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## Simplifying radicals (higher-index roots)

# Simplifying cube root expressions

CCSS.Math:

## Video transcript

Multiply and simplify 5
times the cube root of 2x squared times 3 times the
cube root of 4x to the fourth. So the two things that
pop out of my brain right here is that we can change
the order a little bit because multiplication is
both commutative-- well, the commutative
property allows us to switch the order
for multiplication. And so we can get
the constant terms. We can multiply
the 5 times the 3. And then the other
two things that we're multiplying-- they're
both the cube root, which is the same thing as taking
something to the 1/3 power. So the cube root of
x-- this is exactly the same thing as
raising x to the 1/3. So let's do that. Let's switch the order and
let's rewrite these cube roots as raising it
to the 1/3 power. So I have the 5 and the 3. So that's going to be 5 times 3. And then we have the cube root
of-- do that in a new color. Then we have the cube
root of 2x squared. So this I can rewrite as 2x
squared to the 1/3 power. And then I have the cube
root of 4x to the fourth. So that's the same thing as 4x
to the fourth to the 1/3 power. And now we know from
our exponent properties, if we have two things that are
both raised to the same power and then we take
their product, we could just take their
product first and then raise it to the power. So if I have a to the x
power times b to the x power, this is the same thing as
a times b to the x power. So we can simplify this
part of the expression right over here as 2x squared
times 4x to the fourth to the 1/3 power. And of course, 5 times 3 is 15. And if we simplify what's in
the expression right over here-- once again, it's commutative,
so we can swap the order. And it's associative, and so
we can swap the groupings. How we group them doesn't
matter because it's all multiplication here. This is 2 times 4, which
is 6, times x squared times x to the fourth. x
squared times x to the fourth is x to the sixth power. And it's all of that
to the 1/3 power. And then that is times
that-- oh, sorry. Not 6. 2 times 4 is 8. What am I doing? 2 times 4 is 8. x squared times x to the
fourth is x to the sixth. I think my brain was adding the
exponents and wrote the 6 down. Of course, 2 times
4 is 8, not 6. But we add the exponents. They have the same base. x squared times x to the
fourth is x to the sixth. And we're going to raise
that to the 1/3 power. And then all of
that is times 15. And then we essentially can
use this property again. Actually, not that property. We know that this--
well, actually, yeah, exactly this
property again. We have something
multiplied to a power. This is the exact same thing. This is the exact same
thing as 8 to the 1/3 power times x to the sixth
to the 1/3 power. And then all of that is
being multiplied by 15. And so 8 to the
1/3 power-- that's the same thing as
the cube root of 8. You might recognize that
8 is 2 times 2 times 2. So 8 to the 1/3 power is 2. 8 is 2 to the third. So 2 the third to the
1/3 is 2 to the first. 2 times 2 times 2 is 8. And x to the sixth
to the 1/3 we know from our exponent properties. That's the same thing as x
to the sixth times 1/3 power, or x to the 6
divided by 3 power. Or 6 divided by 3
is 2, or x squared. So that is just x squared. So you have 15 times
2, which gives us 30. So that's these terms
right over here. And then you have this
term right over here. I want to do that in
a different color. And then you have
this term right over-- that's not
a different color. You have this term right
over here is x squared, and you are done. And there's a bunch of
ways you could do it. You might not decide to
use exponent notation. You could say, look,
this is a cube root. I can then take the cube root
of the product of both of them. So you don't have
to write a 1/3 here. You could just
write the cube root of this whole thing over here. And then, depending on how
you want to group it and all the rest, you could do
this in different orders. As long as you get the
right exponent properties, you should get to
this same answer.