What is a simple way to understand this so I don't keep having this problem. :( when I am taking the nth root, the nth root of what? 12 or 24? And I am still unsure of where 16 came from. The original problem is 12 raised to the one fourth equals 24 which is why I thought it was 8, but I guess I need an explanation for nth roots. My next problem is 2x raised to the 1/3 + 5 = 7.1. I know that this is the cube root, but of what 2 or 5? I guess I am a little unsure of how to set these up or how to understand nth roots. I looked in my text, and it doesn't really explain it in a way that I can understand.

Here's one way to think of it: if A^(1/N) = B, then B^N = A.

For example, if X ^ {\frac 14} = 2, then X = 2 ^ 4, so X is 16. Does this help?

The other part of your question has to do with order of operation. Always do the exponents before multiplication (unless there are parentheses or brackets, which override). So, your original question was 12 times x raised to the 1/4, written out as:

12 X ^ {\frac 14} must be thought of as:

12 \( X ^ {\frac 14} \)

This will help you understand nth roots:
Mathwords: nth Root (http://www.mathwords.com/n/nth_root.htm)
Mathwords: Radical Rules (http://www.mathwords.com/r/radical_rules.htm)

As for your second quesiton it's the cubed root of 2x.
( 2x^(1/3) ) + 5 = 7.1

Hope this helps.