 Let's get more radical. So let's try to find the principal square root of negative 3 squared. So by our definition, the square root of negative 3 squared is the non-negative number whose square is negative 3 squared. So paper is cheap. Let's write things down. We want something squared to be the same as negative 3 squared. And again, mathematicians aren't so subtle and the obvious thing we could put inside the parentheses is negative 3. And we might say since negative 3 squared is equal to negative 3 squared, then the principal square root of negative 3 squared is negative 3. Except this is wrong. Mathematicians aren't subtle, but we do like to follow the rules, and the rules say that our principal square root has to be a non-negative number. And in fact, it's vitally important to remember when we write this symbol, the principal square root of n is never negative. So how can we find the principal square root of negative 3 squared? Well, there's two ways of doing this. First, the hard way. Since negative 3 squared is equal to 9, and 3 squared is equal to 9, then the non-negative number whose square is the same as negative 3 squared is 3. And so the principal square root of negative 3 squared is equal to 3. Now the reason that this is the hard way is the following. First, we have to find negative 3 squared. And then we have to somehow determine that 3 squared is equal to the same thing. The easy way stems from the following observation. Since negative a squared is equal to a squared, and a squared is equal to a squared, then a and negative a are both square roots of a squared. And that means that the principal square root of a squared is whichever of these is non-negative. So the square roots of negative 3 squared are negative 3 and negative negative 3, otherwise known as 3. So the principal square root is going to be whichever of these is non-negative. Well, that must be the 3 itself. And the directive to take whichever square root is non-negative leads to the following useful result. For any value of a, the principal square root of a squared is the absolute value of a. How about this square root of negative 25? So a useful idea in math and in life is to give things a name. Let square root of minus 25 equal n. Now we've named square root of negative 25, but remember this is the principal square root of 25. So we know that n is a non-negative number. And n squared must be equal to minus 25. Now let's think about that. Something we should remember is our rules for operating with signed numbers. If a and b are real numbers, then if they have the same sign, their product will be positive. But if they have opposite signs, their product is going to be negative. Now since n squared is equal to n times n, and n has the same sign as n, the product n squared, it's the product of two numbers with the same sign, and it must be positive. And what this means is that it can never be negative. And that means n squared equals minus 25 can't possibly be true for any real number. And so that means square root of minus 25 cannot be a real number. And this leads to an important result. Square root of n is only defined for n greater than or equal to zero. At least for now. Later on, we'll take a look at square root of negative 25 and similar square roots.