we are asked to approximatethe principal root, or the positive square root of45, to the hundredths place. and i'm assuming they don'twant us to use a calculator. because that would be too easy. so, let's see if we canapproximate this just with our pen andpaper right over here. so the square root of 45,or the principal root of 45. 45 is not a perfect square. it's definitely nota perfect square.

let's see, what are theperfect squares around it? we know that it isgoing to be less than-- the nextperfect square above 45 is going to be 49 becausethat is 7 times 7-- so it's less thanthe square root of 49 and it's greater thanthe square root of 36. and so, the square root of36, the principal root of 36 i should say, is 6. and the principalroot of 49 is 7.

so, this value right over hereis going to be between 6 and 7. and if we look at it, it'sonly four away from 49. and it's nine away from 36. so, the differentbetween 36 and 49 is 13. so, it's a total 13 gap betweenthe 6 squared and 7 squared. and this is nine ofthe way through it. so, just as a kind ofapproximation maybe-- and it's not going to work out perfectlybecause we're squaring it, this isn't a linearrelationship--

but it's going to be closer to7 than it's going to be to 6. at least the 45 is9/13 of the way. let's see. it looks like that'sabout 2/3 of the way. so, let's try 6.7 as a guessjust based on 0.7 is about 2/3. it looks like about the same. actually, we could calculatethis right here if we want. actually, let's dothat just for fun. so 9/13 as a decimalis going to be what?

it's going to be 13 into 9. we're going to put somedecimal places right over here. 13 doesn't go into 9but 13 does go into 90. and it goes into 90-- let'ssee, does it go into it seven times-- it goesinto it six times. so, 6 times 3 is 18. 6 times 1 is 6, plus 1 is 7. and then yousubtract, you get 12. so, went into it almostexactly seven times.

so, this value righthere is almost a 0.7. and so if you say, how manytimes does 13 go into 120? it looks like it'slike nine times? yeah, it would gointo 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 apretty good guess. this is 0.69 of theway between 36 and 49. so, let's go roughly 0.69of the way between 6 and 7. so this is once againjust to approximate. it's not necessarily goingto give us the exact answer. we have to use that tomake a good initial guess. and then see how it works. let's try 6.7.

and the really way totry it is to square 6.7. so 6.7 times-- maybe i'llwrite 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'vemoved 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 a4 right over here. and we have two total numbersbehind the decimal point. one, two. so this gives us 44.89. so, 6.7 gets us pretty close. but we're still not probablyright to the hundredth.

well, we're definitely notto the hundredths place. this since we've onlygone to the tenths place right over here. so, if we want to get to 45, 6.7squared is still less than 45, or 6.7 is still less thanthe 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 isreally close already. let's just try it out. 6.71. so once again, we have todo some arithmetic by hand. we are assumingthat 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. and then we have two 0s here. 6 times 1 is 6. 6 times 7 is 42. just have this new 4 here.

plus 40. it's interesting to thinkwhat we got incrementally by adding that onehundredth over there. well, we'll see actuallywhen we add all of this up. you get a 1. 7 plus 7 is 14. 1 plus 6 plus 9 is16, 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 behindthe decimal point. one, two, three, four. so, when you we squared 6.71. 6.71 squared isequal to 45.0241. so 6.71 is a little bit greater. so, let me make it clear now. we know that 6.7 is lessthan the square root of 45.

and we know thatis 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 wesquare this, so 6.7 squared got us 44.89 whichis 0.11 away from 45. and then, if we lookat 6.71 squared, we're only 2.4hundredths above 45. so, this right here is closerto the square root of 45. so if we approximate tothe hundredths place,

definitely want to go with 6.71.