Is expression $(1/x)/(2/x^2)$ is fraction expression or rational expression?

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A rational expression is nothing more than a fraction in which the numerator and/or the denominator are polynomials. Here are some examples of rational expressions.

$$\dfrac{6}{x-1}, \dfrac{x^2-1}{x^2+1}, x^2+3x+1.$$ We know that $\dfrac{\dfrac{1}{3}}{\dfrac{2}{5}}=\dfrac{5}{6}$. So $\dfrac{\dfrac{1}{3}}{\dfrac{2}{5}}$ is rational. Now Similarly we have

$$\dfrac{\dfrac{1}{x}}{\dfrac{2}{x^2}}=\dfrac{x^2}{2x}.$$ So do we can say that $\dfrac{\dfrac{1}{x}}{\dfrac{2}{x^2}}$ is rational expression?

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I am not sure what your definition of a fractional expression from what I've found a fractional expression is the same as a rational expression.

If your saying if the example is just a rational/fractional expression it depends on how far the expression is simplified. If the simplifed result has the variables in the denomonator cancelled out the result can be written as a rational function or a non-rational function.

So let's go back to your example of $\frac{\frac{1}{x}}{\frac{2}{x^2}}$. If you simplify it to $\frac{1}{2}x$ its is not a rational expression but if you choose to write it as $\frac{\frac{1}{2}x}{1}$ then its a rational expression.