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CCSS.Math:

And now I want to go over some
of the other core exponent properties. But they really just fall
out of what we already know about exponents. Let's say I have two
numbers, a and b. And I'm going to raise it to--
I could do it in the abstract. I could raise it to the c power. But I'll do it a little
bit more concrete. Let's raise it to
the fourth power. What is that going
to be equal to? Well that's going
to be equal to-- I could write it like this. Copy and paste this,
copy and paste. That's going to be equal to
ab times ab times ab times ab times ab. But what is that equal to? Well when you just multiply
a bunch of numbers like this it doesn't matter what order
you're going to multiply it in. This right over here is going
to be equivalent to a times a times a times
a times-- We have four b's as well that
we're multiplying together. Times b times b times b times b. And what is that equal to? Well this right over here
is a to the fourth power. And this right over here
is b to the fourth power. And so you see, if you take
the product of two numbers and you raise them
to some exponent, that's equivalent to
taking each of the numbers to that exponent. And then taking their product. And here I just used
the example with 4, but you could do this really
with any arbitrary-- actually any exponent. This property holds. And you could satisfy yourself
by trying different values, and using the same
logic right over here. But this is a general property. That-- let me
write it this way-- that if I have a to
the b, to the c power, that this is going to be equal
to a to the c times b to the c power. And we'll use this to
throughout actually mathematics, when we try to
simplify things or rewrite an expression in
a different way. Now let me introduce you
to another core idea here. And this is the idea of raising
something to some power. And I'll just use example of 3. And then raising
that to some power. What could this
be simplified as? Well let's think about it. This is the same thing
as a to the third-- let me copy and paste that--
as a to the third times a to the third. And what is a to the
third times-- So this is equal to a to the third
times a to the third. And that's going to be equal
to a to the 3 plus 3 power. We have the same
base, so we would add and they're being multiplied. They're being raised
to these two exponents. So it's going to be the
sum of the exponents, which of course is going to
be equal to a-- that's a different color a-- it's going
to be a to the sixth power. So what just happened over here? Well I took two a to the thirds. And I multiplied them together. So I took these two 3s
and added them together. So this essentially
right over here, you could view
this as 2 times 3. That's how we got the 6. When I raise something
to one exponent, and then raised it
to another, that's the equivalent to
raising the base to the product of
those two exponents. I just did it with this
example right over here. But I encourage you try other
numbers to see how this works. And I could to do
this in general. I could say a to the b power. And then-- let me
copy and paste that-- and then I'm going to
raise that to the c power. Well what is that
going to give me? Well I'm essentially
going to have to take c of these,
so one, two, three. I don't know how
large of a number c is, so I'll just
do the dot, dot dot. So dot, dot, dot. I have c of these,
right over here. So what is that
going to be equal to? Well that is going to be
equal to a to the-- well for each of these
c, I'm going to have a b that I'm going
to add together. So let me write this. So I'm going to have a b plus
b plus b plus dot, dot, dot plus b. And now I have c
of these b's, so I have c b's right over here. Or you could view
this as a, this is equal to a to
the c times b power. c or a, you could do
a to the cb power. So very useful. So if someone were to say
what is 35 to the third power, and then that raised
to the seventh power? Well this is going to
obviously be a huge number. But we could at least
simplify the expression. This is going to be equal to
35 to the product of these two exponents. It's going to be 35 to the
3 times 7, or 35 to the 21, or to the 21st power.