WEBVTT

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I have here a bunch of radical
expressions, or square root

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expressions.

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And what I'm going to do is
go through all of them and

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simplify them.

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And we'll talk about whether
these are rational or

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irrational numbers.

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So let's start with A.

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A is equal to the square
root of 25.

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Well that's the same thing as
the square root of 5 times 5,

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which is a clearly
going to be 5.

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We're focusing on the positive
square root here.

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Now let's do B.

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B I'll do in a different color,
for the principal root,

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when we say positive
square root.

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B, we have the square
root of 24.

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So what you want to do, is
you want to get the prime

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factorization of this
number right here.

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So 24, let's do its prime
factorization.

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This is 2 times 12.

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12 is 2 times 6.

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6 is 2 times 3.

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So the square root of 24, this
is the same thing as the

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square root of 2 times
2 times 2 times 3.

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That's the same thing as 24.

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Well, we see here, we have one
perfect square right there.

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So we could rewrite this.

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This is the same thing as the
square root of 2 times 2 times

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the square root of 2 times 3.

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Now this is clearly 2.

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This is the square root of 4.

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The square root of 4 is 2.

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And then this we can't
simplify anymore.

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We don't see two numbers
multiplied by itself here.

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So this is going to be times
the square root of 6.

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Or we could even right this as
the square root of 2 times the

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square root of 3.

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Now I said I would talk
about whether things

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are rational or not.

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This is rational.

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This part A can be expressed
as the ratio of 2 integers.

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Namely 5/1.

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This is rational.

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This is irrational.

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I'm not going to prove
it in this video.

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But anything that is the product
of irrational numbers.

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And the square root of any prime
number is irrational.

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I'm not proving it here.

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This is the square root of 2
times the square root of 3.

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That's what the square
root of 6 is.

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And that's what makes
this irrational.

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I cannot express this as
any type of fraction.

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I can't express this as some
integer over some other

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integer like I did there.

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And I'm not proving it here.

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I'm just giving you a little
bit of practice.

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And a quicker way to do this.

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You could say, hey,
4 goes into this.

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4 is a perfect square.

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Let me take a 4 out.

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This is 4 times 6.

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The square root of 4 is 2, leave
the 6 in, and you would

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have gotten the 2 square
roots of 6.

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Which you will get the hang of
it eventually, but I want to

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do it systematically first.

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Let's do part C.

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Square root of 20.

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Once again, 20 is 2 times
10, which is 2 times 5.

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So this is the same thing as the
square root of 2 times 2,

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right, times 5.

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Now, the square root of 2 times
2, that's clearly just

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going to be 2.

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It's going to be the square
root of this times

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square root of that.

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2 times the square root of 5.

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And once again, you could
probably do that in your head

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with a little practice.

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The square root of the
20 is 4 times 5.

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The square root of 4 is 2.

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You leave the 5 in
the radical.

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So let's do part D.

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We have to do the square
root of 200.

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Same process.

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Let's take the prime
factors of it.

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So it's 2 times 100, which is
2 times 50, which is 2 times

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25, which is 5 times 5.

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So this right here,
we can rewrite it.

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Let me scroll to the
right a little bit.

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This is equal to the square
root of 2 times 2 times 2

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times 5 times 5.

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Well we have one perfect square
there, and we have

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another perfect square there.

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So if I just want to write out
all the steps, this would be

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the square root of 2 times 2
times the square root of 2

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times the square root
of 5 times 5.

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The square root of
2 times 2 is 2.

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The square root of 2 is just
the square root of 2.

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The square root of 5 times 5,
that's the square root of 25,

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that's just going to be 5.

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So you can rearrange these.

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2 times 5 is 10.

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10 square roots of 2.

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And once again, this
is irrational.

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You can't express it as a
fraction with an integer and a

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numerator and the denominator.

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And if you were to actually try
to express this number, it

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will just keep going on and on
and on, and never repeating.

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Well let's do part E.

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The square root of 2000.

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I'll do it down here.

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Part E, the square
root of 2000.

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Same exact process that we've
been doing so far.

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Let's do the prime
factorization.

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That is 2 times 1000, which is
2 times 500, which is 2 times

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250, which is 2 times 125,
which is 5 times 25,

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which is 5 times 5.

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And we're done.

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So this is going to be equal to
the square root of 2 times

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2-- I'll put it in parentheses--
2 times 2, times

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2 times 2, times 2 times
2, times 5 times 5,

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times 5 times 5, right?

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We have 1, 2, 3, 4, 2's, and
then 3, 5's, times 5.

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Now what is this going
to be equal to?

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Well, one thing you might see
is, hey, I could write this

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as, this is a 4, this is a 4.

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So we're going to have
a 4 repeated.

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And so this the same thing as
the square root of 4 times 4

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times the square root of
5 times 5 times the

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square root of 5.

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So this right here
is obviously 4.

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This right here is 5.

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And then times the
square root of 5.

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So 4 times 5 is 20 square
roots of 5.

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And once again, this
is irrational.

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Well, let's do F.

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The square root of 1/4, which
we can view this is the same

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thing as the square root of 1
over the square root of 4,

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which is equal to 1/2.

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Which is clearly rational.

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It can be expressed
as a fraction.

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So that's clearly rational.

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Part G is the square
root of 9/4.

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Same logic.

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This is equal to the square root
of 9 over the square root

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of 4, which is equal to 3/2.

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Let's do part H.

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The square root of 0.16.

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Now you could do this in your
head if you immediately

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recognize that, gee, if
I multiply 0.4 times

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0.4, I'll get this.

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But I'll show you a more
systematic way of doing it, if

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that wasn't obvious to you.

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So this is the same thing
as the square

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root of 16/100, right?

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That's what 0.16 is.

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So this is equal to the square
root of 16 over the square

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root of 100, which is equal to
4/10, which is equal to 0.4.

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Let's do a couple
more like that.

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OK.

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Part I was the square root of
0.1, which is equal to the

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square root of 1/10, which is
equal to the square root of 1

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over the square root of 10,
which is equal to 1 over--

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now, the square root of 10--
10 is just 2 times 5.

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So that doesn't really
help us much.

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So that's just the square
root of 10 like that.

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A lot of math teachers don't
like you leaving that radical

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in the denominator.

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But I can already tell you
that this is irrational.

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You'll just keep getting
numbers.

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You can try it on your
calculator, and

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it will never repeat.

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Your calculator will just give
you an approximation.

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Because in order to give the
exact value, you'd have to

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have an infinite number
of digits.

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But if you wanted to
rationalize this,

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just to show you.

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If you want to get rid of the
radical in the denominator,

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you can multiply this times the
square root of 10 over the

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square root of 10, right?

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This is just 1.

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So you get the square
root of 10/10.

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These are equivalent statements,
but both of them

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are irrational.

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You take an irrational number,
divide it by 10, you still

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have an irrational number.

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Let's do J.

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We have the square
root of 0.01.

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This is the same thing as the
square root of 1/100.

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Which is equal to the square
root of 1 over the square root

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of 100, which is equal
to 1/10, or 0.1.

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Clearly once again
this is rational.

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It's being written
as a fraction.

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This one up here was
also rational.

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It can be written expressed
as a fraction.