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Also unlike addition and multiplication, exponentiation is not associative: for example, (2 3) 2 = 8 2 = 64, whereas 2 (3 2) = 2 9 = 512. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right -associative), not bottom-up [ 27 ] [ 28 ] [ 29 ] (or left -associative).
An expression like 1/2x is interpreted as 1/(2x) by TI-82, [3] as well as many modern Casio calculators [36] (configurable on some like the fx-9750GIII), but as (1/2)x by TI-83 and every other TI calculator released since 1996, [37] [3] as well as by all Hewlett-Packard calculators with algebraic notation.
In mathematics and computer science, optimal addition-chain exponentiation is a method of exponentiation by a positive integer power that requires a minimal number of multiplications. Using the form of the shortest addition chain , with multiplication instead of addition, computes the desired exponent (instead of multiple) of the base .
The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc. Various notations have been used to represent hyperoperations.
y or x · y. Plain text, programming languages, and calculators also use a single asterisk to represent the multiplication symbol, [6] and it must be explicitly used; for example, 3x is written as 3 * x. Rather than using the ambiguous division sign (÷), [a] division is usually represented with a vinculum, a horizontal line, as in 3 / x ...
Under the definition as repeated exponentiation, means , where n copies of a are iterated via exponentiation, right-to-left, i.e. the application of exponentiation times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation.
The same syntactic expression 1 + 2 × 3 can have different values (mathematically 7, but also 9), depending on the order of operations implied by the context (See also Operations § Calculators). For real numbers , the product a × b × c {\displaystyle a\times b\times c} is unambiguous because ( a × b ) × c = a × ( b × c ) {\displaystyle ...
One may take this relation as a definition of the natural operations by choosing S and T to be ordinals α and β; so α ⊕ β is the maximum order type of a total order extending the disjoint union (as a partial order) of α and β; while α ⊗ β is the maximum order type of a total order extending the direct product (as a partial order) of ...