More Simplifying Radical Expressions

More Simplifying Radical Expressions

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.