Wikipedia
In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations ( addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x − 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power $\tfrac{1}{2}$,
$$\sqrt{\frac{1-x^2}{1+x^2}}$$
is also an algebraic expression. By contrast, transcendental numbers like π and e are not algebraic.
A rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations ( commutative properties and associative properties of addition and multiplication, distributive property and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of arithmetic. Thus, $\frac{3x^2 - 2xy + c}{y^3-1}$ is a rational expression, whereas $\sqrt{\frac{1-x^2}{1+x^2}}$ is not.
A rational equation is an equation in which two rational fractions (or rational expressions) of the form $\frac{P(x)}{Q(x)}$ are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.
Usage examples of "algebraic expression".
If he wanted to work with a curve he would instinctively write it down, not as an intersection of planes and cones, but as a series of numbers and letters: an algebraic expression.
This was the algebraic expression of Pythagoras' theorem, of course.