Uploaded by MuchoMath on 21.03.2009

Transcript:

>> Professor Perez: Hey!

This is Professor Perez from Saddleback College.

Today, we're going to work on square roots.

Don't get scared!

Anyway, before we get started, we've got to get Charlie out.

He better be ready to go!

Hey, Charlie, you ready to go?

>> Charlie: Yeah.

>> Professor Perez: We're doing square roots today!

That's right.

These problems are radical!

Anyway, let's get started here.

Right there.

Okay, pay attention Charlie.

What number do we square to get 16?

>> Charlie: 4.

>> Professor Perez: That's right.

You just did a square root!

>> Charlie: What?

>> Professor Perez: Anyway...well pay attention Charlie.

If somebody asks you what number do we square to get 16, it's 4, because 4 squared is 16.

But it is also negative 4 because negative 4 squared is 16, right?

Okay. Now let's talk about the square root.

What is the square root of 16, Charlie?

Well watch this.

We have a 16 and we have our radical symbol there.

That radical symbol with the 16 underneath is saying, what is the square root of 16?

So, when somebody asks you, what is the square root of 16,

you're thinking what number do you square to get 16, and it's 4, right?

But we also saw that negative 4 squared equals 16.

Well, when you use a radical symbol and say, what is the square root of 16,

we give what we call the principal root which is always positive.

One thing to remember, whenever you take the square root of a number,

the answer will always be positive so keep that in mind.

Okay, so the square root of 16 is 4 because 4 squared is 16 Charlie.

All right.

Let's look at some other square roots.

Now, before we move on though, let's look at what we call perfect squares.

Watch. Charlie, what's 0 squared?

>> Charlie: 0.

>> Professor Perez: 0.

That's right.

So 0's considered a perfect square.

Which means when you're asked, what is the square root of 0,

the answer will be 0 because 0 squared is 0.

It's so easy it's confusing.

Now watch this one, Charlie!

1 is a perfect square because 1 squared is 1 therefore the square root of 1 will be 1, right?

You've got to think about that one.

Now, here's this one.

4 is also a perfect square because 2 squared is 4.

Therefore, Charlie, what is the square root of 4, what's the answer?

>> Charlie: 2.

>> Professor Perez: 2.

That's right.

And other perfect squares are 9 because 3 squared is 9,

we have 16 because 4 squared is 16.

25 is a perfect square because 5 squared is 25

and 36 is a perfect square because 6 squared is 36.

And of course, there's a whole lot more, right?

We'll stop there for now.

All right, Charlie, let's do some problems.

What is the square root of 36, Charlie?

>> Charlie: 6.

>> Professor Perez: 6, because 6 squared is 36.

Remember, square roots we always give positive answers, the principal root.

Okay Charlie, how about the square root of 49?

>> Charlie: 7.

>> Professor Perez: 7, because 7 squared is 49.

Now Charlie, what's the square root of 144?

>> Charlie: 12.

>> Professor Perez: 12, because 12 squared is 144, right?

Okay, we're dealing with perfect squares here.

Charlie, what's the square root of 64?

>> Charlie: 8.

>> Professor Perez: 8, because 8 times 8 is 64.

What's the square root of 100?

>> Charlie: 10.

>> Professor Perez: 10, because 10 squared is 100.

Now pay attention to this one Charlie.

Don't get scared!

What's the square root of 10?

Uh-huh, that's right.

10 is not a perfect square, therefore we have to use a calculator.

Now the square root of 9 is 3, so the square root of 10 should be a little bit more than 3,

and Charlie, if you take your calculator out and calculate the square root of 10,

you will get a number 3.162 which is a little bit bigger than 3, right?

In this class at this time, we're only going to deal with square roots of perfect squares, okay?

All right, now, Charlie, we're going to look at what we call a right triangle.

This is a word problem.

Now, a right triangle means it has a right angle.

And the side length opposite the right angle, which is that 13,

Charlie, is called the hypotenuse.

And the 5 and that a are referred to as the legs of the triangle.

Now, when you have a right triangle, we have a theorem which is called the Pythagorean Theorem.

