 votes)

- Okay so this ain't a question, or I think it is? I don't know I'm going to 7th grade this year and I'm really scared and nervous so can yall give me some advice? Thank you it will help me this year, and also why is the video so so SO long? I don't get any of this math it's not hitting my brain yet, can yall help me out, please?(7 votes)
- they're gonna tear ye up bud yer not gonna make it 3 days they're gonna remove all your teeth and eat your legs(6 votes)

- what ever happend to the (-1)^2 at10:27? Wouldn't it just be 1? and if so were did it go?(7 votes)
- (-1)^2 is 1 but in this equation, multiplying by 1 will have no effect on the outcome so it can just be dropped.(2 votes)

## Video transcript

In this video, I want to do a
bunch of examples involving exponent properties. But, before I even do that,
let's have a little bit of a review of what an exponent
even is. So let's say I had 2
to the third power. You might be tempted to
say, oh is that 6? And I would say no,
it is not 6. This means 2 times itself,
three times. So this is going to be equal to
2 times 2 times 2, which is equal to 2 times 2 is 4. 4 times 2 is equal to 8. If I were to ask you what 3 to
the second power is, or 3 squared, this is equal to 3
times itself two times. This is equal to 3 times 3. Which is equal to 9. Let's do one more of these. I think you're getting the
general sense, if you've never seen these before. Let's say I have 5 to
the seventh power. That's equal to 5 times
itself, seven times. 5 times 5 times 5 times 5
times 5 times 5 times 5. That's seven, right? One, two, three, four,
five, six, seven. This is going to be a really,
really, really, really, large number and I'm not going to
calculate it right now. If you want to do it by hand,
feel free to do so. Or use a calculator, but this
is a really, really, really, large number. So one thing that you might
appreciate very quickly is that exponents increase
very rapidly. 5 to the 17th would be even a
way, way more massive number. But anyway, that's a review
of exponents. Let's get a little bit steeped
in algebra, using exponents. So what would 3x-- let me do
this in a different color-- what would 3x times
3x times 3x be? Well, one thing you need to
remember about multiplication is, it doesn't matter what
order you do the multiplication in. So this is going to be the same
thing as 3 times 3 times 3 times x times x times x. And just based on what we
reviewed just here, that part right there, 3 times 3,
three times, that's 3 to the third power. And this right here, x times
itself three times. that's x to the third power. So this whole thing can be
rewritten as 3 to the third times x to the third. Or if you know what 3 to the
third is, this is 9 times 3, which is 27. This is 27 x to the
third power. Now you might have said, hey,
wasn't 3x times 3x times 3x. Wasn't that 3x to
the third power? Right? You're multiplying 3x times
itself three times. And I would say, yes it is. So this, right here, you could
interpret that as 3x to the third power. And just like that, we stumbled
on one of our exponent properties. Notice this. When I have something times
something, and the whole thing is to the third power, that
equals each of those things to the third power times
each other. So 3x to the third is the same
thing is 3 to the third times x to the third, which is
27 to the third power. Let's do a couple
more examples. What if I were to ask you what
6 to the third times 6 to the sixth power is? And this is going to be a really
huge number, but I want to write it as a power of 6. Let me write the 6 to the sixth
in a different color. 6 to the third times 6 to the
sixth power, what is this going to be equal to? Well, 6 to the third, we
know that's 6 times itself three times. So it's 6 times 6 times 6. And then that's going to be
times-- the times here is in green, so I'll do it in green. Maybe I'll make both
of them in orange. That is going to be times
6 to the sixth power. Well, what's 6 to
the sixth power? That's 6 times itself
six times. So, it's 6 times 6 times
6 times 6 times 6. Then you get one
more, times 6. So what is this whole
number going to be? Well, this whole thing-- we're
multiplying 6 times itself-- how many times? One, two, three, four,
five, six, seven, eight, nine times, right? Three times here and then
another six times here. So we're multiplying 6 times
itself nine times. 3 plus 6. So this is equal to 6 to
the 3 plus 6 power or 6 to the ninth power. And just like that,
we/ve stumbled on another exponent property. When we take exponents, in this
case, 6 to the third, the number 6 is the base. We're taking the base to
the exponent of 3. When you have the same base,
and you're multiplying two exponents with the same base,
you can add the exponents. Let me do several more
examples of this. Let's do it in magenta. Let's say I had 2 squared times
2 to the fourth times 2 to the sixth. Well, I have the same base
in all of these, so I can add the exponents. This is going to be equal to 2
to the 2 plus 4 plus 6, which is equal to 2 to
the 12th power. And hopefully that makes sense,
because this is going to be 2 times itself two times,
2 times itself four times, 2 times itself
six times. When you multiply them all out,
it's going to be 2 times itself, 12 times or 2
to the 12th power. Let's do it in a little bit more
abstract way, using some variables, but it's the
same exact idea. What is x to the squared or x
squared times x to the fourth? Well, we could use the property
we just learned. We have the exact
same base, x. So it's going to be x to
the 2 plus 4 power. It's going to be x to
the sixth power. And if you don't believe
