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# Simplifying radical expressions: three variables

A worked example of simplifying the cube root of 27a²b⁵c³ using the properties of exponents. Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to simplify the
cube root of 27a squared times b to the fifth times
c to the third power. And the goal, whenever
you try to just simplify a cube root like
this, is we want to look at the parts of
this expression over here that are perfect cubes,
that are something raised to the third power. Then we can take just
the cube root of those, essentially taking them
out of the radical sign, and then leaving
everything else that is not a perfect
cube inside of it. So let's see what we can do. So first of all, 27--
you may or may not already recognize this
as a perfect cube. If you don't already
recognize it, you can actually do
a prime factorization and see it's a perfect cube. 27 is 3 times 9,
and 9 is 3 times 3. So 27-- its prime factorization
is 3 times 3 times 3. So it's the exact same thing
as 3 to the third power. So let's rewrite this
whole expression down here. But let's write it in terms of
things that are perfect cubes and things that aren't. So 27 can be just rewritten
as 3 to the third power. Then you have a squared--
clearly not a perfect cube. a to the third would have been. So we're just going
to write this-- let me write it over here. We can switch the order
here because we just have a bunch of things being
multiplied by each other. So I'll write the a
squared over here. b to the fifth is not a
perfect cube by itself, but it can be expressed
as the product of a perfect cube and
another thing. b to the fifth is the exact same thing as
b to the third power times b to the second power. If you want to see that
explicitly, b to the fifth is b times b times
b times b times b. So the first three are
clearly b to the third power. And then you have b to
the second power after it. So we can rewrite b to
the fifth as the product of a perfect cube. So I'll write b
to the third-- let me do that in that
same purple color. So we have b to the
third power over here. And then it's b to the
third times b squared. So I'll write the b
squared over here. And we're assuming we're going
to multiply all of this stuff. And then finally, we
have-- I'll do in blue-- c to the third power. Clearly, this is a perfect cube. It is c cubed. It is c to the third power. So I'll put it over here. So this is c to the third power. And of course, we still have
that overarching radical sign. So we're still trying to take
the cube root of all of this. And we know from our
exponent properties, or we could say from
our radical properties, that this is the
exact same thing. That taking the cube root
of all of these things is the same as
taking the cube root of these individual factors
and then multiplying them. So this is the same
thing as the cube root-- and I could separate
them out individually. Or I could say the cube
root of 3 to the third b to the third c to the third. Actually, let's do it both ways. So I'll take them
out separately. So this is the same thing
as the cube root of 3 to the third times the cube
root-- I'll write them all in. Let me color-code it so we don't
get confused-- times the cube root of b to the third
times the cube root of c to the third times
the cube root-- and I'll just group
these two guys together just because we're not
going to be able to simplify it any more-- times the cube
root of a squared b squared. I'll keep the colors
consistent while we're trying to figure
out what's what. And I could have said that
this is times the cube root of a squared times
the cube root of b squared, but that won't
simplify anything, so I'll just leave
these like this. And so we can look at
these individually. The cube root of 3 to the
third, or the cube root of 27-- well, that's clearly
just going to be-- I want to do that in
that yellow color-- this is clearly
just going to be 3. 3 to the third power is
3 to the third power, or it's equal to 27. This term right over
here, the cube root of b to the third--
well, that's just b. And the cube root
of c to the third, well, that is
clearly-- I want to do that in that-- that
is clearly just c. So our whole expression has
simplified to 3 times b times c times the cube root
of a squared b squared. And we're done. And I just want to
do one other thing, just because I did mention
that I would do it. We could simplify it this way. Or we could recognize that
this expression right over here can be written as 3bc
to the third power. And if I take three
things to the third power, and I'm multiplying it,
that's the same thing as multiplying them
first and then raising to the third power. It comes straight out of
our exponent properties. And so we can rewrite
this as the cube root of all of this times
the cube root of a squared b squared. And so the cube
root of all of this, of 3bc to the third power, well,
that's just going to be 3bc, and then multiplied by the cube
root of a squared b squared. I didn't take the trouble
to color-code it this time, because we already figured
out one way to solve it. But hopefully, that
also makes sense. We could have done
this either way. But the important thing is
that we get that same answer.