I'm having trouble understanding when I am supposed to add absolute values when simplifying radicals.

(It's cube root not square root)

XER and YER because it's a cube root and you can take cube root of negatives and positives.

√-108x^7y^9
= √-27•4•x^6•x•x^9
= -3x^2y √4x
= -3x^2|y|√4x

So for this one would it be simplified to: 3x^2y √4x or -3x^2|y|√4x. And why.

(-108 x^7 y^9)^(1/3)

(-27 * 4 * x^6 * x^1 * y^9)^(1/3)

-3 * x^2 * y^3 *(4 x)^(1/3)

Would I need to add || absolute value around any variables?

well if x is negative, then x^1 and x^7 are negative and x^6 is + so you do not want ||

if y^9 is negative then y wants to be - and y^3 will be negative so again not || needed

To understand when to add absolute values when simplifying radicals, you need to consider the rules for simplifying radicals and the properties of the absolute value function.

In this particular example, you are simplifying the expression √(-108x^7y^9) using the rules for simplifying radicals. Let's break down the steps to find the simplified expression:

Step 1: Decompose the radicand: √(-27 * 4 * x^6 * x * x^9)

Step 2: Simplify the perfect square factor: √(-27) * √(4) * √(x^6) * √(x) * √(x^9)

Step 3: Evaluate the square roots of perfect squares: -3 * 2 * x^3 * x * x^4

Step 4: Simplify the expression: -6x^8y√x

Now, let's talk about the absolute value. The absolute value function (|x|) always gives the positive or zero value of any real number x. It means that |x| = x if x is non-negative (x ≥ 0), and |x| = -x if x is negative (x < 0).

In the expression -6x^8y√x, the variables x and y can be either positive or negative. However, the square root (√) of x only gives a real number when x ≥ 0. It means that the expression √x is only defined for non-negative values of x.

Therefore, when simplifying radicals and dealing with square roots, you must consider the argument inside the square root and take the absolute value when it is negative. In this case, the expression √(-108x^7y^9) simplifies to -3x^2|y|√(4x) because the variable y could be negative, so we take the absolute value of y to ensure that √(4x) is defined.

Thus, the simplified expression would be -3x^2|y|√(4x).