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We are asked to approximate the principal root or the the positive square root of 45 to the hundreths place

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and i'm assuming they don't want us to use a calculator cos that would be too easy

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so lets see if we can approximate this

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just with our pen and paper right over here.

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so square root of 45 or the principal root of 45

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45 is not a perfect square,it is definitely not a perfect square

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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.

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lets see it's gonna be 49 cos that is 7 times 7.so it's less than the square root of 49

9
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and it's greater than the square root of 36,and so the square root of 36 or the principal root of 36 i should say is 6 and the

10
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principal root of 49 is 7.so this value right over here is going to be between 6 and 7

11
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and if we look at it it's only 4 away from 49 and it's 9 away from 36.so looks like the difference between 36 and 49 is 13

12
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so it is a total 13 gap between the 6 squared and the 7 squared.and this is 9 of the way through it

13
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so just as a kind of approximation maybe and it may not work out perfectly because we are squaring it.

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this isn't a linear relationship but it is going to be closer to 7 than it's going to be to 6 and at least the square or the 45 is 9/13ths of the away

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so we could try it looks about two thirds of the way so lets try 6.7 as a guess.just based on point 7 it's about 2/3 it looks about the same

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and actually we could calculate this right here if we want.

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so 9/13ths as a decimal is going to be what?it's going to be 13 into 9 we have to some decimal places over here

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13 doesn't go into 9 but it does go into 90 .lets see does it go into it 7 times.it goes into it 6 times

19
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this value is almost 0.7

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so you get it's about .69

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so 6.7 would be a pretty good guess.this is .69 of the way between 36 and 49

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

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so this once again is just to approximate it is not necessarily going to give us the exact answer we have to use that as a good initial guess

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and then see how well it works.so lets try 6.7

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and the real way to try is to square 6.7

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so 6.7 times 6.7

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so we have 7 times 7 is 49

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7 times 6 is 42 plus 4 is 46

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put a zero now we have now moved up a space to the left

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so we now have 6 times 7 is 42

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carry the 4

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6 times 6 is 36 plus 4 is 40

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so 9 plus 0 is nine

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6 plus 2 is eight

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4 plus 0 is four and we have a four right over here

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and we have 2 total numbers behind the decimal point

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1,2

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so this gives us 44.89

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so 6.7 gets us pretty close but we are still not probably right to the hundred.

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we've only gone 10's place right over here

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so if we want to get to 45 this 6.7 squared is still less than 45 or 6.7 is still less than the square root of 45

42
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so lets try 6.71 and increase it a little bit see if we can get from 44.89 to 45

43
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cos this is really close already

44
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lets just try it out 6.71

45
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so once again we have to do some arithmetic by hand

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we are assuming they don't want us to use a calculator here

47
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we have 4 numbers behind the decimal point

48
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so when we squared 6.71

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6.71 squared = 45.0241

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so 6.71 is a little bit greater

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let me make it clear now

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we know that 6.7 is less than the square root of 45 and we know that is less than 6.71

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cos when we square this we get something a little bit over the square root of 45

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but the key here is when we square this

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so 6.7 squared got us 44.89 which is a 11 hundreths away from 45

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and then if we look at 6.71 squared we're only 2.4hundreths above 45 so this right here is closer to the square root of 45

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so if we approximate to the hundreths place definitely wanna go with 6.71