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In mathematics, an algebraic expression is an expression built up from constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number).^{[1]} For example, 3x^2 - 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power \tfrac{1}{2},
is also an algebraic expression.
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, 3x^2 - 2xy + c 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.
Algebra has its own terminology to describe parts of an expression:
1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, x, y - variables
By convention, letters at the beginning of the alphabet (e.g. a, b, c) are typically used to represent constants, and those toward the end of the alphabet (e.g. x, y and z) are used to represent variables.^{[2]} They are usually written in italics.^{[3]}
By convention, terms with the highest power (exponent), are written on the left, for example, x^2 is written to the left of x. When a coefficient is one, it is usually omitted (e.g. 1x^2 is written x^2).^{[4]} Likewise when the exponent (power) is one, (e.g. 3x^1 is written 3x),^{[5]} and, when the exponent is zero, the result is always 1 (e.g. 3x^0 is written 3, since x^0 is always 1).^{[6]}
The table below summarizes how algebraic expressions compare with several other types of mathematical expressions.
A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x^{2} + 4x + 4. An irrational algebraic expression is one that is not rational, such as √x + 4.
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