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The first 3 powers of 2 with all but last digit odd is 2 4 = 16, 2 5 = 32 and 2 9 = 512. The next such power of 2 of form 2 n should have n of at least 6 digits. The only powers of 2 with all digits distinct are 2 0 = 1 to 2 15 = 32 768 , 2 20 = 1 048 576 and 2 29 = 536 870 912 .
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).
Analogously, the inverses of tetration are often called the super-root, and the super-logarithm (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function =, the two inverses are the cube super-root of y and the super-logarithm base y of x.
Leonard Eugene Dickson studied generalizations of Waring's problem for seventh powers, showing that every non-negative integer can be represented as a sum of at most 258 non-negative seventh powers [2] (1 7 is 1, and 2 7 is 128). All but finitely many positive integers can be expressed more simply as the sum of at most 46 seventh powers. [3]
2. Denotes the additive inverse and is read as minus, the negative of, or the opposite of; for example, –2. 3. Also used in place of \ for denoting the set-theoretic complement; see \ in § Set theory. × (multiplication sign) 1. In elementary arithmetic, denotes multiplication, and is read as times; for example, 3 × 2. 2.
The sequence of powers of ten can also be extended to negative powers. Similar to the positive powers, the negative power of 10 related to a short scale name can be determined based on its Latin name-prefix using the following formula: 10 −[(prefix-number + 1) × 3] Examples: billionth = 10 −[(2 + 1) × 3] = 10 −9
On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on (−∞,0]. Hence, zero is the (global) minimum of the square function. The square x 2 of a number x is less than x (that is x 2 < x) if and only if 0 < x < 1, that is, if x belongs to the open interval (0,1).
Zero to the power of zero, denoted as 0 0, is a mathematical expression that can take different values depending on the context. In certain areas of mathematics, such as combinatorics and algebra, 0 0 is conventionally defined as 1 because this assignment simplifies many formulas and ensures consistency in operations involving exponents.