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# Exponent properties with quotients

Video transcript

Let's do some exponent examples that involve division. Let's say I were to ask you
what 5 to the sixth power divided by 5 to the
second power is? Well, we can just go to the
basic definition of what an exponent represents and say 5
to the sixth power, that's going to be 5 times 5 times
5 times 5 times 5-- one more 5-- times 5. 5 times itself six times. And 5 squared, that's just 5
times itself two times, so it's just going to
be 5 times 5. Well, we know how to simplify
a fraction or a rational expression like this. We can divide the numerator and
the denominator by one 5, and then these will cancel out,
and then we can do it by another 5, or this 5 and
this 5 will cancel out. And what are we going
to be left with? 5 times 5 times 5 times 5 over
1, or you could say that this is just 5 to the fourth power. Now, notice what happens. Essentially we started with six
in the numerator, six 5's multiplied by themselves in
the numerator, and then we subtracted out. We were able to cancel out
the 2 in the denominator. So this really was equal to 5
to the sixth power minus 2. So we were able to subtract
the exponent in the denominator from the exponent
in the numerator. Let's remember how this relates
to multiplication. If I had 5 to the-- let me do
this in a different color. 5 to the sixth times 5 to the
second power, we saw in the last video that this is equal
to 5 to the 6 plus-- I'm trying to make it color coded
for you-- 6 plus 2 power. Now, we see a new property. And in the next video, we're
going see that these aren't really different properties. They're really kind of same
sides of the same coin when we learn about negative
exponents. But now in this video, we just
saw that 5 to the sixth power divided by 5 to the second
power-- let me do it in a different color-- is going to
be equal to 5 to the-- it's time consuming to make it color
coded for you-- 6 minus 2 power or 5 to the
fourth power. Here it's going to be
5 to the eighth. So when you multiply exponents
with the same base, you add the exponents. When you divide with the same
base, you subtract the denominator exponent from
the numerator exponent. Let's do a bunch more of these
examples right here. What is 6 to the seventh
power divided by 6 to the third power? Well, once again, we can
just use this property. This going to be 6 to the 7
minus 3 power, which is equal to 6 to the fourth power. And you can multiply it out this
way like we did in the first problem and verify that
it indeed will be 6 to the fourth power. Now let's try something
interesting. This will be a good segue
into the next video. Let's say we have 3 to the
fourth power divided by 3 to the tenth power. Well, if we just go from basic
principles, this would be 3 times 3 times 3 times 3, all of
that over 3 times 3-- we're going to have ten of these--
3 times 3 times 3 times 3 times 3 times 3. How many is that? One, two, three, four, five,
six, seven, eight, nine, ten. Well, if we do what we did in
the last video, this 3 cancels with that 3. Those 3's cancel. Those 3's cancel. Those 3's cancel. And we're left with 1 over--
one, two, three, four, five, six 3's. So 1 over 3 to the sixth
power, right? We have 1 over all of
these 3's down here. But that property that I just
told you, would have told you that this should also be equal
to 3 to the 4 minus 10 power. Well. What's 4 minus 10? Well, you're going to get
a negative number. This is 3 to the negative
sixth power. So using the property we just
saw, you'd get 3 to the negative sixth power. Just multiplying them out,
you get 1 over 3 to the sixth power. And the fun part about
all of this is these are the same quantity. So now you're learning a little
bit about what it means to take a negative exponent. 3 to the negative sixth power
is equal to 1 over 3 to the sixth power. And I'm going do many, many more
examples of this in the next video. But if you take anything to the
negative power, so a to the negative b power is equal
to 1 over a to the b. That's one thing that we just
established just now. And earlier in this video, we
saw that if I have a to the b over a to the c, that this is
equal to a to the b minus c. That's the other property
we've been using. Now, using what we've just
learned and what we learned in the last video, let's do some
more complicated problems. Let's say I have a to the third,
b to the fourth power over a squared b, and all of
that to the third power. Well, we can use the property
we just learned to simplify the inside. This is going to be equal
to-- a to the third divided by a squared. That's a to the 3 minus
2 power, right? So this would simplify
to just an a. And you could imagine, this
is a times a times a divided by a times a. You'll just have an a on top. And then the b, b to the fourth
divided by b, well, that's just going to be
b to the third, right? This is b to the first power. 4 minus 1 is 3, and then all of
that in parentheses to the third power. We don't want to forget about
this third power out here. This third power is this one. Let me color code it. That third power is that one
right there, and then this a in orange is that
a right there. I think we understand
what maps to what. And now we can use the property
that when we multiply something and take it to the
third power, this is equal to a to the third power times
b to the third to the third power. And then this is going to be
equal to a to the third power. times b to the 3 times 3 power,
times b to the ninth. And we would have simplified
this about as far as you can go. Let's do one more of these. I think they're good practice
and super-valuable experience later on. Let's say I have 25xy to
the sixth over 20y to the fifth x squared. So once again, we can rearrange
the numerators and the denominators. So this you could rewrite as
25 over 20 times x over x squared, right? We could have made this bottom
20x squared y to the fifth-- it doesn't matter the order we
do it in-- times y to the sixth over y to the fifth. And let's use our newly learned
exponent properties in actually just simplify
fractions. 25 over 20, if you divide
them both by 5, this is equal to 5 over 4. x divided by x squared-- well,
there's two ways you could think about it. That you could view as
x to the negative 1. You have a first power here. 1 minus 2 is negative 1. So this right here is equal to
x to the negative 1 power. Or it could also be
equal to 1 over x. These are equivalent. So let's say that this
is equal into 1 over x, just like that. And it would be. x
over x times x. One of those sets of x's would
cancel out and you're just left with 1 over x. And then finally, y to the sixth
over y to the fifth, that's y to the 6 minus 5 power,
which is just y to the first power, or just
y, so times y. So if you want to write it all
out as just one combined rational expression, you have 5
times 1 times y, which would be 5y, all of that over
4 times x, right? This is y over 1, so 4 times x
times 1, all of that over 4x, and we have successfully
simplified it.