Following these threads: Thread 1 and Thread 2, I am still confused about the way we define functions with fractional exponents.
For example, $f(x)=(x^2)^\frac{3}{2}$. Is it true to say that $f(x)=x^3$ when $x\geq0$ and $f(x)=-x^3$ when $x<0$? In general, should we define $f(x)=(x^k)^\frac{a}{b} = (\sqrt[b]{x^k})^a $ or $f(x)=(x^k)^\frac{a}{b} = (\sqrt[b]{x^{ka}}) $?
Thank you!