What does rational expression mean?

Question

I need further understanding on rational expressions. I link here to the original website ClickMe

Answer 1

A rational expression involving a set of numbers (or variables, or other things you can add and multiply) is an expression using those numbers and the operations addition, subtraction, multiplication and division.

Since integral powers are just repeated multiplications, they are allowed in rational expressions.

So $$ \frac{3x^2}{x-1} $$ is a rational expression in $x$.

Sometimes an expression that does not look like a rational expression has a (more or less) equivalent form that is. So $$ \sqrt[3]{(x - 3)^3} $$ isn't a rational expression, but it's equal to the rational expression $x-3$.

I think that may explain the "reducible" and "irreducible" examples in the definition, but I don't like that distinction.

With square roots (as in the example you cite) you have to be careful about positive square roots when necessary - but that's a whole other kind of question. The fact that $|x|$ is either $x$ or $-x$ does not make $|x|$ a rational expression, since you need the full logic "if $x \ge 0$ then $|x| = x$ else ..." to calculate it, and the "if" part isn't a rational expression.