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# Approximating square roots to hundredths

Video transcript

We are asked to approximate
the principal root, or the positive square root of
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 the square root of 45,
or the principal root of 45. 45 is not a perfect square. It's definitely not
a perfect square. 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 49 because
that is 7 times 7-- so it's less than
the square root of 49 and it's greater than
the square root of 36. And so, the square root of
36, the principal root of 36 I should say, is 6. And the principal
root of 49 is 7. So, this value right over here
is going to be between 6 and 7. And if we look at it, it's
only four away from 49. And it's nine away from 36. So, the different
between 36 and 49 is 13. So, it's a total 13 gap between
the 6 squared and 7 squared. And this is nine of
the way through it. So, just as a kind of
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. At least the 45 is
9/13 of the way. Let's see. It looks like that's
about 2/3 of the way. So, let's try 6.7 as a guess
just based on 0.7 is about 2/3. It looks like about the same. Actually, we could calculate
this right here if we want. Actually, let's do
that just for fun. So 9/13 as a decimal
is going to be what? It's going to be 13 into 9. We're going to put some
decimal places right over here. 13 doesn't go into 9
but 13 does go into 90. And it goes into 90-- let's
see, does it go into it seven times-- it goes
into it six times. So, 6 times 3 is 18. 6 times 1 is 6, plus 1 is 7. And then you
subtract, you get 12. So, went into it almost
exactly seven times. So, this value right
here is almost a 0.7. And so if you say, how many
times does 13 go into 120? It looks like it's
like nine times? Yeah, it would go
into it nine times. 9 times 3. Get rid of this. 9 times 3 is 27. 9 times 1 is 9, plus 2 is 11. You have a remainder of 3. It's about 0.69. So 6.7 would be a
pretty good guess. This is 0.69 of the
way between 36 and 49. So, let's go roughly 0.69
of the way between 6 and 7. So this is once again
just to approximate. It's not necessarily going
to give us the exact answer. We have to use that to
make a good initial guess. And then see how it works. Let's try 6.7. And the really way to
try it is to square 6.7. So 6.7 times-- 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. Put a 0 now because 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 probably
right to the hundredth. Well, we're definitely not
to the hundredths place. This since we've only
gone to the tenths place right over here. So, if we want to get to 45, 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. Let me do this in a new color. I'll do 6.71 in pink. So, let's try 6.71. Increase it a little bit. See if we go from 44.89 to 45. Because this is
really close already. 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, we have 1 times 1 is 1. 1 times 7 is 7. 1 times 6 is 6. Put 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 0s here. 6 times 1 is 6. 6 times 7 is 42. Just have this new 4 here. 6 times 6 is 36, plus 4 is 40. Plus 40. It's interesting to think
what we got incrementally by adding that one
hundredth over there. Well, we'll see actually
when we add all of 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 one,
two, three, four numbers behind
the decimal point. One, two, three, four. So, when you we squared 6.71. 6.71 squared is
equal to 45.0241. So 6.71 is a little bit greater. So, let me make it clear now. We know that 6.7 is less
than the square root of 45. And we know that
is less than 6.71. Because when we square this,
we get something a little bit over the square root of 45. But the key here is when we
square this, so 6.7 squared got us 44.89 which
is 0.11 away from 45. And then, if we look
at 6.71 squared, we're only 2.4
hundredths above 45. So, this right here is closer
to the square root of 45. So if we approximate to
the hundredths place, definitely want to go with 6.71.