And it's only applied to right triangles like this and that theorem is up there.

c squared equals a squared plus b squared.

Now we're going to use that theorem to solve for that side length a, right?

Find that side length a. Okay?

Now, in the theorem, c represents the side length opposite the right angle

which is called the hypotenuse, the longest side.

The a and the b represent what we call the legs of the triangle.

And here we have it labeled there.

The hypotenuse is 13, Charlie, the legs are 5 inches and a and we're trying

to find a. So here we go, Charlie.

There's the theorem.

c squared equals a squared plus b squared, our c is that 13, our a is that leg right there,

and the other leg we'll call that 5 okay?

Now, Charlie, what's 13 squared?

>> Charlie: 169.

>> Professor Perez: That's a tough one there, and 5 squared is 25.

Now, we're going to solve for a squared.

And so, what do we do?

>> Charlie: Subtract 25.

>> Professor Perez: That's right, and so we get 169 subtract 25 is?

>> Charlie: 144.

>> Professor Perez: 144.

That's right.

And on the right hand side, we're left with a squared.

So 144 equals a squared, Charlie.

So, Charlie, what number do you square to get 144?

>> Charlie: 12.

>> Professor Perez: That's right.

So, 12 should equal a. Well, if 12 equals a, that means a equals 12

and so that's our missing side length, and you can see, if you take 12 squared

which is 144 plus 5 squared which is 25 you'll get 169.

And if you take the square root of 169, you get 13 so the theorem does work.

That's a tough problem there, right?

Okay, we're going to do more of those later.

Anyway, let's do some square root problems now.

Here we go, Charlie.

The square root of 4 plus the square root of 9.

We don't need calculators for this.

What's the square root of 4?

>> Charlie: 2.

>> Professor Perez: And the square root of 9?

>> Charlie: 3.

>> Professor Perez: And what's 2 plus 3?

>> Charlie: 5.

>> Professor Perez: Very nice!

Okay, here's another one.

Square root of 16 subtract the square root of 25.

What's the square root of 16?

>> Charlie: 4.

>> Professor Perez: Square root of 25?

>> Charlie: 5.

>> Professor Perez: Very nice.

What's 4 subtract 5?

>> Charlie: Negative 1.

>> Professor Perez: Negative 1.

No calculator required.

Let's do another one Charlie, now don't get scared.

3 square roots of 64 means 3 times the square root of 64.

2 square roots of 49 means 2 times the square root of 49, Charlie.

So, we have 3 times, what's the square root of 64?

>> Charlie: 8.

>> Professor Perez: 8, subtract 2 times the square root of 49 which is?

>> Charlie: 7.

>> Professor Perez: 7.

There we go.

And 3 times 8 is?

>> Charlie: 24.

>> Professor Perez: 2 times 7?

>> Charlie: 14.

>> Professor Perez: And 24 subtract 14?

>> Charlie: 10.

>> Professor Perez: Is 10.

Very nice there Charlie!

All right, let's do one more.

Now, don't get scared.

Notice we have fractions underneath the radical sign.

So, what fraction do we square to get 25 over 9, Charlie?

>> Charlie: 5 thirds?

>> Professor Perez: That's right.

A lot of people like to think, well the square root of 25 is 5 and the square root

of 9 is 3, so that is 5 thirds, right?

Because 5 thirds squared means 5 thirds times 5 thirds which is 25 over 9, right?

So the square root of 25 over 9 is 5 thirds.

What's the square root of 81 over 16, Charlie?

>> Charlie: 9 over 4.

>> Professor Perez: Very nice there!

9 over 4. And here, we have to find the LCD which is?

>> Charlie: 12.

>> Professor Perez: Okay, now we're going to use some Kung-Fu.

Now remember, I'll work this one for you.

3 goes into 12 4 times, 4 times 5 is 20.

4 goes into 12 3 times, 3 times 9 is 27.

And remember, it's over 12 because these are twelfths.

And 20 plus 27 is 47, and it's over 12.

That's our answer, 47 twelfths.

That was some good Kung-Fu there!

Now notice, we did all of these without a calculator, right?

So anyway, you better be ready to do these without a calculator on the next exam.

