Algebraic expressions

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).^[1] For example, ${\displaystyle 3x^{2}-2xy+c}$ is an algebraic expression. Since taking the square root is the same as raising to the power ${\displaystyle {\tfrac {1}{2}}}$,

${\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}$

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, ${\displaystyle {\frac {3x^{2}-2xy+c}{y^{3}-1}}}$ is a rational expression, whereas ${\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}$ is not.

A rational equation is an equation in which two rational fractions (or rational expressions) of the form ${\displaystyle {\frac {P(x)}{Q(x)}}}$ 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.