Simplifying radicals

Using exponent rules to simplify radicals or square roots

Welcome to the presentation
on simplifying radicals.

So let's get started with getting a little terminology out of the way.

You're probably just wondering what a radical is and I'll just let you know.

I've got to get the pen settings right.

A radical is just that.

Or you're probably more familiar calling that the square root symbol.

So with the terminology out of
the way,

let's actually talk about what it means to simplify a radical.

And some people would argue that what we're going to actually be doing

is actually making it more complicated.

But let's see.

Let me erase that.

So if I were to give you the square root of 36,

you'd say hey, that's easy.

That's just equal to 6 times 6

or you'd say the square root of 36 is just 6.

Now, what if I asked you what
the square root of 72 is?

Well, we know that
72 is 36 times 2, right?

So let's write that.

Square root of 72 is the same thing as the square root of 36 times 2.

Right? We just rewrote seventy-two as thirty-six times two.

And the square root, if you remember from level 3 exponents.

square root is the same thing as something to the one half power.

So let's write it that way.

And I'm just writing it this way just to show you how this radical simplification works,

and that it's really not a new concept.

So this is the same thing as
36 times 2 to the one half power.

Right? Because it's just a square root
is the same thing as one half power.

And we learned from the exponent rules that when you multiply two numbers

and then you raise that to the one half power,

that that's the same thing as raising each of the numbers to the one half power

and then multiplying. Right?

Well that right there, that's the same thing as saying the square root is 36 times the square root of 2.

And we already figured out what
the square root of 36 is.

It's 6.

So that just equals 6 times
the square root of 2.

And you're probably wondering why I went through this step of changing the radical,

the square root symbol, into the one half power.

And I did that just to show you that this is just an extension of the exponent rules.

It isn't really a new concept.

Although, I guess sometimes it's not so obvious that
they are the same concepts.

I just wanted to
point that out.

So let's do another problem.

I think as we do more and more problems, these will become more obvious.

The square root of 50.

Well, the square root of 50 --

50 is the same thing as 25 times 2.

And we know, based on what we just did and this is really just an exponent rule,

The square root of 25 times 2 is the same thing as the square root of 25

times the square root of 2.

Well we know what the
square root of 25 is.

That's 5.

So that just equals 5 times
the square root of 2.

Now, you might be saying, "Hey,
Sal, you make it look easy,

but how did you know to split 50
into 25 and 2?"

Why didn't I say that 50 is equal to the square root of 5 and 10?

Or that 50 is equal to the square root
-- actually, I think 1 and 50?

I don't know what
other factors 50 has.

Well, anyway, I won't go
into that right now.

The reason why I picked 25 and
2 is because I wanted a factor of 50--

I actually wanted the largest factor of 50 that is a perfect square.

And that's 25.

If I had done 5 and 10, there's really nothing I could have done with it,

because neither 5 nor 10 are perfect squares

and the same thing's with 1 and 50.

So the way you should think
about it,

think about the factors of the original number

and figure out if any of those factors are perfect squares.

And there's no real
mechanical way.

You really just have to learn
to recognize perfect squares.

And you'll get familiar
with them, of course.

They're 1, 4, 9, 25, 16,
25, 36, 49, 64, et cetera.

And maybe by doing this module, you'll actually learn to recognize them more readily.

But if any of these numbers are a factor of the number under the radical sign

then you'll probably want to factor them out.

And then you can take them out of the radical sign

like we did up in this problem.

Let's do a couple more.

What is 7 times the square root of 27?

And when I write the 7 right
next to it,

that just means times the square root of 27.

Well, let's think about what
other factors of 27 are,

and whether any of them are a perfect square.

Well, 3 is a factor of 27, but
that's not a perfect square.

9 is.

So, we could say 7 --

that's equal to 7 times the square root of 9 times 3.

And now, based on the rules we just learned,

that's the same thing as 7 times the square
root of 9

times the square root of 3.

Well that just equals 7 times 3
because the square root of 9 is 3

times the square root of 3.

That equals 21 times
the square root of 3.

Done.

Let's do another one.

What is nine times the square root of eighteen?

Well, once again, what are the factors of eighteen?

Well do we have 6 and 3?

1 and 18?

None of the numbers I mentioned
so far are perfect squares.

But we also have 2 and 9.

And 9 is a perfect square.

So let's write that.

That's equal to 9 times the
square root of 2 times 9.

Which is equal to 9 times the
square root of 2 --

that's a 2, times the square root of 9.

Which equals 9 times the square
root of 2 times 3, right?

That's the square root of 9 which equals

27 times the square root of 2.

There we go.

Hopefully, you're starting to
get the hang of these problems.

Let's do another one.

What is 4 times the
square root of 25?

Well, twenty-five itself is a perfect square.

This is kind of a problem that's so easy that it's a bit of a trick problem.

25 itself is a perfect square.

The square root is 5, so this is just equal to 4 times 5,

which is equal to 20.

Square root of 25 is 5.

Let's do another one.

What's 3 times the
square root of 29?

Well 29 only has two factors.

It's a prime number.

It only has the
factors 1 and 29.

And neither of those numbers
are perfect squares.

So we really can't simplify
this one anymore.

So, this is already in
completely simplified form.

Let's do a couple more.

What about 7 times the square root of 320?

So, let's think about 320.

Well we could actually do it in steps when we have larger numbers like this.

I can look at it and say, well
it does look like 4 --

actually it looks like 16 would go into this because 16 goes into 32.

So let's try that.

So that equals 7 times the
square root of 16 times 20.

Well, that just equals 7 times the square root of 16

times the square root of 20.

7 times the square root of 16.

The square root of 16 is 4.

So 7 times 4 is 28.

So that's 28 times the
square root of 20.

Now are we done?

Well actually, I think I can factor 20 even more

because 20 is equal to 4 times 5.

So I can say this is equal to 28 times the square root of 4 times 5.

The square root of 4 is 2 so
that could just take the 2 out

and that becomes 56 times
the square root of 5.

I hope that made sense to you.

And this is actually a
pretty important technique

I just did here.

Immediately when I look at 320.

I don't know what the largest
number is that goes into 320.

It actually turns
out that it's 64.

But just looking at the number, I said, well I know that 4 goes into it.

So I could have just pulled out 4,

and then said, "Oh, that's equal to 4 times 80."

And then I would have had to work with 80.

In this case, I saw 32 and I was like, it looks like 16 goes into it

and I factored out 16 first.

And when I took out the square root of 16, I multiplied the outside by 4

and that's how I got the 28.

But then I reduced the number
on the inside and said,

"Oh, well that still is divisible by a perfect square.

It's still divisible by 4." And
then I kept doing it

until I was left with essentially, a prime number or a number that couldn't be reduced anymore under the radical.

And it actually doesn't
have to be prime.

So hopefully, that gives you a good sense of how to do radical simplification.

It's really just an extension of the exponent rules that you've already learned,

and hopefully as you do the module, you'll get good at it.

Have fun!