Before we get into it, you may or may not have been taught this already, but I will briefly mention it here because I think it may help clear up some confusion with the rules of nth roots. When you have a root, you can express that as an exponent and it means the same things. And you do that by considering the index to be a fraction exponent. Probably not a very good choice of words, but this should be clear with this short demonstration:
2√x = x1/2
If you get confused or muddled up with the following properties and rules, try thinking of them in terms of this example, and then you should be able to apply the rules of exponents that you already know that help you sort things out. Square root notation is just a fancy, specialized way of working with fractional exponents.
So, here we go.
To start with, we have to state some assumptions and generalizations. We let x and y be real numbers, and m and n are natural numbers. In this case, the following properties are true as long as the expressions are not undefined:
1. (n√x)n = x
Like I said, most of these make sense if you think about them for a second. In this case, consider that most (if not all) arithmetic functions have opposites. Addition is opposite to subtraction, multiplication to division. In this case (let's assume n = 2 for a second) square root is opposite to squaring. Or, cube root is opposite to cubing. And so on. Square root of 16 is 4. 4 squared is 16. They cancel each other out, and you're left with x. Makes sense? E.g. √x2 = x
2. n√xy = n√xn√y
Again, this one kinda makes immediate sense. Kind of like multiplication of a polynomial, you multiply everything inside the brackets by what is on the outside (ie. 2 x (x + 3) = 2x + 6 ). In this case, the root is applied to everything beneath the root sign. It is important to make sure that you realize that this law holds only for multiplying (and dividing... see #3) terms beneath the root sign, but does not work for adding or subtracting terms. So, 3√xy = 3√x3√y
3. n√(x/y) = (n√x) / (n√y)
This is a follow-up to #2. As long as the terms beneath the root sign are multiplying or dividing, you can re-write the expression like this, with a radical sign on the top dividing by a radical sign on the bottom. Again, this doesn't apply if you are adding or subtracting x and y beneath the radical sign.
4. a) if n is even: n√xn = |x|
This is connected to how you can square a positive or negative number and get the same POSITIVE result. So, if you try 2√(5)2 you get positive 25. Same thing if you try -5... positive 25. If the n term is even, you will always get the positive, or ABSOLUTE VALUE, of x. However...
b) if n is odd: n√xn = x
In this case, you CAN finish with a negative number. If you try 3√(5)3 you get 5, but if you try 3√(-5)3 you get -5. Write it out long-hand to see why. -5 x -5 x -5 = -125. Cube root of -125 is -5. There is only one answer.
5. m√(n√x) = mn√x (Sorry, this one is hard to type out)
Once again, this one is kind of like multiplication, and the rules of exponents. If you have (x,2)2, the rules of exponents say that you can multiply the exponents to have x4. In this case, the numbers are in a different place, but the same kind of rule holds. 2√(3√x) = 2x3√x = 6√x.
You will likely find most use of these properties in simplifying larger expressions, making them easier to work with. Such as, pulling out perfect square or cube from a larger number:
E.g. √72 = √9√ 8 = √9 x √4 x √2 = 3 x 2 x √2 = 6√2
In my next post, I will explain a concept that always seems to give students difficulties... conjugates, and rationalizing the denominator.
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