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Factor out the
greatest common factor. And the expression they give us
is 4x to the fourth y plus 8x to the third y. When they say to factor out the
greatest common factor they're essentially telling us, find
the greatest common factor of 4x to the fourth and
8x to the third y and factor it out
of this expression. Or kind of undistribute it. And to find that
greatest common factor-- and I always put
it in quotes when we speak in algebraic terms. Because we don't
really know what x and y are, whether
they're positive or negative or whether they're greater
than or less than 1. So it's not always going to be
the greatest absolute number. But it's the greatest
in it contains the most terms of these two expressions,
these two monomials. So if we were to essentially
factor out 4x to the fourth y it would look like this. We would do the
prime factorization of 4, which is just 2 times 2,
times x to the fourth, which is x times x times
x times x, times y. We just expanded
it out as a product of its basic constituents. Now let's do the
same thing for 8-- I'll color code it--
8x to the third. Let me do it in similar colors. So in this situation we
have 8x to the 1/3 y. So the prime factorization
of 8 is 2 times 2 times 2. It's 2 times 2 times 2. Prime, or I should say
the factorization of x to the third, or
the expansion of it, is just times x
times x times x. x multiplied by
itself three times. And then we are multiplying
everything by a y here, times y. So what factors are
common to both of these? And we want to include as
many of them as possible to find this greatest
common factor. So we have two 2's
here, three 2's here. So we only have two 2's
in common in both of them. We have four x's here,
only three x's here. So we only have
three x's in common. Three x's and three x's. And we have a y
here and a y here. So y is common to
both expressions. So the greatest common factor
here is going to be 2 times 2. So it's going to be 2 times 2
times x times x times x times y. Or 4x to the third y. So this is what we
want to factor out. So that means we can write this
thing as-- if we factor out a 4x to the third y,
where essentially we have to divide each of
these by 4x to the third. We're factoring it out. So let me rewrite this. So this is 4x to the fourth
y plus 8x to the third y. And we're going to divide each
of these by 4x to the third y. And hopefully this
makes sense to you. If we were to
multiply this out, we would distribute this 4x to
the third y on each terms. And then it would cancel
with the denominator. You would have the same thing in
the numerator and denominator. And then you would get
this expression over here. So hopefully this
makes sense that these are the exact same expression. But when you write
it this way, then it becomes pretty clear that
this is 4x to the third y. And then you just simplify
each of these expressions. 4 cancels with 4.
x to the fourth divided by x to the third is x. y divided by y is just 1. So you have x plus
8 divided by 4 is 2. x to the third divided
by x to the third is 1. Y divided by y is 1. So x plus 2. Another way to see what's left
over when you factor it out is if you were to take
out the common factor. So we took out this and this. What was left over
in 4x to the fourth y when we took this stuff out? When we undistributed it? Well, the only
thing that was left was this x right over here. Let me do that in another color. The only thing that
was left was this x. So that's why we just
have that x over there. When we factored everything
out of the 8x to the third y we factored all this
other stuff out. We factored out
4x to the third y. We factored it out, so
all we had left was the 2. Now in general, you
don't always have to go through this process. You could have done it this way,
but this really hopefully makes it clear exactly
what we're doing. You could have said,
look, 4x to the fourth y plus 8x to the third y. You could have said,
well let's see, the largest number that's
divisible into both 4 an 8 is 4. So let's factor
out a 4 out here. The largest multiple of x that's
divisible into x to the fourth and x to the third, well that's
going to be x to the third. And you put an x to
the third out here. And you say, well,
the largest thing that's divisible both
into y and y is just y. So you could have done it a
little bit faster in your head. So you factor out
a 4x to the third and you say, OK, if I
take out a 4 out of here, then this becomes a 1. If I take an x to the third
out of x to the fourth I'll just have an x left over. And then if I take a
y out of the y then I just have a 1 there. So this term becomes x. And then if I take
a 4x to the third y out of here, if I
take a 4 out of an 8 I just have a 2 left over. If I take an x to the third
out of x to the third, that's just 1. If I take y out of
y, that's just one. So I'm just left with x plus 2. Eventually you'll just do
this in your head a little bit faster. But hopefully this
makes everything clear.