 So remember the symbol nth root of a refers to the principal nth root of a If it exists this number will either be non-negative if n is even or The same sign as a if n is odd It's easy to determine whether a number is positive or negative But if the radicand is a variable expression, we must be more careful and use the absolute value Remember the symbol absolute value of a indicates a non-negative number whose magnitude is the same as a So for example, let's say we want to simplify the square root of a squared Since a squared is a squared then a is a square root of a squared And we might write square root of a squared equals a But we would be wrong In order for this to be the square root of a squared using our radical symbol It must also be non-negative because remember square root of n using our symbol is the non-negative Number whose square is n and we don't know whether a is positive or negative So to guarantee we actually have a non-negative number. We'll use the absolute value It's important to understand that we're not using the absolute value because we have a root We're using the absolute value because for square roots. We have to have a non-negative number So if I want to simplify the cube root of x cubed since x cubed is x cubed then x is a cube root of x cubed And we can write the cube root of x cubed is equal to x Do we need absolute value here? Well, remember if x is a real number that x cubed will have the same sign as x and Since we're dealing with a principal cube root. We need to have the principal cube root have the same sign as what we started with And so x and x cubed have the same sign and we don't need to do anything In fact, we can go a little bit further since absolute value of x is always Non-negative, but might have a different sign from x to the n It is not generally correct to write nth root of x to the n is the absolute value of x Instead we must consider whether the root must be positive If n is even then the nth root of a must be non-negative So the nth root of x to the n is the absolute value of x But if n is odd then the nth root of a must have the same sign as a So the nth root of x to the n will just be x with no absolute value In other words, you have to use the absolute value for even index roots And you can't use the absolute value for odd index roots For example, let's try to simplify square root 50 y to power 6 We can begin by factoring by perfect squares So remember that as long as a and b are non-negative, the square root of ab is the square root of a times the square root of b It's also useful to remember that c squared is non-negative for any real numbers c So y squared y squared and y squared are all positive And so we can break apart this square root into a product of the individual factors Square root of 25 is 5 Square root of y squared is the absolute value of y Square root of 2 is just square root of 2 And we can combine these three absolute values of y into a single absolute value of y to the third We can find the fourth root of x to the 10th y to the 6th And we can simplify by factoring out perfect fourth powers We can break up our radical into a product of radicals The fourth root of a fourth power is x or y But since our index is even we have to use the absolute value And we'll rewrite this as absolute value of x squared y times the leftovers Or we could take the cube root of 8x to the fifth. We can remove perfect cube factors We can break our radical into a product of radicals The cube root of 8 is 2 and the cube root of x cubed is x And remember if the index of the root is odd, don't use absolute value And finally we have the leftover bit cube root of x squared How about a quotient? So it's useful to remember for a and b non-negative The square root of a over b is the square root of a over the square root of b And so that means we can split this square root of 50 y to the fifth over 18x to the third As the square root of 50 y to the fifth over the square root of 18x to the third Now if you listen carefully to what I just said, you'll realize there's a bit of a problem We don't actually know whether 50 y to the fifth or 18x to the third are positive So we might not be able to do this And if you're concerned about this, you should be But we'll see how we can resolve this once we get to the end of the problem We'll simplify the numerator We'll factor the radicand by perfect squares The square roots of 25 y squared and y squared are going to be 5 The absolute value of y and the absolute value of y And we have our leftover square root of 2 y The absolute value of y absolute value of y can be multiplied to be the absolute value of y squared And remember c squared is non-negative for any real number So we don't really need the absolute value in this case Meanwhile, we can also simplify the denominator. So we'll factor 18x to the third by perfect squares This breaks up as a square root of 9, square root of x squared, square root of 2x and we can simplify And we get our denominator And here's an important idea. We should answer questions in the same language. They were asked In this case, the original question was the square root of a quotient But as written the answer is a quotient of square roots So what we should do is to recombine this quotient of square roots and rewrite it as a square root of a quotient And here's the important idea. Remember that we had some concerns about being able to split our square root of a quotient into a quotient of square roots because we didn't know if the radicand was positive or negative But if the original expression has any meaning Then the original radicand has to be non-negative, which means that x and y have to have the same sign Now it's possible that they're both negative But in the final simplified form as long as they have the same sign the radicand will still be non-negative