Remember the domain is all the values x can have and still the function is defined at this point !Find the domain:
f(x) = cube root of (x + 4)
f(x) = fourth root of (x squared + 3x)
f(x) = x - 5 / (square root of (x squared - 9))
What is under an even-th root should be positive. For odd-th root, it's ok. That is to say you don't care about the sign of x+4 in the cube (3) root (and hence the domain is...), but you'll care about the sign of x²+3x under the fourth root, or the sign of x²-9 under the square (2) root.
When you have a quotient, the denominator can't be 0. So find the values of x for which it is 0. Then remoove them from the possible values of x.
Now please, give it a try, or ask questions if explanations weren't clear enough.