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CCSS.Math:

if you're watching a movie and someone is attempting to do fancy mathematics on a chalkboard you'll almost always see a symbol that looks like this this is a radical symbol and this is used this is used to the square root and we'll see other types of roots as well but your question is well what does this thing actually mean and now that we know a little bit about exponents we'll see that the square root symbol or the root symbol or the radical is not so is not so hard to understand so let's start with an example so we know that 3 to the 2nd power is what 3 squared is what well that's the same thing as 3 times 3 that's going to be equal to 9 but what if we went the other way around what if we started with a 9 and we said well what time's itself is equal to 9 we only know that answer is 3 but how could we how could we use a symbol that tells us that so as you can imagine that symbol is this is going to be the radical here so we can write the square root of 9 and when you look at this way you say ok what squared is equal to 9 and you would say well this is going to be equal to this is going to be equal to 3 and I want you to really look at these two these two these two equations right over here because this is the essence of the square root symbol if you say the square root of 9 you're saying what time's itself is equal to 9 and well that's going to be 3 and 3 squared is equal to 9 I can do that again I can do that many times I can write 4 4 squared is equal to 16 well what's the square root of 16 going to be well it's going to be equal to it's going to be equal to 4 let me do it again and actually let me start with the square root what is the square root of 25 going to be well this is the number that times itself is going to be equal to 25 or the number where if I were squared I get 225 well what number is that well that's going to be equal to 5 why because we know that 5 squared is equal to Phi squared is equal to 25 now I know that there's a little nagging feeling that some of you might be having because if I were to take if I were to take negative three and square it and square it I would also get positive nine and the same thing if I were to take negative four and I were to square the whole thing I would also get positive sixteen or negative five and F R squared I would also get positive 25 so why couldn't this thing right over here why can't the square root the positive 3 or negative 3 well depending on who you talk to that's actually a reasonable thing to think about but when you see a radical simple light a symbol like this people usually call this the principal root principle root principle principle square root square root and another way to think about it it's the positive this is going to be the positive square root if someone wants the negative square root of 9 they might say something like this they might say the negative let me scroll up a little bit they might say something like the negative square root of 9 well that's going to be equal to negative 3 and what's interesting about this is well if you square both sides of this of this equation if you were to square both sides of this equation what do you get well negative anything negative squared becomes a positive and then the square root of 9 squared well that's just going to be 9 and on the right hand side negative 3 squared well negative 3 times negative 3 is positive 9 so it all works out nine is equal 9 is equal to 9 and so this is an interesting thing actually let me let me write this a little bit more algebraically now if we were to write if we were to write the principal root of 9 is equal to X this is this there's only one possible excerpt that satisfies it because the standard convention what most math Matt what most mathematicians have agreed to view this radical symbol as is that this is the principal square root this is the positive square root so there's only one X here the only 1x that would satisfy this and that is X is equal to 3 now if I were to write x squared is equal to 9 now this is slightly definitely x equals 3 definitely satisfies this this could be x equals 3 but the other thing the other X that satisfies this is X could also be X's could also be equal to negative 3 because negative 3 squared is also equal to 9 so these two things these two statements are almost equivalent although when you're looking at this one there's two X's that satisfy this one well there's only one X that satisfies this one because this is the positive square root if people wanted to write something equivalent where you would have two x's that could satisfy it you might see something like this plus or minus square root of 9 is equal to X and now X could take on positive 3 or negative 3