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# Approximating square roots to hundredths

CCSS.Math:

## 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 with our pen and paper right over here so square root of 45 so the square root of 45 or the principal 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'll 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 the principal root of 49 is 7 so this value right over here is going to be between it's going to be between 6 & 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 + 7 squared and this is this is 9 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 is add a linear relationship but it's going to be closer to 7 then it's going to be to 6 and at least the square or the 45 is 9 13th of the way so we could try we could try let's see it looks like a lot so that's about that's about two-thirds of the way so let's try six point seven let's try six point seven as a guess just based on 0.7 is about two-thirds 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 913 says the 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 913 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 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 zero point seven and so if you say how many times is 13 going 120 it looks like it's like nine times yeah it would go into it 9 times 9 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 so you get it's about point six nine so six bits so six point seven would be a pretty good guess this is 0.6 nine of the way between 36 and 49 so let's go roughly 0.69 of the way between six and seven so this is once again just relaxing it's not necessarily give us the exact answer we have to use that is to make it a good initial guess and then see how well it works so let's try let's try six point seven let's try six point seven and the really way to try it is to square six point seven so the six point seven six point seven times six point maybe don't write the multiplication symbol there six point seven times six point seven so we have 7 times 7 is 49 7 times 6 is 42 plus 4 is 46 put a 0 now because we're now and we're now moved up two 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 1 2 so this gives us 44 point eight nine so six point seven guess it's pretty close but we're still not it's still not probably right to the hundred well what's definitely not to the hundreds place this is we've only gone to the tenths place right over here so if we want to get to 35 this is this 6.7 squared is still less than the square root or I should say 6.7 squared is still less than forty five or six point seven is still less than the square root of 45 so let's try six point seven one so let's try six point let me do this in a new color I'll do six point seven one in pink so let's try six point seven one increase it a little bit see if we can go from forty four point eight nine to forty five because this is really close already and if this is well let's just try it out six point seven one 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 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 this new 4 here 6 times 6 6 times 6 is 36 plus 4 is 40 plus 40 so if you really well it's interesting to think what we got incrementally by adding that that 100 over there because this part over here 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 1 2 3 4 numbers behind the Deaf 2 3 4 so when you B squared six point seven one six point seven one squared is equal to 45 point zero two four one so six point seven one is a little bit greater so let's let me make it clear now we know that six point seven is less than the square root of 45 and we know that that is less than six point seven one because when we square this we get something a little bit over the square root of 45 but the key here is is when we squared this so six point seven squared so let's six point seven squared goddess forty four point eight nine which is 1111 hundredths away from forty five so this is and then if we look at six point seven one squared we're only two point four hundredths above forty five so this right here is closer to the square root of forty five so if we approximate to the hundredths place definitely want to go with six point seven one