In this video, I'm going to do some more examples of
simplifying radical expressions.
But these are going to involve adding and subtracting
different radical expressions.
And I think it's a good tool to have in your toolkit in
case you've never seen it before.
So let's do a few of these.
So let's say I have 3 times the square root of 8-- we
learned before that's actually the principal square root of
8, or the positive square root of 8-- minus 6 times the
principal square root of 32.
So let's see what we can do to simplify this.
So first of all, 8, we can write that as 2 times 4.
When 4 is a perfect square, you might
already recognize that.
We could further factor that into 2 times 2.
But I don't think we need to.
So we can rewrite 3 square root of 8 as 3 times the
square root of 4 times the square root of 2.
This is the same thing as the square root of 4 times 2,
which is the square root of 8.
So this term is the same thing as that term.
And then, let's look at 32.
We want to do the square root of 32.
32 is 2 times 16.
Once again, 16's a perfect square, so
we could stop there.
If you didn't realize that, you would factor
that as 4 times 4.
You'd see that twice.
You could even go even further down to 2 times 2 and all of
that, but you see immediately that's a perfect square, so we
can stop there.
So this second expression can be written as minus 6 times
the square root of 16 times the square root of 2.
This right here-- I want to be clear-- is the same thing as
the square root of 16 times 2.
You can separate out.
The square root of 16 times 2 is the square root of 16 times
the square root of 2.
We saw that with our exponent properties.
Now, what does this first term simplify to?
This is 3 clearly.
This right here is a 2.
So you have 3 times 2 times the square root of 2.
That is 6 times the principal root of 2.
And then from that we're going to subtract-- well, what's
this term right here?
That is positive 4.
So 6 times 4 is 24 times the square root of 2.
And we're not done yet.
If I have 6 of something and I'm going to subtract from
that 24 of that same something, what do I have?
I have 6 square roots of 2 and I'm going to subtract from
that 24 square roots of 2, well, this is going to be
equal to 6 minus 24 is negative 18 square roots of 2.
And hopefully, this doesn't confuse you.
Remember, if we had 6x minus 24x, we would have minus 18x
or negative 18x.
Now, instead of an x, we just have a square root of 2.
6 of something minus 24 of something will get us negative
18 of that something.
Let's do another one.
Let's say I have the square root of 180 plus 6 times the
square root of 405.
So this is really an exercise in being able to simplify
these radicals, which we've done before.
But you can never get too much practice doing that.
So let's just do the factorization
of these right here.
So 180 is 2 times 90, which is 2 times 45,
which is 5 times 9.
And we can factor 9 down more into 3 times 3 to realize it's
a perfect square, but we could leave it like that.
So this first term right here we can write as the square
root of 2 times 2 times the square root of 5 times the
square root of 9.
I'm going to put the square root of 9 out front.
So square root of 2 times 2 times the square root of 5
times the square root of 9.
Now, what is this second term equal to?
So let's factor it out.
405.
That is 5 times-- I think it's 81.
But just to verify, 405, 5 doesn't go into 4, so
let's go into 40.
5 goes into 40 eight times.
8 times 5 is 40.
Subtract.
You get a 0.
Bring down the 5.
5 goes into 5 one time.
Right, 81 times.
81 is 9 times 9.
You could factor more if we were trying to do the fourth
root or something like that, but we want to just do a
square root.
We have a 9 and a 9, so no need to factor any more.
So this second expression right here is plus 6 times the
square root of 9 times 9 times the square root of 5.
So what is this?
This is 3.
This is 2.
This is the square root of 4.
So it's 3 times 2 is 6.
So we have 6 square roots of 5.
Plus-- what's this right here?
The square root of 9 times 9, the square root of 81.
That's, of course, just 9.
So 6 times 9 is 54, so plus 54 square roots of 5.
And then, what do we have left?
We have 6 of something plus 54 of something.
That's going to be equal to 60 of that
something just like that.
Let's just do one more and we're going to have some
abstract quantities here.
We're going to deal with some variables.
But I really just want to do it to show you that the
variables don't change anything.
Let's say if we have the square root or the principal
root of 48a.
And I'm going to add that to the square root of 27a.
So once again, let's just factor the 48 part.
We'll leave the a aside.
So 48 is 2 times 24, which is 2 times 12.
Sorry, 2 times 12, which is 3 times 4.
So we could rewrite this first expression here as the square
root of 2 times 2 times the square root of 4 times the
square root of 3.
Now, you might have done it a quicker way.
You might have just factored into 3 and 16 and immediately
realized that 16 is a perfect square.
But I did it just kind of the brute force way.
You'd get the same answer either way.
And, of course, not just the square root of 3, you also
have the square root of a there.
So I'll just put the a right over here.
I could put it in a separate square root, but both of these
aren't perfect squares, so I'll leave both of these under
the radical sign.
Now, 27 is 3 times 9.
9 is a perfect square root, so we can stop there.
So this second term, we can rewrite it as the square root
of 9 times the square root of 3a.
And in both of these you can kind of view it I'm skipping
an intermediate step.
The intermediate step, I could have written that first
expression as the square root of 9 times 3a and then
gone to this step.
But I think we have enough practice realizing that 9
times 3a, all of that to the 1/2 power, or taking the
principal root of all of that is the same thing as taking
the principal root of 9 times the principal root of 3a.
So that's the step I skipped in both of these.
But hopefully, that doesn't confuse you too much.
And so, this term right here is going to be a 2.
This term right here is going to be a 2.
So this is going to be 4 times the square root of 3a.
And then this over here, this right here, is a 3.
So this is going to be plus 3 times the square root of 3a.
4 of something plus 3 of something will be equal to 7
of the something.
Anyway, hopefully, you found that useful.