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alright ladies and gentlemen and welcome

back to this Athenian stranger tutorial

video where today we are going to learn

how to take a square root entirely by

hand let's take a look at this law start

here by evaluating the square root of 38

out to two decimal places okay the

problem we're presented with is that 38

is not a perfect square so it's not 2

times 2 is 4 3 3 times 3 is 9 4 times 4

16 5 times 5 25 6 times 6 36 7 times 7

49 it's not a perfect square of any

whole number so we have to approximate

it and here's how we're going to do that

instead of writing the square root below

it will write 38 and then we'll put the

division symbol

what we're looking for here is a perfect

route of some number that is less than

or equal to 38 so I'll give you with

what it is can be six times six equals

36 okay it can't be 7 because 7 times 7

is 49

- big right so it's got to be six times

six we want a number that is smaller the

product of these two numbers is smaller

than 38 so we'll put in here at 6 and 38

divided by 6 is 6 6 times 6 is 36

and we're left with a remainder of two

and here's the cool part this is where

it differs from division on how you do

this put a dot all right and now we're

going to invent and bring down two zeros

all right set this template up just like

I'm showing you next to 200 down here we

will write a less than or equal to

symbol and to the left of that symbol

we'll put a line to the left of that

line we'll put

multiplication symbol to the left of

that symbol we'll put another line and

then right here in this space we need to

double six

6 times 2 is 12 so this is 1/2 what

we're looking for here and it's now a

three-digit number whose last digit goes

here it's better to just see it happen

all right

so what we're looking for is a

three-digit number whose last digit can

be placed here and when you multiply

that three-digit number by that last

digit you get a number that is less than

or equal to 200 well it's just 1 right

because if you did a 122 times 2 you

would get a number that was bigger than

200 so it's 120 1 times 1

okay 121 times 1 is 121 put that here

and now perform the subtraction

10 minus 1 is 9 9 minus 2 is 7

so now we're looking at 79 and we will

bring down two more zeros so our next

step is to evaluate up 7900 but we have

accomplished something awesome this

number right here

one is the tenths place of the

approximated square root so we'll go in

here and put 6.1 here's the trick you've

got six point one and I want to remove

the decimal place so we're going to

double

sixty-one

okay what is the double of 61 well that

would be 61 times two

all right so that is two and twelve so

here's what I'm going to place my less

than or equal to symbol right here and

then a little line multiplication symbol

little line and then to the left of that

I'll write 122

okay what is the next step

the next step is to find a whole number

that I can tack on to the end making

this a four-digit number and then

whatever that whole number is I'm gonna

multiply it this four-digit number by

that whole number terminating ones place

digit that I've found and when I

multiply it together it's got to be less

than or equal to 79 all right so in this

example we'll try six before we put it

let's try it one two two six times six

see what that is six times six is

thirty-six carry the three 6 times 2 is

12 plus 3 is 15 carry the 1 6 times 2 is

12 plus 1 is 13

carry the one six times one is six plus

one is seven

all right so you see here we couldn't do

one two to seven times seven because it

would be bigger than 7900 this is as

close as we're gonna get so we'll go up

here and put a six here and a six here

and that means that six is the next

digit in the approximated square root so

we'll go up here and put a six now if

you wanted to continue this process you

would double

six one six so we're just when we double

these numbers we're taking the numbers

without the decimal so we would the next

step in this process would be to take

six one six times two

and that would give me 12 carry the 1 3

and 12 okay one two three two and we'll

need that here in a moment but we would

take this the next step in the process

would be to take seven three five six

and write it here seven three five six

if you just if you

wanted to continue this out for our

purposes in math we don't need to

usually do that but you could okay so 10

minus 6 would be 4 9 minus 5 would be 4

8 minus 3 would be 5 5 and that's it

so then we would bring down two more

digits and now we would be looking at

fifty four thousand four hundred and we

would put less than or equal to symbol

number

multiplication symbol number and then we

would write 7 3 5 6 or rather 1 2 3 2

okay we would write 1 2 3 2 and we would

be looking for that final digit this is

getting big right it gets the problem

itself gets big so we'll just stop at 6

point 1 6 and this is the cool thing if

we take our calculator here

put in 38

and then we take the square root we see

that we got it right it's six point one

six so that's how we would take a square

root entirely by hand so we can say the

square root of 38 is approximately equal

to six point one six all right well I

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