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>> This is part four of square roots and radicals,
and we're gonna be dealing with multiplying square roots.
First of all, recall from part one, or part two, I think,
if X is greater than zero, so as long as we're not dealing
with negative values of X, then the square root
of X squared is equal to X, and that's to make sure that,
remember when you see the square root symbol it means the
principal square root.
That makes sure you don't get a negative number.
Alright. Now we're gonna have a new statement.
What about if we don't have any restrictions?
If I just have the square root of X times the square root of X?
What does that equal?
It always equals X. Doesn't matter
if X is positive or negative.
Okay. No restrictions on X. Now, you could also read
that as the square root of X squared.
That means the same thing, doesn't it,
because the square root of X squared means square root
of x times square root of X. So that also equals X. Okay?
But that's different.
This is different.
Because, I've got the X squared underneath the radical sign.
So it's a real slight, subtle difference.
Now, it doesn't matter what's underneath there.
So if I have the square root of seven times the square root
of seven, using that definition I just gave you,
if you have the square to the same thing, times itself,
then you have just the number underneath the square
root symbol.
So it's seven.
Now, what if I have the square root of one hundred
and three times the square root of one hundred and three?
Well, that's just a hundred three.
And, in fact, if you have the square root of some junk,
whatever's underneath, times the same--
times the square root of the exact same junk,
you just get all that junk.
Whatever happens to be under there.
So, if I have, you know,
five X cubed Y underneath the square root times another five X
cubed Y, I'm just gonna get the five X cubed Y. Now you could
also write it like this.
What if you have a bunch of stuff,
underneath the square root, and you're squaring it sort
of like an end does it.
Squaring and taking the square root, well actually,
if you take the square root first and then square it,
you get right back to where you started,
which will be this stuff.
If you've got the square root of nineteen
and you're squaring it, you get nineteen.
Right. So everybody get the idea?
This is the multiplication property for square roots.
If A is greater than zero, greater than or equal to,
and B is greater than or equal to zero, then the square root
of A times the square root of B equals the square root of AB,
which means, square root of A times B. Now, it's important
that A and B are not negative numbers to use this.
Okay? We're not going to work with numbers that aren't real
when there's a minus sign underneath the square root.
Okay, so let's look at an example.
What's the square root of three times the square root of seven?
Well, you just multiply three times seven
to get the square root of twenty-one.
Easy, right?
Now, how about the square root
of two times the square root of eight?
Well, it's the square root of two times eight,
which is the square root of sixteen.
And then what's the square root of sixteen?
Four. Okay.
What if I had the square root
of five times the square root of five?
You know you shouldn't multiply those together
because we can go back to our first property in this video
and you've got the square root of like XM square root of X,
the answer should just be five.
But if you did it the long way, you might of written square root
of five times five, which is square root of twenty-five,
and eventually you would get five, right?
That would be the long way to do it.
So if you don't have to multiply them together,
and then take the square root, it's much easier
to just do it this way.
Especially if it's not so easy to multiply in your head.
Like if you had the square root
of forty-seven times the square root of forth-seven.
Instead of using the multiplication property,
just think, oh, that's just gonna be forty-seven.
Why do forty-seven times forty-seven
and get this big old number, and then try to figure
out what the square root is?
You're right back to where you began with.
Here's some problems for you to try.
So put the video on pause and multiply,
and then simplify if you can.
Right. Let's do the first one.
Well, you've got to do the square root
of three times square root of five,
that's just the square root of three times five,
or square root of fifteen.
Everybody okay there?
Alright. Number two.
You're gonna multiply those two together, so you have A times A
to the fifth underneath the square root,
which is the same thing as A to the sixth.
But that can be simplified further,
because now you have an even exponent on your variable,
so remember you take half of the exponent.
So the answer to that is A cubed.
Alright. What about number three?
Well, you've got three things to multiply,
so you have to multiply two times three times seven,
which is forty-two.
So this is just the square root of forty-two.
Okay. Last one here.
Gonna go underneath, so I'll have plenty of room here.
So I have to multiply all this together, right,
so I have to do two X cubed, times eighteen X to the fifth.
And that gives me thirty-six X to the eighth.
And now, each of those are perfect squares.
Thirty-six is a perfect square,
and X to the eighth is a perfect square
because it has an even exponent.
So the square root for thirty-six is six,
and the square root of X to the eighth is X to the fourth.
Let's do one more.
How about if we had the square root of twenty-three X
to the fifth, times the square root
of twenty-three X to the seventh.
Hopefully you notice right off the bat with that
that you have two twenty-threes under there,
so that's just gonna be twenty-three.
I'm not gonna multiply that out even, I'm just gonna write
that as twenty-three times twenty-three.
I don't feel like doing that multiplication,
but let's do the X to the fifth times X to the seventh.
So, I've got the twenty-three times twenty-three.
Right? Times X to the twelfth.
So, the X to the twelfth is a perfect square.
And then, if you're taking the square root
of twenty-three times twenty-three,
that's just gonna be a twenty-three.
Right? And now for X to the twelfth,
you're gonna take half the exponents,
so it'll be twenty-three X to the sixth.
Alright. Now if you wanted
to multiply twenty-three times twenty-three,
you're perfectly welcome to do that.
And then you'd have to realize that the square root
of that number you get, is twenty-three.
So, this is how to multiply square roots and how
to simplify, and now what we're going to do is
in the next video, figure out how to simplify something
like the square root of eight.
Eight's not a perfect square right?
By using this rule for multiplying the square roots,
and the trick will be, there's a little teaser here,
is to see if you can factor what's underneath the square
root as a perfect square root times something else.
So, eight could be written as four times two.
Notice how I've written it
as a perfect square times something else.
Four times two.
And if I use my rule for multiplying going backwards,
that must have been the square root
of four times the square root of two.
Square root of four is just two, isn't it?
And voila, so that's a little preview.
We're gonna have to practice writing numbers underneath the
square roots as perfect squares and something else, if possible.
Doesn't always work, like square root of fourteen,
the only way you could factor that is two times seven,
and neither of those is a perfect square.
So, square root of fourteen is simplified, whereas,
square root of eight is not.
But square root of eight can be written as two squared,
so you would just simplify.
Okay, that's a little preview for you for the next video.