How should one go about visualizing fractional exponents? In the basic sense $3^2$ is $3\cdot3$. This logic is easy to follow but I am trying to get a grip on how this works with decimals so for example $2^{1/{12}}$ is $2$ times what number? The struggle is trying to figure not what is a $1/12$ of $2$ but what is $1/12$ time of two. Is it even conceptually correct to even try to understand in terms of multiplication?
$x^{a/b} = \sqrt[b]{x^a}$
I see the logic in this relationship but fail to understand what number it is $a/b$.
Attempts include trying to find the $1/12$ of $2$ and then try to multiply that by $2$ but this yields an incorrect answer e.g. $(1/12) x 2 = 0.166667$ then $2$ times $0.16667$ which gives an incorrect answer of $.33333$ when using the calculator $2^{1/12} = 1.059463$.
Visualizing whole numbers like this is easy $3^2 = 3\cdot3$ but no sure if this a correct way for fractional exponents because $1/12 = .08333$ but $2 \cdot .08333$ also yields an incorrect answer. Looking for insights on how interpret this.
