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.