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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
hope this video was helpful if it was
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