>> This is Part Six of square roots and radicals.
And we're going to continue simplifying.
So, let's look at this problem.
In Part Five, we did a little bit easier problem,
so we're stepping it up a little bit.
Square root of 48x to the 8, n to the 5.
All right.
So, what I'm going to do is write this so that I'm going
to write this as 48 is a perfect square, times something else.
Now, you can actually write 48
as either 4 times 12 or 3 times 16.
And it's best if you could pick the biggest square root.
So, I'm going to write it as --
I'm sorry, the biggest factor that's a perfect square,
16 times 3.
This will be the quickest way to do it.
And x to the 8, well, that is a perfect square
because it has an even exponent.
And n to the 5, it's an odd exponent, so I need to write it
as an even exponent, right?
So, n to the 4 times n. So breaking this out so
that you can see the perfect squares.
So we have square root of 16, square root of x to the 8,
square root of n to the 4.
We could pull those out, there's a 4,
an x to the 4 and an n squared.
So, that gets multiplied out in front, 4x to the 4, n squared.
Now, what's left inside is a 3n.
Now, using this method I've given you, what's important is
that you do pick the largest perfect square factor
of 48, which is 16 times 3.
But if you don't you just have
to make sure whatever you have left underneath the square root,
there's no more factors that are perfect squares, let's say of 3.
All right, let's say what if I did just ask you
to do the square root of 48
and you just couldn't think of what it was.
I've given you some ideas how to do it, and you just can't think
of a perfect square factor of 48.
You could always go and make a factor tree to prime factor it.
So, if you did it this way, I might say that's 6 times 8.
And let's see, that's 2 times 3 and then 8 is 4 times 2.
And this is 2 times 2.
So, notice, I'm going to circle all the leaves here.
And notice I've got 1, 2, 3, 4 factors of 2 and a factor of 3.
So, the key is, if you have no idea where to start,
you could write the prime factorization down.
So, 2 times 2 times 2 times 2 times 3, okay?
And then if you have two of the same factor,
that means it's a perfect square.
So, from these two factors, 2 times 2, a 2 gets to go out,
and from these two factors a 2 gets to go out.
And then you multiply those.
So, that's another way of getting the answer.
2 times 2 would be on the outside, 4 square roots of 3.
Okay? So -- whoops.
Square root of 48, you could have done it the way I showed
you how to do it, the first way, 16 times 3 is 48, take out a 4
and you're going to get 4 square roots of 3.
You could do it by completely prime factoring
or you could always take just a few steps to get that answer.
So, let's say you did know that 4 when into it,
which is a perfect square, and you wrote 12,
then you would have the 2 that you could take
out from the square root of 4.
And let's see.
Then I would have a 2 out, but then the trick is realizing
that 12 has a perfect square factor of 4 as well.
So I have 4 times 3 again.
You would have to take out another 2
and 2 times 2 is 4 square roots of 3.
So, just keep in mind that, you know, there's more than one way
of getting the final result of 4 square roots of 3 in this case.
So there's three different ways you might come about it.
All right, let's work another problem.
4x squared m times the square root of 63x squared m to the 7.
Okay, so when we break this down, it's not so bad.
On the outside, we have the 4x squared m. Now, you have to look
for a perfect square factor in 63.
So again, you could go ahead and do a factor tree to figure
that out, or hopefully, you realize that's 9 times 7.
9's a perfect square.
x squared is already a perfect square, isn't it,
because it has an even exponent.
And m to the 7, I need to write as m to the 6 times n. I need
to write with an even exponent,
so you sort of break off this m. Okay, so now we're going
to circle the perfect squares and we're going
to put the square root on top.
So, I could take the square root of 9 is 3, the square root
of x squared is x, and the square root
of m to the 6 is m cubed.
Keep in mind, we're making these assumptions
that all variables are not negative, okay?
Otherwise it gets more complicated with absolute value.
All right.
So, whatever's out in front gets multiplied
by whatever's up on top in here.
So, if you can't do that in your head,
I'm going to write it out the long way.
I've got 4 x squared m times 3 xm cubed.
And what's left inside is a 7 times an m, which is 7m.
So all right.
So we've got 4 times 3, that's 12.
I've got an x squared times an x, that gives me an x cubed.
I have an m times time an m cubed, that's an m to the 4,
and a square root of 7m.
And so that's the answer.
So if you just work through it slowly, you get there.
All right.
Here's one more for you to try.
Simplify this.
Okay? Negative 5 x to the 4 square root
of 8 x squared y to the 9.
Put the video on pause and try it on your own.
Okay, hopefully you're back.
So, what do we have here?
On the outside, we have this negative 5 x to the 4,
and on the inside, for 8, we could write that as 4 times 2.
x squared is already a perfect square.
And we're going to break off a y from the y to the 9 to write
that as y to the 8 times y. And then we're going
to do the square root of the perfect squares
and put them up on top.
So, what do we have?
I'm going to multiply whatever's in front of the square root
by what I've written up on the top.
So I have a negative 5 times 2 is negative 10.
An x to the 4 times x is x to the 5.
And I also have a y to the 4.
What's left inside is a 2Y.
All right.
How about you doing a little practice.
Try these two problems by putting the video on pause.
All right.
So, let's go ahead and work them.
Hopefully you're going
to get the same answer I'm going to get.
So, what do we have here?
45. You have to think
of a perfect square factor, and 45 is 5 times 9.
So I'm going to write that as 9 times 5.
You might notice I usually write the perfect square first.
You don't have to, though.
And what about x cubed?
It's odd, so I need to write that as x squared times x. So,
we could pull out a 3 from the square root of 9 and pull
out an x from the square root of x squared.
And then multiplying by what's out in front.
That gives me a negative 18x in front of the square root.
And underneath is 5x.
So that's the answer for that one.
And the second one, we can break 28 into 4 times 7,
because 4 is the perfect square.
And x squared is already a perfect square.
So, I can take out a 2 from the square root of 4 and take an x
from the square root of x squared.
And that's going to give me, I multiply the 2x times the 4x,
8x squared is what goes out on the front,
times the square root of 7.