me, what is x squared? x squared is equal
to x times x. And if you were going to
multiply that times x to the fourth, you're multiplying it by
x times itself four times. x times x times x times x. So how many times are you now
multiplying x by itself? Well, one, two, three, four,
five, six times. x to the sixth power. Let's do another one of these. The more examples you see,
I figure, the better. So let's do the other
property, just to mix and match it. Let's say I have a to the third
to the fourth power. So I'll tell you the property
here, and I'll show you why it makes sense. When you add something to an
exponent, and then you raise that to an exponent, you can
multiply the exponents. So this is going to be a to the
3 times 4 power or a to the 12th power. And why does that make sense? Well this right here
is a to the third times itself four times. So this is equal to a to the
third times a to the third times a to the third times
a to the third. Well, we have the same base, so
we can add the exponents. So there's going to be a to
the 3 times 4, right? This is equal to a to the 3
plus 3 plus 3 plus 3 power, which is the same thing
is a the 3 times 4 power or a to the 12th power. So just to review the properties
we've learned so far in this video, besides
just a review of what an exponent is, if I have x to the
a power times x to the b power, this is going
to be equal to x to the a plus b power. We saw that right here. x squared times x to the fourth
is equal to x to the sixth, 2 plus 4. We also saw that if I have x
times y to the a power, this is the same thing is
x to the a power times y to the a power. We saw that early on
in this video. We saw that over here. 3x to the third is the same
thing as 3 to the third times x to the third. That's what this is
saying right here. 3x to the third is the same
thing is 3 to the third times x to the third. And then the last property,
which we just stumbled upon is, if you have x to the a and
then you raise that to the bth power, that's equal to
x to the a times b. And we saw that right there. a
to the third and then raise that to the fourth power is the
same thing is a to the 3 times 4 or a to the
12th power. So let's use these properties
to do a handful of more complex problems. Let's say
we have 2xy squared times negative x squared
y squared times three x squared y squared. And we wanted to
simplify this. This you can view as
negative 1 times x squared times y squared. So if we take this whole thing
to the squared power, this is like raising each of these
to the second power. So this part right here could
be simplified as negative 1 squared times x squared squared,
times y squared. And then if we were to simplify
that, negative 1 squared is just 1, x squared
squared-- remember you can just multiply the exponents-- so
that's going to be x to the fourth y squared. That's what this middle
part simplifies to. And let's see if we can merge
it with the other parts. The other parts, just to
remember, were 2 xy squared, and then 3x squared y squared. Well now we're just going ahead
and just straight up multiplying everything. And we learned in multiplication
that it doesn't matter which order you
multiply things in. So I can just rearrange. We're just going and multiplying
2 times x times y squared times x to the fourth
times y squared times 3 times x squared times y squared. So I can rearrange this, and I
will rearrange it so that it's in a way that's easy
to simplify. So I can multiply 2 times 3, and
then I can worry about the x terms. Let me do it in this color. Then I have times x times x to
the fourth times x squared. And then I have to worry about
the y terms, times y squared times another y squared times
another y squared. And now what are
these equal to? Well, 2 times 3. You knew how to do that. That's equal to 6. And what is x times x to the
fourth times x squared. Well, one thing to remember is
x is the same thing as x to the first power. Anything to the first power
is just that number. So you know, 2 to the first
power is just 2. 3 to the first power
is just 3. So what is this going
to be equal to? This is going to be equal to--
we have the same base, x. We can add the exponents, x to
the 1 plus 4 plus 2 power, and I'll add it in the next step. And then on the y's, this
is times y to the 2 plus 2 plus 2 power. And what does that give us? That gives us 6 x to the
seventh power, y to the sixth power. And I'll just leave you with
some thing that you might already know, but it's
pretty interesting. And that's the question of what
happens when you take something to the zeroth power? So if I say 7 to the zeroth
power, What does that equal? And I'll tell you right now--
and this might seem very counterintuitive-- this is equal
to 1, or 1 to the zeroth power is also equal to 1. Anything that the zeroth power,
any non-zero number to the zero power is going
to be equal to 1. And just to give you
a little bit of intuition on why that is. Think about it this way. 3 to the first power-- let me
write the powers-- 3 to the first, second, third. We'll just do it the
with the number 3. So 3 to the first power is 3. I think that makes sense. 3 to the second power is 9. 3 to the third power is 27. And of course, we're trying to
figure out what should 3 to the zeroth power be? Well, think about it. Every time you decrement
the exponent. Every time you take the exponent
down by 1, you are dividing by 3. To go from 27 to 9,
you divide by 3. To go from 9 to 3,
you divide by 3. So to go from this exponent to
that exponent, maybe we should divide by 3 again. And that's why, anything to
the zeroth power, in this case, 3 to the zeroth
power is 1. See you in the next video.