x^n when n is not an integer - getting to the root of things

What happens if the n in the function x^n is not an integer? We know from the rules of combining exponents that x^½ times x^½ is x. This means that x^½ is the same as the square root of x. As you slide the slider, you will see that some graphs are plotted for both positive and negative values of x and some graphs for positive values only. Why? Challenge - For each power of x you explore check and uncheck the absolute value functions checkbox. How would you characterize the difference between f(x) and absolute value of f(x) – which is written as | f(x) | In general, do you believe that |f(x)| and f(|x|) are the same or different? Why?