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# Simplifying square roots (variables)

A worked example of simplifying radical with a variable in it. In this example, we simplify 3√(500x³). Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

What I want to do in
this video is resimplify this expression, 3 times the
principal root of 500 times x to the third, and take
into consideration some of the comments that
we got out on YouTube that actually give some
interesting perspective on how you could simplify this. So just as a quick review of
what we did in the last video, we said that this
is the same thing as 3 times the
principal root of 500. And I'm going to do it a
little bit different than I did in the last video, just
to make it interesting. This is 3 times
the principal root of 500 times the principal
root of x to the third. And 500-- we can
rewrite it, because 500 is not a perfect square. We can rewrite 500
as 100 times 5. Or even better, we could rewrite
that as 10 squared times 5. 10 squared is the
same thing as 100. So we can rewrite this first
part over here as 3 times the principal root
of 10 squared times 5 times the principal
root of x squared times x. That's the same thing
as x to the third. Now, the one thing I'm
going to do here-- actually, I won't talk about
it just yet, of how we're going to do it
differently than we did it in the last video. This radical right
here can be rewritten as-- so this is going to
be 3 times the square root, or the principal
root, I should say, of 10 squared times
the square root of 5. If we take the square root
of the product of two things, it's the same thing as taking
the square root of each of them and then taking the product. And so then this
over here is going to be times the square root
of, or the principal root of, x squared times the
principal root of x. And the principal root
of 10 squared is 10. And then what I said
in the last video is that the principal
root of x squared is going to be the absolute
value of x, just in case x itself is a negative number. And so then if you
simplify all of this, you get 3 times
10, which is 30-- and I'm just going to
switch the order here-- times the absolute value of x. And then you have
the square root of 5, or the principal root of 5,
times the principal root of x. And this is just going to be
equal to the principal root of 5x. Taking the square
root of something and multiplying that times the
square root of something else is the same thing as just
taking the square root of 5x. So all of this simplified
down to 30 times the absolute value of x times
the principal root of 5x. And this is what we
got in the last video. And the interesting
thing here is, if we assume we're only
dealing with real numbers, the domain of x right
over here, the x's that will make this
expression defined in the real numbers--
then x has to be greater than or equal to 0. So maybe I could
write it this way. The domain here is that
x is any real number greater than or equal to 0. And the reason
why I say that is, if you put a negative number
in here and you cube it, you're going to get
another negative number. And then at least
in the real numbers, you won't get an actual value. You'll get a square root
of a negative number here. So if you make this-- if
you assume this right here, we're dealing with
the real numbers. We're not dealing with
any complex numbers. When you open it up
to complex numbers, then you can expand the
domain more broadly. But if you're dealing
with real numbers, you can say that x is going to
be greater than or equal to 0. And then the absolute
value of x is just going to be x, because it's not
going to be a negative number. And so if we're assuming
that the domain of x is-- or if this expression is
going to be evaluatable, or it's going to have
a positive number, then this can be written as 30x
times the square root of 5x. If you had the situation
where we were dealing with complex
numbers-- and if you don't know what a complex number
is, or an imaginary number, don't worry too much about it. But if you were
dealing with those, then you would have to keep
the absolute value of x there. Because then this
would be defined for numbers that
are less than 0.