Simplifying roots of negative numbers

we're asked to simplify the principle square root of negative 52 and we're going to assume because we have a negative 52 here inside of the radical that this is the principle this is the principle branch of the complex square root function that we can actually put input negative numbers in the domain of this function that we can actually get imaginary or complex results so we can rewrite negative 52 as negative 1 times 52 so this can be re-written as the principal square root of negative 1 negative 1 times negative 1 times 52 and then if we assume that this is the principal branch of the complex square root function we can rewrite this this is going to be equal to the square root of negative 1 times the print or I should say the principal square root of negative 1 times the principal square root times the principal square root of 52 now I want to be very very clear here you can do what we just did if we have the principal square root of the product of two things we can rewrite that it's the principal square root of each and then we take the product but you can only do this or I should say you can only do this if either both of these numbers are positive or only one of them is negative you cannot do this if both of these were negative for example you could not do this you could not say you could not say the square root of fit or the principal square root of 52 is equal to negative 1 times negative 52 you could do this so far I haven't said anything wrong 52 is definitely negative 1 times negative 52 but then since these are both negative you cannot then say you cannot then say that this is equal to the square root of negative 1 times the square root of negative 52 in fact I invite you to continue on this train of reasoning you're going to get a nonsensical answer this is not ok this is not ok you cannot do this right over here and the reason why you cannot do this is that this property does not work when both of these numbers are negative now with that said we can do it if only one of them are negative or both of them are positive obviously now the principle square root of negative 1 if we're talking about the principle branch of the complex square root function is I so this right over here does simplify to I and then let's think if we can simplify the square root of 52 ne and to do that we can think about its prime factorization see if we have any perfect square sitting in there so let me so 52 is 2 times 26 and 26 is 2 times 2 times 13 so we have 2 times 2 there or 4 there which is a perfect square so we can rewrite this as equal to this is equal to well we have our I now square root of the principal square root of negative 1 is I the other square root of negative 1 is negative I but the principal square root of negative 1 is I and then we're going to multiply that times the square root times the square root of 4 times 13 4 times 13 and this is equal to and this is going to be equal to I times the square root of 4 I times the square root of 4 or the principal square root of 4 times the principal square root of 13 the principal square root of 4 is 2 so this all simplifies and we can switch the order over here this is equal to 2 times the square root of 13 2 times the principal square root of 13 I should say times I times I and I just switched around the order it makes it a little bit easier to read if I put the I after the numbers over here but I'm just multiplying I times 2 times the square root of 13 that's the same thing as multiplying 2 times the principal square root of 13 times I and I think this is about as simplified as we can get here