Estimating Square Roots - Problem 2

A lot of students look at problems like this and they say, “Hey wait a second, 48 doesn’t have a square root.” Well you’re kind of on the right track. There is a number that times itself gives you the answer 48, it’s just that that number isn’t an integer. We know the perfect square numbers which are numbers which are numbers whose square roots are integers like 49 is equal to 7 times 7. 49 is really close to 48.

So when this problem asks me to say between what two consecutive integers is a square root of 48 the number that comes to my head right away is 49 because 49 is equal to 7 times 7.

So this is going to be some answer between 6 and 7. Because 6² is 36, 7² is 49 and 48 falls in between those two numbers, in between 36 and 49. Also notice that 48 is really close to 49 which tells me my answer I can be more specific it’s not just between 6 and 7 it's actually really close to 7.

You could try using the calculator just to check your work. Same idea with this one I’m not going to deal with the negative sign for now I’m just going to look at the 150 because I know that 12² is equal to 144 and 13² is equal to 169 and 150 falls between those two values. So the square root of 150 is going to be between 12 and 13 except for I have to also deal with the negative because I want the negative square root of 150. No problem I’m just going to stick negative signs there and there. The square root of -100 or the negative square root of 150 is some number that falls between -12 and -13.

Last but not least, I’m going to use the same process for 200. It’s a little bit bigger, think about numbers that are perfect squares that are around 200. Like I know that 14² is 196. Gosh I’m going to have that memorized. I also know that 15²is equals to 225 and 200 falls in between those values. So when the problem says between what two consecutive integers is the square of 200, I’ll say it’s between 14 and 15.

I can also be more specific than that if I needed to and I could say it's pretty darn close to 14, right? Because 200 is pretty darn close to 196, it’s closer to 196 than it is to 225, so my answer is pretty close to 14.

Once you get the hang of this and as long as you guys know your perfect square numbers, these problems are pretty nifty like nobody on the street, I bet you, knows not exactly what the square root of 200 is. Square root of 200 is 14.14, but no one really knows that. But you could figure out that the square root of 200 is pretty darn close to 14, a little bigger than 14 using this kind of logic. It’s pretty cool, I think it’s a neat skill to have.