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# Approximating square roots

Video transcript

what I want to do in this video is get...a little bit of experience a little of two examples of trying to roughly estimate the square root of non-perfect squares so let's say that I had... If I want to estimate the square root of 32 In particular I'm just curious between what two integers will this square root lie one way to think about it is 32 is in between what two perfect squares we see 32 is actually let me...we should 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 what's the next perfect square after 32 well 32 is less than 36 so we can say 32 is less than 6 squared so if you will take the square root of all of these sides right here we could say that instead of here we have all of the value squared but instead if we take the square root we can say 5 is going to be less than the square root of 32 which is less than 6 notice, to go from here to here*3 all we did is we squared things we raised everything to the 2nd power but the inequality should still hold so the square root of 32 should be between 5 and 6, it's going to be 5. something let's do another example let's say we want to estimate we want to say what between what two integers is the square root of 55 well we can do the same idea lets square it so if we square the square root of 55 we just gonna get the 55 we're just going to get let me use the same color let ne uset that same color 55 ok 55 is between which two perfect squares so the perfect square that is below 55 or the greatest perfect square 55 that is less than 55 let's see, 6 squared is 36 then 7 squared is 49 8 squared is 64 so it will be 49 or we can write that a 7 squared what's the next perfect square above it well let's figure 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 it clear what's going on 55 is the squared root of 55 squared that's kind of by definition it's gonna be the square root of 55 sqared and so the square root of 55 is gonna to be between what it's going to be between 7 ad 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 answer if someone ask what is the square root of 55 at first you like "oh, I don't know what that is I don't have a calculator etc." and you go wait wait that is going to be between 49 and 64 so it's gonna to be 7. something you could even get a rough estimate of 7.1 based on how far away it is from 49 and from 64 you can begin to approximate things let's do one more example let's say we wanna to figure out where does the square root of 123 lie and once again like always you could just pause the video and try to think about it your self between what two integers does this lie well if we were to square it