Anyway, that's if for now.

We'll see you again soon!

This is Professor Perez from Saddleback College.

Today, we're going to work on square roots.

Don't get scared!

Anyway, before we get started, we've got to get Charlie out.

He better be ready to go!

Hey, Charlie, you ready to go?

>> Charlie: Yeah.

>> Professor Perez: We're doing square roots today!

That's right.

These problems are radical!

Anyway, let's get started here.

Right there.

Okay, pay attention Charlie.

What number do we square to get 16?

>> Charlie: 4.

>> Professor Perez: That's right.

You just did a square root!

>> Charlie: What?

>> Professor Perez: Anyway...well pay attention Charlie.

If somebody asks you what number do we square to get 16, it's 4, because 4 squared is 16.

But it is also negative 4 because negative 4 squared is 16, right?

Okay. Now let's talk about the square root.

What is the square root of 16, Charlie?

Well watch this.

We have a 16 and we have our radical symbol there.

That radical symbol with the 16 underneath is saying, what is the square root of 16?

So, when somebody asks you, what is the square root of 16,

you're thinking what number do you square to get 16, and it's 4, right?

But we also saw that negative 4 squared equals 16.

Well, when you use a radical symbol and say, what is the square root of 16,

we give what we call the principal root which is always positive.

One thing to remember, whenever you take the square root of a number,

the answer will always be positive so keep that in mind.

Okay, so the square root of 16 is 4 because 4 squared is 16 Charlie.

All right.

Let's look at some other square roots.

Now, before we move on though, let's look at what we call perfect squares.

Watch. Charlie, what's 0 squared?

>> Charlie: 0.

>> Professor Perez: 0.

That's right.

So 0's considered a perfect square.

Which means when you're asked, what is the square root of 0,

the answer will be 0 because 0 squared is 0.

It's so easy it's confusing.

Now watch this one, Charlie!

1 is a perfect square because 1 squared is 1 therefore the square root of 1 will be 1, right?

You've got to think about that one.

Now, here's this one.

4 is also a perfect square because 2 squared is 4.

Therefore, Charlie, what is the square root of 4, what's the answer?

>> Charlie: 2.

>> Professor Perez: 2.

That's right.

And other perfect squares are 9 because 3 squared is 9,

we have 16 because 4 squared is 16.

25 is a perfect square because 5 squared is 25

and 36 is a perfect square because 6 squared is 36.

And of course, there's a whole lot more, right?

We'll stop there for now.

All right, Charlie, let's do some problems.

What is the square root of 36, Charlie?

>> Charlie: 6.

>> Professor Perez: 6, because 6 squared is 36.

Remember, square roots we always give positive answers, the principal root.

Okay Charlie, how about the square root of 49?

>> Charlie: 7.

>> Professor Perez: 7, because 7 squared is 49.

Now Charlie, what's the square root of 144?

>> Charlie: 12.

>> Professor Perez: 12, because 12 squared is 144, right?

Okay, we're dealing with perfect squares here.

Charlie, what's the square root of 64?

>> Charlie: 8.

>> Professor Perez: 8, because 8 times 8 is 64.

What's the square root of 100?

>> Charlie: 10.

>> Professor Perez: 10, because 10 squared is 100.

Now pay attention to this one Charlie.

Don't get scared!

What's the square root of 10?

Uh-huh, that's right.

10 is not a perfect square, therefore we have to use a calculator.

Now the square root of 9 is 3, so the square root of 10 should be a little bit more than 3,

and Charlie, if you take your calculator out and calculate the square root of 10,

you will get a number 3.162 which is a little bit bigger than 3, right?

In this class at this time, we're only going to deal with square roots of perfect squares, okay?

All right, now, Charlie, we're going to look at what we call a right triangle.

This is a word problem.

Now, a right triangle means it has a right angle.

And the side length opposite the right angle, which is that 13,

Charlie, is called the hypotenuse.

And the 5 and that a are referred to as the legs of the triangle.

Now, when you have a right triangle, we have a theorem which is called the Pythagorean Theorem.

And it's only applied to right triangles like this and that theorem is up there.

c squared equals a squared plus b squared.

