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# Comparing irrational numbers with radicals

## Video transcript

I have six numbers here and you see that five of them are irrational there they involve the square root of a non perfect square and my goal or our goal in this video is without a calculator see if we can sort these numbers from least to greatest and like always pause this video and see if you can do that so I'll give you a hint the hint is it's very hard without a calculator you know square root of 2 it's going to be one point something something square root three is going to be one point something something how do we do this but you just have to realize if I have some number let's say I have some number a that is greater than zero and if we know that a is less than B well then a squared is going to be less than B squared if one number if one positive number is less than another positive number then the square of this positive number is going to be less than the square of that number so one thing that we could do when we're when we are comparing all of these irrational numbers that involve square roots of non perfect squares well let's compare their squares because their squares are not going to be irrational numbers it's going to be much easier to compare and then we can order them because if we order the squares then we can we tell us what happens if we order their square roots so what am I talking about well I'm just going to square each of these so if I take this to the second power this is going to be four square roots of two times 4 square roots of 2 you can change the order of multiplication that's 4 times 4 times the square root of 2 times the square root of 2 now 4 times 4 is 16 square root of 2 times square root of 2 well that's just going to be 2 so it's going to be 16 times 2 which is equal to 32 now what about 2 square roots of 3 well same idea it's 2 let's square it let's square it and I'll do this one a little bit faster so if we square 2 square roots of 3 this is going to be 2 squared times square root of 3 squared so it's going to be 2 squared times the square root of 3 squared well 2 squared is going to be 4 our square root of three squared is going to be three so this is going to be equal to this is going to be equal to twelve that's this thing this thing squared and if you don't if this step seems a little bit confusing if you have the product of two things raised to a power that's the same thing as raising each of them to that power and then taking the product and you can actually see I kind of worked it out here why that actually makes sense notice when I just changed the order of multiplication you had four times four or four squared times square root of two squared which is going to be two so let's keep doing that so what is this value squared well it's going to be 3 squared which is nine times square root of 2 squared which is 2 9 times 2 is 18 what's the square root of 17 squared well that's just going to be 17 then in blue this is just going to be this is just going to be 17 what is 3 square roots of 3 squared well it's going to be 3 squared which is 9 times square root of 3 squared the square root of 3 times the square root of 3 is 3 so it's going to be 9 times 3 or 27 and what is 5 squared well that's this is pretty straightforward that's going to be 25 so let's order them from least to greatest so which of them when I square it gives me the smallest value well the SMAW compared 32 to 12 to 18 to 17 to 27 to 25 12 is the smallest value so if their square is a small so these are all positive numbers and this is going to be the smallest value out of all of them so let me write that first so 2 square roots of 3 so I've covered that one now what's next well now I have this value 17 is the next smallest is the next smallest square so it's square root is going to be the next smallest or the next square root so it's going to be 2 square roots of 3 then square root of 17 that is this one here then we go to 18 so if we look at its square root would numbers we were originally trying to sort that be square roots three square roots of three let's the right-side 3 square roots of 2 3 square roots of 2 we got that one covered then the next one is going to be 25 when you look at the squares so the next value out of our original set the next largest one is going to be 5 so then we get to 5 we've covered that one then the next one let's see we have 27 and 32 left so the next largest one it's going to be the 27 is the next largest square so the next largest number out of the ones we care about is 3 square roots of 3 so 3 square roots of 3 we cover that one and then we finish with this is the largest value for square roots of 2 for square roots of 2 and we're done that was pretty neat without a calculator we were able to sort these irrational numbers these vat of these these numbers well not all of them are irrational but the ones that involved a square root of something that is not a perfect square