Hey, guys.
Welcome to Algebra I.
Today's lesson
will focus on
simplifying square roots
of whole numbers.
All that you've learned
about perfect squares
and square roots,
it's all gonna come in handy
in this one.
Ready? Let's go.

(Describer) Standing at a screen, she holds a stylus.

Now, before we get into
the meat of this lesson,
I've got to jog your memory
about perfect squares
and square roots, okay?
Let's look at these two.

(Describer) ...on the screen.

The first one's telling me
to find the square root of 9.
That little symbol
over the 9,
it's technically called
a "radical."
It's often called a "root,"
sometimes a "square root."
That little three in there,
it's a "cubed root."
Get to those later.
This one's asking me
to find the square root of 9,
which essentially means what
number times itself equals 9?
You think back.
You run down numbers.
It's 3.
Because 3 times 3 is 9.
So, I'd say, "All right.
The square root of 9 is 3"...

(Describer) She writes.

I'm gonna write
a little "because" here...
"3 times 3 equals 9."
You don't have to write
the "because."
I want you to know why
the answer is 3. Okay?

(Describer) The next one.

So, 36?
You think, what number--
This is asking you to find
the square root of 36.
What number times itself
equals 36?
You run down the list.
You're thinking.
Maybe you're checking
in your calculator.
And it's 6...

(Describer) She writes.

because...
6 times 6 is 36.
Okay?
That's essentially what it means
when you take a square root.
You figure out what number
times itself is gonna equal
whatever that number is
under your radical.
Look at this next one with me.
The square root of 50. Hmm.

(Describer) She changes the screen.

Okay. When you're thinking
about that--you're thinking,
you're thinking,
you're thinking.
You're like, "Wait.
There's got to be something."
There really isn't.
There's no whole number that you
can multiply times itself
and it'll equal 50.
There's not.
Sometimes people say,
"Yeah, there is.
It's 25, right?"
What'd they do
if they told me it's 25?
They just divided 50 by 2.
There's no number you can
multiply by itself and get 50.
When you have to take
the square root of a number
that's not a perfect square,
you have to simplify
the radical.
You represent it
in another way that's simpler,
but you don't get
a whole number for your answer.
To know exactly how to do that,
you have to be familiar
with perfect squares.
We know numbers
go on for infinity.
They have technically
no beginning and no end,
but there are some
that we commonly use.
Let's generate that list.
I'm gonna ask you
to give an answer.
1 times 1. That's 1.
That's a perfect square.
2 times 2 is 4.
3 times 3? That's 9.
Okay, I'm squaring these.
Four squared?
What's 4 times 4?
Sixteen.
Five squared?
That's 25.
Six squared, 6 times 6?
Thirty-six.
Seven squared?
Forty-nine.
Eight squared?
Sixty-four.
Nine squared?
Eighty-one.
Ten squared?
A hundred.
You're probably like,
"When is she gonna stop?"
In a second.
Eleven squared?
121.
Just for kicks,
it's not really common,
but 13 squared?
It's actually 169.
If you can be familiar
with these,
at least up to 11 squared,
to 121,
you'll be good
working your way through these.
It may seem overwhelming,
but you're gonna work
with these numbers so much
that it's gonna become
second nature to you.
But while we're learning it,
you might want to write a list,
so while we're working
with these problems,
you have a list of the most
commonly used perfect squares.
Now that we have this list,
let's go back
to that square root of 50.
Now, how we simplify this,
let's read our directions.
"Write the numeric expression
in simplest radical form."
Sounds crazy, I know,
but what
that's asking me to do
is to rewrite this problem
as a product of factors, okay?
I'm gonna start out
by doing this.
I want to figure out
factors of 50,
and I want one of those factors
to be a perfect square, okay?
I've got my handy-dandy list
of the commonly used
perfect squares.
Just start running them down.
Can you factor 50--
would any
of these numbers work,
where any of these numbers
times something else is 50?
Let's look.
Mm-hmm. Twenty-five.
Sometimes there's more
than one answer.
There may be more than one way
I could get 50
by using one of these numbers,
but pick the largest number.
In this case,
I know I can multiply
25 times 2 and get 50.
So, I'm gonna go back
to the problem
and rewrite that as 25 x 2.

(Describer) ...under a radical.

I essentially replace 50
and wrote it
by its factors instead.
And I know that that 25
is the factor
that's a perfect square.
Now that I have it written
like this,
I'm gonna break this up
and rewrite it
as the square root of 25
times the square root of 2.
Didn't change the meaning.
Math says, "That's fine.
You can make that jump."
It started out with
both numbers under the radical,
and I separated them.
Now I want to take
the square root of the number
that is the perfect square.
So, 25. I know
that is a perfect square.
What's the square root of 25?
What number times itself
is 25? Five.
I'm gonna rewrite that as 5.
Two's not a perfect square.
I'm not gonna take it.
You could put it
in your calculator,
and you'll get a long decimal
that never ends
and never repeats.
Your calculator
will stop it at one point,
but in real life,
it keeps going.
I don't want to take
the square root of 2.
I don't want to write
that long decimal.
I'm keeping it
as the square root of 2.
So, the square root of 50
simplifies to
5 square roots of 2.
And that is my answer. Okay?
This is one of those problems
we have to do a few times
to get the hang of it.

(Describer) She changes the screen.

