Welcome to the presentation on simplifying radicals.
So let's get started with a little terminology
out of the way.
You're probably just wondering what a radical is and
I'll just let you know.
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.
So if I were to give you the square root of 36.
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.
Let's write that.
Square root of 72 is the same thing as the square
root of 36 times 2.
And the square root, if you remember from level 3
exponents, square root is the same thing as something
to the 1/2 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 1/2 power.
Because it's just a square root is the same thing as 1/2 power.
And we learned from the exponent rules that when you
multiply two numbers and then you raise that to the 1/2
power, that that's the same thing as raising each of the
numbers to the 1/2 power and then multiplying.
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 1/2 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, 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 is 50.
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
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 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 9 times the square root of 18?
Well once again, what are the factors of 18?
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.
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 25 itself is a perfect square.
This problem is so easy, 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?
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.
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 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.