algebra precalculus - Converting Numbers to Exponential Form

can someone explain to me how problems such as:

are converted into exponential form? I am interested in the logic and method behind this, rather than the answer. Much thanks in advance.

Answer

a) $x^{\frac{1}{2}}$

b) $y^{-1}$

c)$(x+3)^{\frac{1}{6}}$

d) $(2x-3)^{-11}$

e) $y^\frac{7}{3}$

Notice that $\frac{1}{x^m}=x^{-m}$ and $\sqrt[m] x=x^{\frac{1}{m}}$