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For this video we are going to work on square roots.
So at first they may seem a little bit mysterious,
after all, how do you start with something
like a square root of 16 and all of a sudden get 4?
Well the key to really understanding something
like a square root is really understanding
the squaring process.
So for a moment we are just going to take
numbers and square them.
When you have a number, say 3, and you square
it, or raise it to the power of 2.
What you are really saying here is that you
are taking 3, and you are multiplying it by itself
Now the multiplication process is usually
a little bit more familiar, so 3 times 3,
this would give us a number like 9.
And again, let's try this with another number.
So 5 squared, is the same as 5 times 5, which is 25.
Now if you can understand that process, square
roots are going to be great.
The reason is, with square roots we essentially
working in the other direction.
So when I see something like the square root of 9,
I'm thinking, what multiplied by itself
would essentially give me a 9?
Well we already did the squaring process,
so think of this as 3 squared.
9, 3 squared, same thing.
Now the square and the square root will take
care of each other, and we'll just be left
with 3.
And again, another way I can think of this
is, well the number multiplied by itself,
that would give me 9 here, is the 3.
Alright so let's see this process again.
Taking the square root of 25.
Well, just like our work shows, the number
that's multiplied by itself that would give
us 25, is 5.
There we go.
So with that idea in mind, we can do many
many more types of square roots.
Let's try a few more, just to make sure we
have down the process, and then we'll get
into some more difficult ones near the end.
So what number multiplied by itself, would
give us a 36?
Hopefully this one is familiar. 6 times 6
is the same as 36, exactly the same. Looks good.
So I can think of this as 6 squared, and the
answer is 6.
Now you may not need to write down all of
the steps after doing a bunch of these as
practice, and that's OK.
I'm gong to continue writing down all the
steps, so you can see really whats working
in the background.
So here I have the square root of 64, 8 times 8
is equal to 64.
So I can look at this as 8 squared, or the
answer is just 8.
Alright, the numbers are getting a little
bit bigger here.
No worries, we'll continue on.
Let's see, 49 is 7 times 7. So I have the
square root of 7 squared, or 7.
And one more, looks like we have 100 here.
Well that's the same as 10 times 10, or 10 squared.
10.
Nice.
Now some roots you can't necessarily break them
down into two nice number that would multiply
and give you the number underneath the root.
So in some of these more difficult ones, we'll
have to attack them in a slightly different way.
We'll see what can come out of the root,
and maybe what has to stay underneath there.
Let's take the first one.
If I look at 18, and I'm thinking of a number
that multiplies by itself to give me 18, nothing
really comes to mind.
After all, 4 times 4 is 16, that's too small,
5 times 5 is 25 and that's way too big.
So what I know about this guy is that what
ever number it is, its somewhere between 4 and 5.
Now the interesting thing is I could break
18 into two numbers, let's say 9 and 2. Now
those are not the same, so I'm not done with
this root just yet.
But notice this 9, was in some of our earlier examples.
That is something that we can take the square root of.
And the neat part about square roots, is we
can split them up over multiplication.
So I can think of this as two different roots.
I'm looking at the square root of 9 multiplied
by the square root of 2.
Now this guy, we can do that guy just fine.
Square root of 9 is 3, because 3 times 3 would
give us that 9.
So here I have a simplified square root, which
is simply 3 times the square root of 2.
Now the square root of 2, I'm just going to
leave as it is, because, again I can't think
of two numbers exactly the same that when
multiplied would give me that 2.
So we consider this the simplified one.
Let's try that again with another one.
This one is the square root of 24.
What you immediately what to do, if you can't
find two numbers that are the same, is to
just think of numbers that are squared numbers
that do divide into 24.
So I start thinking, let's see can I divide
into 24. Oh, 4 goes in there. And 4 is one
I can easily take the square root of.
So square root of 4, multiplied by the square
root of 6. Or 2 times the square root of 6.
Not bad.
Now larger numbers such as 144, sometimes
you have to do a little bit of breaking down
in order to figure out how to attack them
using these roots.
If you have a very large number, try seeing
what smaller numbers will divide into it.
For example, 144, I know that's the same as
2 times 72.
Neither one of those are one of our squared
numbers, but we can keep breaking this down.
72 is the same has 2 times 36.
Now as I break this down, I'm constantly looking
for squared numbers, and I think we've finally
got one here.
36 is the same as 6 squared, or 6 times 6.
So I can take that number, along with the
other numbers we've broken it down into and
put those underneath our square root. So 2
times 2 is the same as 4.
That'll take these two.
And a 36.
Alright, so by looking at some of those smaller
numbers now I can really look at this problem
as the square root of 4, and the square root
of 36, and both of these aren't that bad.
We've already done the square root of 4, a
couple of times, this is just 2.
And the square root of 36 is 6.
Of course because 6 times 6 is equal to 36.
One last step. 12
So what this really means in the grand scheme
of things is that when I multiply 12 by itself,
12 squared, it should equal 144.
Or in the other direction, the square root
of 144 is equal to 12.
And there you go!
That's just the basics about square roots,
and simplifying the more difficult ones.
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