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Radicals and Absolute Values - Concept
Since any even-numbered root must be a positive number (otherwise it is imaginary), absolute value must be used when simplifying roots with variables, which ensures the answer is positive. When working with radical expressions this requirement does not apply to any odd root because odd roots exist for negative numbers. Additionally, absolute value is not needed if an even number of a variable come out of the root - the answer must be positive.
The absolute value and square roots, so what we're going to do now is talk about absolute values and sometimes what happens is we actually need an absolute value when we give our answer and so what we're going to do is look at a number of examples and talk about when we need them and when we don't.
Okay so it's starting off, square root of 4, easy example we know the square root of 4 is 2 because 2 times 2 is equal to 4. Okay next, square root of -4, we need 2 numbers that will give us -4 that's not going to happen okay this is a not a real number okay later on we'll actually talk about how we can do this but we're not there yet okay? So square root of 3 squared. 3 squared is 9, square root of nine is 3. Square root of negative 3 squared. When we square a negative number, we actually get a positive so -3 times -3 is 9 square root of 9 is again 3. Okay, cube root of 8 has three 2's in eight so this comes out to be 2 and the cube root of negative 8 is -2. Negative 2 times itself 3 times is negative eight so what we have looked at here is when we have a odd root okay cube root here we can have a positive answer or a negative answer.
Okay when we have a even root all of our answers have to be positive okay? So these are numeric representations but now we're going to variables. Okay, so let's go over here the square root of x squared okay that is made up of 2 x's so we know that we can simplify this as an x. The problem is is that we don't know if x is positive or negative right? Say x was negative 3, just like we had over here. What happens is we are squaring it so that will become positive and then taking the square root of it so it's actually going to stay positive because we don't know if x is positive or negative, we have to put absolute value signs from the outside to make that term positive okay, so square root of x cubed. We can take out one x and we're still left with one x on the inside but we know that this has to be positive because we can't take a negative out of the square root so once again we have to put in absolute value signs okay because wherever comes out of the square root has to be positive. Okay, what about square root of x to the fourth. Okay we know this is 4 xs so we can take out 2 of them leaving us with x squared. Do we need absolute values in this case? No because x squared is always going to be become positive okay? So whenever we need absolute values is basically a one sort of rule of thumb that I always use is whenever you're taking an even root, so here this is the square root there's actually a little invisible 2 out here and whenever taking a even root and we have an add power on a variable, okay so here we have a single x, single x, x squared. If we had to take out an x cubed we would need an absolute value. If we take out an x the fourth we won't because the fourth will always be positive.
We actually never need a absolute value when we're dealing with an odd root and we'll do one example more example say cube root of x to the third. x to the third is 3 x's so this actually equal to x but because we're dealing with an odd root the cube root we do not need an absolute value because we can actually get positive or negative numbers to come out of this okay? So basically whenever you are dealing with the absolute sorry the square root of a variable, if you have an even root and you get out and odd power you always include absolute values.