Examples of Irrational Algebraic Functions

In my textbook, it says that an irrational algebraic function is a function in which the independent variables appear under a radical sign or in a power with a rational number for its exponent.

I understand the variable should be under the square root.

However, I wasn't sure about it being to the power of a rational number.

One example would be x^1/2. However, x^2 would work because the power is a rational number, but isn't it considered a quadratic?

Could someone please explain this to me?

Thank you very much.

An Algebraic Function is the root of a polynomial equation. Thus a solution to $$F(x)^5+xF(x)^2+x=0$$ is an algebraic function.

A Rational Function is the quotient of two polynomials. This $$G(x)=\frac {x^5+x^2+1}{x^2-2}$$ is a rational function.

It's clear that every rational function is algebraic (since $$G(x)$$ satisfies $$q(x)G(x)-p(x)=0$$).

I've never heard the term used before, but it seems fair to say that an Irrational Algebraic Function is any Algebraic Function which isn't rational.

For example: $$H(x)=\sqrt x$$ is an irrational algebraic function. It's algebraic because it satisfies the polynomial $$H(x)^2-x=0$$. It's not rational because there are no two polynomials $$p(x),q(x)$$ such that $$H(x) = \frac {p(x)}{q(x)}$$ (if there were, we'd have $$p(x)=q(x)\times H(x)$$ but then the degree of $$p(x)$$ would not be an integer).

Not every Irrational Algebraic function has such a simple form, however. This follows from Abel's theorem on the Unsolvability of the general quintic equation.

While you are correct in saying that, for example, the exponent 2 is a rational number, I think what your textbook means to say is that the power should be a fraction, i.e. not an integer. In other words, something that could be written under a radical sign in the manner of $$x^{1/2} = \sqrt{x}$$.