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Hi.
In this video, I'm going to take you step by step through
the process of simplifying a radical when it's something
other than just a square root.
Here we have a cube root, specifically the cube root of
48, x to the fourth, y squared, z to the 8.
Since this is a cube root, to simplify parts, we want to see
if we can rewrite each of these things using 3 as an
exponent because the cube root of something cubed cancels
out, simplifies very nicely for us.
So first off, let's take a look at 48.
That number there, again, we need to break up to rewrite it
as things raised to the third power.
So let's see what numbers actually make up 48.
What we want to do is take 48 and break it up
into its prime factors.
There are a lot of ways to do this, but in this example, I'm
going to use a simple tree diagram.
I want any two numbers that happen to multiply to 48.
The first two I think of are 4 and 12.
Since neither of these are prime numbers, I'm going to go
ahead and break them up and see what
multiplies to create them.
4 is 2 times 2.
Those are both prime, so I get to stop.
12 is 3 times 4.
3 is a prime number, so there's nothing else to break
up there, but 4, again, is 2 times 2.
So each of these values that we have here are prime.
These are the prime factors of 48.
If we multiply them together, we get 48.
So instead of writing it as 48, I'm going to write it as
the product of all of these numbers.
So let me switch my pen color again.
This means instead of just 48, I'm going to
end up having 2 cubed.
Remember, we're trying to write this in powers of 3, and
I happen to have three of them.
2 times 2 times 2 times one more 2--
because you notice, there were four of them here--
times 3, the last factor I had.
All of those multiply together to give me 48, and I
specifically wanted to write it grouping things together
into groups of 3 if possible, so right there, 2 cubed.
Next up, we have the xs.
x to the fourth, and I want to write this using 3 as an
exponent, meaning grouping them together
into groups of 3.
Since there are four xs being multiplied together, I can
think of this as x cubed, three of them, times the one
that's left over.
All of those together make up my x to the fourth.
Next up, y squared.
Well, since there are only two of them, there's no way for me
to use 3 as an exponent, so I have to just leave it as is.
So I'm going to leave that y squared as is.
Next up, we have z to the eighth.
That's what I need to break up, grouping it together into
sort of, if you will, clumps of 3.
I've run out of room here so I'm going to slide this over a
little bit so I've got some more room to work with.
So z to the eighth, putting it together into groups of 3.
Since I have eight of them, I can rewrite this as z cubed--
that's going to use up three of them--
times z cubed, which is another three, so six total.
That leaves me two left over, z squared.
The idea is that here, I have a total of 3 plus 3 plus 2, so
8 zs multiplied together.
3 plus 3 plus 2 is 8, not 6.
And that's going to make up the z to the eighth that I had
in my original problem.
So this is all that I had under the original radical, so
in the radicand, just broken up conveniently enough,
putting everything together with an
exponent of 3 if possible.
So sort of breaking it up into its component parts.
And this whole amount is what's inside my cube root, so
I'm going to go put it underneath a giant cube root
radical here.
Now, each one of these parts that is raised to the third
power, it's going to simplify.
What I mean by that is, switching pen color again,
that the cube root of 2 cubed is 2.
I can take that cube root.
It's a perfect cube so everything
will work out nicely.
The cube root of x cubed is x.
Again, I can take that cube root nicely and have it work
out perfectly.
Cube root of z cubed is z.
And again, the cube root of z cubed is z.
So each one of these parts, since I can take the cube root
of it very nicely, very perfectly, I'm going to go
ahead and do that and pull those parts
outside of the radical.
That's going to give me outside of my radical 2, which
is coming from this here, times x, which is the cube
root of x cubed, times z, the cube root of z cubed, times z
again, the cube root of that z cubed.
And that's everything that I can
simplify out of the radical.
What's left inside my radical, my cube root here?
Well, I have this 2, this 3, this x, this y squared, and
this z squared right here, so let me put
all of those inside.
2 times 3 times x times y squared times z squared,
basically everything I hadn't underlined in red up above.
That's what's left inside the radical.
Now it's just a matter of simplifying this and writing
it using exponents or
multiplying the numbers together.
On the outside, I'm going to have 2xz squared times the
cube root of 6xy squared, z squared.
This is what's referred to as simple radical form, meaning
that I have simplified everything out of the radical
that I possibly can, and I'm left with the stuff inside the
radical that I couldn't take a perfect cube of.
So I end up with this stuff here.
Thank you very much.