so when you start thinking about taking square roots of negative numbers or even in the future you'll do imaginary numbers and complex numbers and all the rest you have to expend the definition of what this radical means, so when you are taking the square root of really of any negative number you'll really saying this is no longer the traditional principal square root function you've now talking this is the principal complex square root function, this is now to find for complex inputs or the domain it can also generate imaginary or complex output or you should call that the range and if you assume that, then really straight from this you get that the square root of negative x is going to be equal to i times the square root of x and this is only and i'm going to make this clear because i just told you that this will be false if both a and b are negative, so this is only true, so we can apply this we can apply this we can apply when x is greater than or equal to zero, so if x is greater than or equal to zero the negative x is clearly a negative number or i guess it can be zero, it's a negative number and then we can apply this right over here if x was less than zero then we'll be doing all of this nonesense up here and we will start to get nonesense equal answeres and if you look at it this way you'll say hey look i can be the square of negative 1 if we were taking the if it's the principal branch of the complex square root function, then you could rewrite this right over here as square root of negative 1 times the square root of x and so really, the real fault in this logic when people say hey negative 1 can't be equal to 1, the real faughlt is using this property, when both a and b when both of these are negative numbers that will come up with something that is unambiguously false, if you expend the definition of complex or expend the definition of principle root include negative numbers in the domain and including and to include imaginary numbers then you can do this you can say the the square root of negative x is the qsuare root of negative 1 times or instead (say) the principle square root of negative x, i should be particulare in my words, is the same thing as the principal square root of negative 1 times the principal square root of x when x is greater than or equal to zero and i don't want confuse you, if x is greater than or equal to zero this is clearly, this negative x, that is clearly negative or i guess you say a non positive number.