Now we're going to use that theorem to solve for that side length a, right?

Find that side length a. Okay?

Now, in the theorem, c represents the side length opposite the right angle

which is called the hypotenuse, the longest side.

The a and the b represent what we call the legs of the triangle.

And here we have it labeled there.

The hypotenuse is 13, Charlie, the legs are 5 inches and a and we're trying

to find a. So here we go, Charlie.

There's the theorem.

c squared equals a squared plus b squared, our c is that 13, our a is that leg right there,

and the other leg we'll call that 5 okay?

Now, Charlie, what's 13 squared?

>> Charlie: 169.

>> Professor Perez: That's a tough one there, and 5 squared is 25.

Now, we're going to solve for a squared.

And so, what do we do?

>> Charlie: Subtract 25.

>> Professor Perez: That's right, and so we get 169 subtract 25 is?

>> Charlie: 144.

>> Professor Perez: 144.

That's right.

And on the right hand side, we're left with a squared.

So 144 equals a squared, Charlie.

So, Charlie, what number do you square to get 144?

>> Charlie: 12.

>> Professor Perez: That's right.

So, 12 should equal a. Well, if 12 equals a, that means a equals 12

and so that's our missing side length, and you can see, if you take 12 squared

which is 144 plus 5 squared which is 25 you'll get 169.

And if you take the square root of 169, you get 13 so the theorem does work.

That's a tough problem there, right?

Okay, we're going to do more of those later.

Anyway, let's do some square root problems now.

Here we go, Charlie.

The square root of 4 plus the square root of 9.

We don't need calculators for this.

What's the square root of 4?

>> Charlie: 2.

>> Professor Perez: And the square root of 9?

>> Charlie: 3.

>> Professor Perez: And what's 2 plus 3?

>> Charlie: 5.

>> Professor Perez: Very nice!

Okay, here's another one.

Square root of 16 subtract the square root of 25.

What's the square root of 16?

>> Charlie: 4.

>> Professor Perez: Square root of 25?

>> Charlie: 5.

>> Professor Perez: Very nice.

What's 4 subtract 5?

>> Charlie: Negative 1.

>> Professor Perez: Negative 1.

No calculator required.

Let's do another one Charlie, now don't get scared.

3 square roots of 64 means 3 times the square root of 64.

2 square roots of 49 means 2 times the square root of 49, Charlie.

So, we have 3 times, what's the square root of 64?

>> Charlie: 8.

>> Professor Perez: 8, subtract 2 times the square root of 49 which is?

>> Charlie: 7.

>> Professor Perez: 7.

There we go.

And 3 times 8 is?

>> Charlie: 24.

>> Professor Perez: 2 times 7?

>> Charlie: 14.

>> Professor Perez: And 24 subtract 14?

>> Charlie: 10.

>> Professor Perez: Is 10.

Very nice there Charlie!

All right, let's do one more.

Now, don't get scared.

Notice we have fractions underneath the radical sign.

So, what fraction do we square to get 25 over 9, Charlie?

>> Charlie: 5 thirds?

>> Professor Perez: That's right.

A lot of people like to think, well the square root of 25 is 5 and the square root

of 9 is 3, so that is 5 thirds, right?

Because 5 thirds squared means 5 thirds times 5 thirds which is 25 over 9, right?

So the square root of 25 over 9 is 5 thirds.

What's the square root of 81 over 16, Charlie?

>> Charlie: 9 over 4.

>> Professor Perez: Very nice there!

9 over 4. And here, we have to find the LCD which is?

>> Charlie: 12.

>> Professor Perez: Okay, now we're going to use some Kung-Fu.

Now remember, I'll work this one for you.

3 goes into 12 4 times, 4 times 5 is 20.

4 goes into 12 3 times, 3 times 9 is 27.

And remember, it's over 12 because these are twelfths.

And 20 plus 27 is 47, and it's over 12.

That's our answer, 47 twelfths.

That was some good Kung-Fu there!

Now notice, we did all of these without a calculator, right?

So anyway, you better be ready to do these without a calculator on the next exam.

Anyway, that's if for now.

We'll see you again soon!