Approximating square roots to hundredths

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.