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# Approximating square roots

CCSS.Math:

## Video transcript

what I want to do in this video is get a little bit of experience a little less you see a few examples of trying to roughly estimate the square root of non perfect squares so let's say that I had if I wanted to estimate the square root of 32 and in particular I'm just I'm just curious between what two integers will this square root lie well one way to think about it is 32 is in between what perfect squares we see 32 is actually let me make sure I have some space for future examples so 32 what's the perfect square below 32 so the greatest perfect square below 32 is 25 32 is greater than 25 that's 5 squared so maybe I should write it this way so 5 squared is less than 32 and then 32 what's the next perfect square after 32 well 32 is less than 36 which is less than it so we could say 32 is less than 6 squared so if you were to take the square root of all of these sides right over here we could say that instead of here we have all of the value squared but instead if we took the square root we could say 5 is going to be less than the square root of 32 which is less then which is less than 6 notice to go from here to here to go from here to here and here to here all we did is we square things we raised everything to the second power but the inequality should still hold so the square root of 32 should be between 5 & 6 it's going to be 5 point something let's do another example let's say we wanted to estimate we want to say what between what two integers is the square root of 55 well we can do the same idea let's square it so if we square the square root of 55 we're just going to get to 55 we're just going to get let me do that same color 55 and so okay 55 is between which two perfect squares so the perfect square that is below 55 or I guess a the greatest perfect square that is less than 55 let's see 6 squared is 36 then 7 squared is 49 8 squared is 64 so it would be 40 it would be 49 I could write that as seven squared let me write that that is the same thing as seven squared and what's the next perfect square above it well we just figured it out 7 squared is 49 8 squared is larger than 55 it's 64 so this is going to be less than 64 which is 8 squared and of course 55 just to make make it clear what's going on 55 is the square root of 55 squared that's kind of by definition it's going to be the square root of 55 squared and so the square root of 55 is going to be between what it's going to be between 7 and 8 so 7 is less than the square root of 55 which is less than 8 so once again this is just an interesting way to think about what would you ask if someone said the square root of 55 and let's first you like oh I don't know what that is I don't have a calculator etc etc but like oh wait wait that's going to be between 49 and 64 so it's going to be seven point something it's going to be seven point something and you could even get get a rough estimate of seven point what based on how far away it is from 49 and 64 you can begin to approximate things let's do one more example let's say we wanted to figure out where does the square root of 123 lie and once again like always encourage you to pause the video and try to think about it yourself between what two integers does this lie well if we were to square it you get to 123 and what's the perfect square that is less that is the greatest perfect square less than 123 let's see 10 squared is 100 11 squared is 121 12 squared is 144 so 11 squared so 123 so we could write 121 is less than 123 which is less than 144 that's 12 squared so if we take the square roots we could write that 11 is less than the square root of 123 which is less than 144 so once again what's the square root of 123 it's going to be 11 point something and in fact it's going to be closer to 11 then it's going to be 212 123 is a lot closer to 121 than it is to 144 so it might be I don't know 11.1 something like that I don't know if that's exactly right we would have to check that on the calculator but hopefully this gives you a website it's going to be it actually will be less than 144 but if we want to think about what consecutive integers is that'd be between that's going to be a 12 right over there almost almost minima no well anyway you get the idea hopefully you enjoyed that