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>> This is part 3 of square roots and radicals
and I'm just going to go
over how we might approximate the square root of a number
that is not a perfect square.
For instance, we do know the perfect squares, remember,
like 0 squared, 1 squared, 2 squared, 3 squared, 4 squared,
etc. So, this goes on forever.
So, 0, 1, 4, 9, 16, those, those are called perfect squares
because they are squares of whole numbers, okay?
So starting with the whole number 0, 1, 2, 3, 4,
etc. and that's how we decide what the perfect squares are.
Alright, so the question is, what if we take something
and we want to take the square root of it,
but it's not a perfect square?
So, first of all, like square root 4,
that is a perfect square, so it's 2, square root of 25.
It's 5, etc., right?
No problem.
The problem is what do you do
with the square root of let's say, 19?
What the heck is that?
Okay well, the first thing is to note
that 19 is not a perfect square because there is no number
when you multiply it by itself, you are going to get 19,
in other words no whole number.
So, if it's not a whole number in here,
it means it's not rational.
So, we'd say, this an irrational number.
But we can estimate approximately where it is.
So, we're going to look at the square of 19 again.
And we're going to think of it this way.
We're going to try to figure
out approximately what 2 numbers it's in between.
You might even think about where would it be on a number line?
Where would square root of 19 be on the number line?
So, let's see, this is 0, you know 1, 2, 3, 4, 5,
etc. I don't know where it is.
Well, We're going to try to think
of the perfect square just before 19
and the perfect square right after 19.
So of course you have to know your perfect squares.
I am going to list this over here again.
Starting with 4, you know, the 0 and 1, those are the,
well I'll start with 1, 1 squared, and then 4, 9, 16, 25,
36, notice I am just doing 5 squared, 6 squared, 7 squared,
8 squared, 9 squared, etc. You know, this goes on and on
and on, the rest of the perfect squares.
So, 19 is between which one of those?
Which 2 of those numbers?
Well it's in between 16, right?
And 25, 19 the number 19, which is underneath the square root,
is in between 16 and 25.
So, therefore the square root of 16 is smaller
than the square root of 19, which is smaller
than the square root of 25.
So, square root of 16, well, that's a perfect square, 16,
so square root of 16 is 4, right?
And the square root of 25 we can simplify,
so therefore we know square root of 19 is somewhere
between the number 4 and 5.
So, you know, we know it's somewhere in this area, right?
Square root of 19 has to be somewhere in there.
And you could approximate it, you know, I don't know exactly
where it is, but it looks like it might be a little bit close
to 4, you know maybe it's here for instance.
That might be the square root of 19.
Alright, so that's the technique
so let's do it on another number.
Let's say we wanted to know where the square root of 40 was.
>> Okay? 40 is not a perfect square
so what we do is think of, hmm,
what's the perfect square just below 40?
And actually he did, that one you're going
to go automatically get the next one.
The square root of 36 is 6.
So, if we're trying to find 2, we're trying
to find 2 whole numbers it's in between.
So, it's in between 6 and 7, right?
Let's just stop right there.
Or if you want, you can put in the 49.
So basically you can say the square root of 40 is between,
oops, I don't know what's going on with my tablet here.
Square root of 40 is between 6 and 7,
so we're not getting exactly, I'm not saying it's 6.5 or 6.2
or 6.9, etc., although we can do that as well,
right now I'm just saying how are we going to figure
out a couple of numbers it's in between.
Alright, so what I want you to tell me is I want you to fill
in the blanks with whole numbers.
So we're looking for consecutive whole numbers.
Fill in the blanks with consecutive whole numbers
to approximate the square root.
Alright, so the square root of 12, you have to think
of the perfect square under 12.
Well that would be the square root of 9, right?
So this means, you should be in between 3 and 4, makes sense?
And the reason over here is because the square root
of 9 is less than the square root of 12, which is less
than the square root of 16.
That's the reasoning, okay, how about square root of 70,
you think, the perfect square under 70, that's 64,
so it must be in between 8 and 9.
Same reasoning because square root of 64 is less
than square root of 70, is less than squared 81,
so that's the reason, and then now you've approximately,
approximately, know where the square root of 70 is.
And it's you know, probably it's 8 point something.
Now what you can do is actually get out your calculator,
put in the square root of 70 and see what comes up.
So to check yourself, you can actually use a calculator,
do the square root of 70 and my calculator I have 8.366600265
and it really does go on forever and ever.
But, you can see it's a little bigger than 8, right?
So it makes sense that we estimated correctly the square
root of 70 is between 8 and 9.
But here's the thing to really pay attention to,
this is not exactly square root of 70, 8.37 or 8.366600265,
it really does go on forever and ever.
So if you didn't note that that's just an approximation,
if I took 8.3666, just say I just went that far and I squared
that in my calculator, I get 69.99999556, so you know what,
it's pretty darn close, but it's not exactly 70, right?
So when you are using the calculator it gives
you approximations.
When I'm saying it's between 8 and 9,
that's also simply an approximation.
The exact answer for the square root of 70 is just
as you see it, the square root of 70,
the calculator will only give you an approximation,
pretty good approximation too.
Alright, let's do this.
Let's use a calculator to approximate each square root
to the nearest thousand.
Alright, so why don't you try that with your calculator.
Alright, and I'm going to do to the nearest thousands.
You should have gotten 3.873,
remember the thousands is 3 places after your decimal point.
Now on you calculator it probably comes
out to 3.87298 blah, blah, blah, but remember you're going
to have to round that up to 3 at the thousands place.
Alright, next step, take that approximation, 3.873,
and square it, so you're going to take 3.873
and square it, and I have 15.000129.
So, it's not exactly 15, it's a little bit bigger than 15,
but it was a pretty good approximation, just 3.873,
but I want to make sure you understand,
that that's not exactly the square root of 15.
Alright, try it for the square root of 30.
[ Typing sounds ]
>> If you approximate, you should get 5.477
and so now we're going to take that 5.477 in your calculator
and square it, so 5.477 squared, is exactly 29.997529.
That's just a little bit under 30, but it's pretty close right?
It's just not exact.
Notice when I rounded up, like for the square root 15,
when I squared that number it was a little over 15.
For the square root of 30, when I kept it the same,
I did not round up to the thousands place, when I squared
that approximation, I get a number just a little bit less
than 30.
Good approximations,
but do remember these are approximations,
they are not exact.
And if you have a number for the square root,
like 2 times the square root of 3,
this means 2 times the square root of 3.
So to get this answer I am going to have
to do 2 times an approximation for the square root of 3.
So let's use, let's go to the nearest thousands again
for the square root 3.
If you approximate, you get 1.732.
And so, if I multiply that by 2, 2 times 1.732, you get 3.464,
so that's a pretty good approximation, right?
But, if you wouldn't have approximated first, alright,
in other words you put in the square root of 3
into your calculator, you leave it and you multiply by 2,
you will get 3.464101615, it's not exactly the same,
it's still pretty close.
So it kind of depends if you approximate and then multiply it
by 2 or you leave it the square root of 2 in your calculator
and then do the approximation.
Alright, so that's how to approximate square roots.