Approximating Square Roots

u16_l1_t1_we2 Approximating Square Roots

We are asked to approximate the principle
root,

or the positive square root 45, to the
hundredths

place, and I'm assuming they don't want us
to

use a calculator because that would be too
easy.

So let's see if we can approximate this,

just with our pen and paper right over
here.

So square root of 45, so the square root
of 45, or the principle root of 45.

45 is not a, 45 is not a perfect square.

It's definitely not a perfect square.

But, we know, let's see what are the
perfect squares around it?

We know that it is going to be less than,

the next perfect square above 45 is going
to be.

Let's see it's going to be 49, because
that is 7 times 7.

So, it's less than the square to 49.

and it's greater than, the square root of
36.

And so the square root of 36, the
principle root of 36 I should say, is

6 and the principle root, the principle
root of 49 is, 7.

So this value right over here is going to
be between.

It's going to be between 6 and 7.

And if we look at it, it's only 4 away

from 49 and it's 9 away, it's 9 away from
36.

So it looks like, so the difference
between 36 and 49 is.

It's 13, so it's a total 13 gap between
the, 6 squared and

7 squared and this is, this is 9 of the
way through it.

So just as a kind of an approximation,
maybe and it's not

going to work out perfectly because we're
squaring it, this isn't a

linear relationship, but it's going to be
closer to, 7 than it's

going to be to 6 and it, at least the
square or the.

45 is 9 13th of the way.

So we could try, we could try, lets see,
it looks like a, that's about, that's

about two thirds of the way, so lets try
6.7, lets try 6.7 as a guess.

Just based on, 0.7 it's about two thirds,
it looks like about the same.

And actually we could calculate this right
here if

we want, actually let's do that just for
fun.

So, 9 13th as a decimal is going to be
what.

It's going to be 13 into 9 we are going to
put some decimal places

right over here, 13 doesn't go into 9, 13
does go into 90 and it goes into

90 as let's see it does go into 7 times It
goes into it 6 times.

So 6 times 3 is, 6 times 3 is 18.

6 times one is 6 plus one is 7.

And then you subtract and get twelve.

So went into it almost exactly 7 times, so
this value right here, almost 0.7.

So if you say how many times 13 goes into
120?

It looks like it's like 9 times, yeah, it
would go into it 9 times.

9 times, 9 times 3, get rid of this, 3
times 3 is 27.

9 times 1 is 9, plus 2 is 11, you have a
remainder of 3.

So you get, it's about .69, so 6, so 6.7
point would be a pretty good guess.

This is .69 of the way between 36 and 49.

So lets go roughly .69 of the way between
6 and 7.

So this is once again, just a approximate,

it's not necessarily give us the exact
answer, we

have to use that as to make a good initial
guess and see how well it works.

So lets try, lets try, 6.7.

Let's try 6.7 and the really way to try it
is to square 6.7.

So 6.7, 6.7 times 6 point, maybe I'll
write the multiplication symbol there.

6.7 times 6.7.

So we have 7 times 7 is 49.

7 times 6 is 42, plus 4, is 46, so the 0
now, cause

we're now in, we're now moved up, to,
we've moved a space to the left,

so now we have 6 times 7 is 42, Carry the
4.

6 times 6 is 36, plus 4 is 40.

And so 9 plus 0 is 9, 6 plus 2 is 8, 4

plus 0 is 4, and then we have a 4 right
over here.

And we have two total numbers behind the
decimal point, one, two.

So this gives us 44.89.

So 6.7 gets us pretty close, but we're
still

not, we're still not probably right to the
hundreds,

well, it's definitely not to the hundreds
place since

we've only gone To the tenth place right
over here.

So if we want to get to 45, the, the 6.7
squared is still less

than, the square root, or I should say,
6.7 squared is still less than 45.

Or 6.7 is still less than the square root
of 45.

So, let's try 6.71.

So let's try 6 point, let me do this in a
new color.

I'll do 6.71 in pink.

So let's try 6.7.

We'll increase it a little bit.

See if we can go from 44.89 to 45, cuz
this is really close already.

And if this is.

Well let's just try it out.

6.71.

So once again, we have to do some
arithmetic by hand.

We are assuming that they don't want us to
use a calculator here.

So if 1 times 1 is 1, 1 times 7 is 7, 1
times 6 is 6.

A 0 here, 7 times 1 is 7, 7 times 7 is 49,
7 times 6 is 42, plus 4 is 46.

And then we have two zeros here, 6 times 1
is 6, 6 times 7 is 42.

Used to have this new four new, 6 times 6,
6 times 6 is 36, plus 4 is 40.

Plus 480.

So if you really, well, it's interesting
to think

what we got incrementally by adding that,
that one

hundredth over there because this part
over here, well,

we'll see, actually, when we add all this
up.

You get a 1.

7 plus 7 is 14.

1 plus 6 plus 9 is 16, plus 6.

Is 22.

2 plus 6 plus 2 is 10.

And then 1 plus 4 is 5 and then we bring
down the 4.

And we have 1, 2, 3, 4 numbers behind the
decimal point.

1, 2, 3, 4.

So when we squared 6.71, 6.71 is equal to
45.02.

41, so 6.71 is a little bit greater.

So let's, let me make it clear now.

We know that, 6.7 is less than the square
root of 45, and we know that, that is less

than 6.71 cause when we square this we get

something a little bit over the square
root of 45.

But he key here is, is when we square

this, so 6.7 squared, so let's, 6.7
squared guys.

44.89 which is eleven hundredths, eleven
hundredths away from 45.

So, this is and then if we look at 6.71
squared, we're only

2.400ths above 45, so this right here is
closer to the square root of 45.

So if we approximate, to the hundreths
place, definitely want to go with 6.7.