How can you express radicals as multiplication/addition?

Question

How can you express radicals as multiplication/addition?

Most mathematical operations clearly reduce to multiplication/addition (same thing), but how do you do that for exponentials/radicals? Thank you.

Example: $x^{1/2}$ = ?

Answer 1

$x^k; k\in \mathbb N$ is $\underbrace{x\cdot x\cdot .... \cdot x}_{k\text{ times}}$

With $x^0 = 1$ and $x^{-k} = \frac 1{x^k}$.

$x^{\frac 1k}$ the real number $y$ (if any... it is assume $x > 0$ and $y > 0$) so that $\underbrace{y\cdot y \cdot...\cdot y}_{k \text{ times}}= y^k = x$.

And $x^{\frac mk} = (x^{\frac 1k})^m$ or in other words if $y$ is the $y$ so that $\underbrace{y\cdot y \cdot...\cdot y}_{k \text{ times}}= x$ then $x^{\frac mk} = \underbrace{y\cdot y\cdot ...\cdot y}_{m \text{ times}}$.

Need to keep in mind: 1) The doesn't define $x^{v}$ where $v$ is not rational. 2) if $x < 0$ then ... this doesn't always work. 3) There are might be more than one $y$ so that $y^k =x$. $x^{\frac 1k}$ refers specifically to the positive one, and 4) This makes a lot of assumptions that will need to be proven such as i) that such a $y$ so that $y\cdot ..... y=x$ actually exists; that there is exactly $1$ such thing and not many; ii) that if $\frac mk = \frac jl$ that $(x^{\frac 1k})^m = (x^{\frac 1l})^j$ etc.