# Is it possible to represent a root as a simple rational function with an exponent?

Using the following function:$$y=\frac{mx^p+b}{d}$$... where $m$, $p$, and $b$ may be any integer
... where $d$ may be any integer $\gt0$
... and where $x$ may be any rational number $\ge0$

Is it possible to manipulate the values of the available variables to make:$$y=\sqrt[r]{x}$$... where $r$ may be any integer?
... or perhaps where $r$ may be any integer $\ge0$, or $\gt0$, if necessary?

I'm working on some computer algorithms, and it'd be very beneficial to know if roots can be represented without requiring additional operations (such as $\log$).

But if I do need to add an additional operation to the function, in order to be able to represent a wider range of possible operations (such as roots), I may be inclined to do so. I'd really prefer to avoid having to use non-integer variable values if at all possible - for performance tuning. So far I've been able to represent the bulk of basic math operations (for rational numbers) with only integer values for these variables (apart from $x$ which is somewhat special) and I'd like to keep it this way if I can.

• Note that $\sqrt[r]0=0$ and $\frac{m0^p+b}d=\frac bd$ – Simply Beautiful Art Dec 16 '15 at 21:29

No, because $\frac{mx^{p}+b}{d}$ is rational but $y = \sqrt[r]{n}$ in general will not be (I assume you do not mean to reuse $x$ here).
• I did intend to reuse $x$ there - that was the point. But I do see what you are saying, that makes good sense. I know it's a rudimentary question, but hearing it put in those terms makes all the difference - I think I understand it now. Thanks for the help! – Giffyguy Dec 16 '15 at 19:17