"Write the numeric expression
in simplest radical form."
They want me to simplify
the square root of 12.
What do we do first?
First, we want to rewrite 12
as a product of its factors.
I want one of those factors
to be a perfect square.
I'm looking in my list
of the ones we use most often
in Algebra I.
Is there any number
in this list
that I can use
as a factor of 12?
You could use 4
because 4 times 3,
that's 12.
I'm gonna rewrite that 12.
Got my square root sign.
Gonna rewrite 12 as 4 times 3.
Notice I always write
the perfect
square factor first
because it helps me
get through.
After I've rewritten this
as the square root
of 4 times 3,
I'm gonna separate it.
I'm gonna write, it's the same
as the square root of 4
times the square root of 3.
That first number,
because of how we're writing it,
you always take
the square root of it.
What is the square root of 4?
It's 2.
Two square roots of 3.
That is my answer for this one.
Some people also say this,
"2 root 3."
"Two square roots of 3."
"Two times
the square root of 3."
There are different ways
to say it,
but it means this: 2 root 3.
Look at another one.
All right.
"Write the numeric expression
in simplest radical form."
This time we have
the square root of 20.
Let's search this list.
Is there any number in here
that's a factor of 20?
If there's more than one,
pick the biggest one.
Did you spot it?
It's 4 again--
4 times 5.
That'll give you 20.
So, I'm gonna rewrite this
as the square root
of 4 times 5.
Now that I've represented it
like that,
I'm gonna chunk it,
separate it.
I'm gonna write it
as the square root of 4
times the square root of 5.
What is the square root of 4?
It's 2.

(Describer) She writes that.

The square root of 5--
5 isn't a perfect square.
I don't want to take it
because I don't want
the decimal.
My answer for this one,
2 times the square root of 5.
Do you see the process
I'm taking every time?
I've got my problem.
I find which one of
these perfect squares
would work as a factor.
I rewrite the problem using
that perfect square factor.
I chunk it up.
I simplify.
Let's try another one.
This time,
they want me to simplify
the square root of 32.
Let's look at our list.
We're thinking
or using the calculator,
whichever you want to do.
Did you find it?
It's 16.
16 times 2, that is 32.
What am I gonna do?
Rewrite that square root
of 32...
as 16 times 2.
Now that I know my factors,
I'm gonna chunk it up.
The square root of 16
times the square root of 2.
What is the square root of 16?
You're right. It's 4.
So, 4 square roots of 2.
And you're all done.
Okay?
All right.
Go and try a few.
Press pause.
Take a few minutes.
Work through these.
Take your time.
Don't press play
until you're ready.
When you want to see
what I got, press play.

(Describer) Title: Write the numeric expression in simplest radical form.
Number One: the square root of 500.
Number Two: the square root of 18.
Number Three: the square root of 75.
Number Four: the square root of 48.

(female describer)
Write the numeric expression
in simplest radical form.
Number 1:
the square root of 500.
Number 2:
the square root of 18.
Number 3:
the square root of 75.
Number 4:
the square root of 48.
Let's see how your answers
compare with mine.
So, square root of 500.
That simplifies...
to 10 square roots of 5.
Square root of 18.
That's 3 square roots of 2.
Square root of 75.
That's 5 square roots of 3.
Square root of 48:
4 square roots of 3.
Okay? Let me show you
how I got those.

(Describer) She switches the screen to the first problem.

Okay.
The square root of 500.
Those two zeros are a hint
that one of the factors is 100,
another number
with two zeros on the end.
I know I can rewrite
the square root of 500...
Need my pen, or I'm not gonna
be writing much of anything.
I can rewrite that
as 100 times 5.

(Describer) ...under a radical.

Remember,
I always write the factor
that's
the perfect square first
to help me move all the way
through this problem.
Now that I know
these are the factors,
I'm gonna chunk them up.
So, the square root of 100
times the square root of 5.
What is the square root
of 100?
Ten. Very good.
So, 10 square roots of 5.
And you're all done.
That's how I got
that first one.
Remember the process.
We write that number under
the radical by its factors,
chunk it up, simplify.
All right,
let's see number two.
Here I needed to simplify
the square root of 18.
Look at our list.
And it's 9.
That's the perfect
square factor for 18.
I'm gonna rewrite
the square root of 18...
as the square root
of 9 times 2,
because 9 times 2
that's gonna give me 18.
Now that I know the factors,
chunk it.
So, the square root of 9
times the square root of 2.
What is the square root of 9?
Good. It's 3.
So, 3 square roots of 2.
That's how I got
that second one.
You notice when I write
my final answer,
I don't write
that multiplication dot?
You don't see it
when you see your answers
in Algebra I.
You know when you see
two terms next to each other,
you're multiplying them
together.
It's not wrong to write it.
The third one.
Whenever I see numbers
like 25, 50, 75, 100,
I always think quarters.
Because in these kinds
of problems,
25 is basically gonna be
the factor you're looking for.
I know that 75 is gonna factor
to 25 times 3.
So, I'm gonna rewrite that
as 25 times 3.
I rewrote it.
Now it's time to chunk it.
The square root of 25
times the square root of 3.
And what is
the square root of 25?
It's 5.
I just bring down
that square root of 3.
That's how I got that third one,
5 root 3.
All done with that one.
Let's look at how I got that
last one. Square root of 48.
So, you look at your list.
This one I gave you
kind of a thinker.
You may have considered
that 4 times 12 is 48,
and you're not wrong.
4 times 12 is 48.
But there's a bigger number
in this list
that's also a factor of 48.
It's 16.
That's actually
the biggest factor.
16 times 3
is gonna give me 48.
So, I'm gonna rewrite this
as the square root
of 16 times 3.
Okay?
Now I'm gonna chunk it.
So, the square root of 16
times the square root of 3.
And what is the square root
of 16? It's 4.
So, 4 root 3.
And that's how I got
that last one. Okay.
All right, guys.
Hope you're feeling comfortable
about how to simplify radicals
involving whole numbers
and square roots.
Hope to see you
back here soon. Bye.

(Describer) Accessibility provided by the U.S. Department of Education.