Tip:
Highlight text to annotate it
X
>> This is part seven of square roots and radicals.
I'm gonna do a little bit on multiplying square roots again
where we might have a little bit larger numbers.
So we're going to use this property.
If a is greater than or equal to 0 and b is greater than or equal
to 0, then the square root of a times the square root
of b equals the square root of ab.
We can use that but sometimes it's easier not
to actually multiply a times b. Here's an example.
Square root of 21 times square root of 14.
I could go ahead and multiply those two numbers together,
but I get kind of a large number.
And then it's actually going to be harder to figure
out what the perfect square factor of that larger number is.
So instead, when you see
that you're multiplying two numbers look and see
if there's a common factor in each of them.
And 7 goes into both numbers.
So how about if I write 21 as 3 times 7.
So notice I'm putting the common factor as the second factor.
And then for 14, that's 7 times 2.
This one I'm putting the 7 first and the reason is
so I can see those two 7s right next to each other.
Well that means we've got a perfect square
on that the 7 times 7 which I can bring out.
And so my answer is 7.
That's on the outside and 3 times 2 is 6.
Now, you would have gotten the same answer
if you multiplied 21 times 14
and then factored it as 49 times 6.
Okay? Or by prime factoring
and getting 3 times 7 times 7 times 2.
It's just why do that extra arithmetic
if it's not necessary.
All right.
Let's do another problem.
All right.
Square root of 35 times the square root of 55.
Hmm. Can you figure out the common factor for both of those?
5? So I'm gonna write 35 at 7 times 5 and 55 as 5 times 11
and that way I can see I've got two factors of 5
so a 5 pops out, right?
That's the square root of 5 times 5.
So 5 comes out and what's left inside is 7 times 11
which is 77.
So see how that's working?
This also works if you have something with variables.
For instance, square root of 22x cubed so the square root
of 22x cubed times the square root of 33x
so first I'm gonna deal with the number parts.
So the common factor in both 22 and 33 is 11.
So how about I write 22 as 2 times 11 and 33 as 11 times 3.
And now, I've got x cubed times x. That'll give me x
to the fourth which is a perfect square.
So what can I do here?
I can take out 11 times 11 that means 11 pops out.
And for x to the fourth an x square pops out.
So that gives me on the outside 11x squared times the square
root of 2 times 3 is 6.
All right.
What about if we had square root
of 6x cubed times the square root of 2x, let's see,
to the fifth times the square root of how
about 5x to the seventh, okay?
All right.
So what do I have here?
I've got for the numbers 6 times 2 times 5.
I'm looking to see if any of them have a common factor.
And well, between the 6 and the 2 I see it.
So how about I rewrite it -- the 6 is 3 times 2 then I have
that 2 and I've also got the 5.
All right.
So I can see at least I'll be able to pull a 2 out.
Now, what about all the x's.
I've got x cubed times x to the fifth times x to the 7th.
So remember you add exponents.
That's gonna give you x to the 15th.
But that's not a perfect square so the trick is to say, well,
I know it's x to the 15th.
I want to write that as x to the 14th times x. So I can pull
out this perfect square.
So from both of those 2s I can pull out a 2 and the square root
of x to the 14 is x to the 7th.
So therefore, on the outside I have 2x to the 7th
and the inside I've got 3 times 5 times x. That's 15x.
How about this one?
12x cubed, I mean the square root
of 12x cubed times the square of 3x.
So with the numbers, now, if you want you could just go ahead
and do 12 times 3 and get square root of 36.
That's perfectly fine or you could do it
that factoring method I've been doing.
I'm gonna just do 36 because 12 times 3 is so simple.
And how about x cubed times x that'll give me x to the fourth.
So let's see, these both are perfect squares so a six comes
out and an x square comes out and here's the trick,
remember if everything comes out there's just 6x squared.
You don't really see it but there's kind
of like an invisible one in here if you want,
so it's not like there's nothing, but the square root
of one is one so you would have 6x squared times one
which is just still just 6x squared.
Now, you could've done this 12x cubed.
You could have written that as you said, well, between 12
and 3 there's a factor of 3, right?
So you might have written it as 4 times 3 times that other 3
and x to the fourth and then hopefully you'll notice
that the square root of 4 is 2.
Here's 2, 3 so I can take out a 3 and from x
to the fourth I could take out an x squared.
Notice you're still going to get that 6x squared.
What about if you have a number in front
like negative 3x squared,
square root of 10 times 5 square root of 30.
Okay? Well, as usual the numbers outside in front
of the square roots get multiplied together
so I have the negative 3 times 5, negative 15x squared
and now we have to deal with what's inside the square root
and again, you could just do 10 times 30 and then figure
out what the perfect square is in there
or I could say what's the common factor and it's 10, right?
So 10 I'm just going to keep as 10
and 30 I'm gonna write as 10 times 3.
So I could take out from these two 10s this one 10
and on the outside I have negative 15 times 10.
So it's negative 150x squared and this time I'm left
with a square root of 3.
You're thinking, good grief, does it ever end?
Well, not really.
There's just all sorts of problems that could come up
and so you've got more than one way of coming up with the answer
and this is a trick to making your arithmetic easier.