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In mathematics, exponentiation, denoted b n, is an operation involving two numbers: the base, b, and the exponent or power, n. [1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [1] = ⏟.
[6] [7] [a] The parentheses can be omitted if the input is a single numerical variable or constant, [2] as in the case of sin x = sin(x) and sin π = sin(π). [a] Traditionally this convention extends to monomials; thus, sin 3x = sin(3x) and even sin 1 / 2 xy = sin(xy/2), but sin x + y = sin(x) + y, because x + y is not a monomial ...
The plus–minus sign, ±, is used as a shorthand notation for two expressions written as one, representing one expression with a plus sign, the other with a minus sign. For example, y = x ± 1 represents the two equations y = x + 1 and y = x − 1. Sometimes, it is used for denoting a positive-or-negative term such as ±x.
Also, characterisations (1), (2), and (4) for apply directly for a complex number. Definition (3) presents a problem because there are non-equivalent paths along which one could integrate; but the equation of (3) should hold for any such path modulo 2 π i {\displaystyle 2\pi i} .
The graph always lies above the x-axis, but becomes arbitrarily close to it for large negative x; thus, the x-axis is a horizontal asymptote. The equation d d x e x = e x {\displaystyle {\tfrac {d}{dx}}e^{x}=e^{x}} means that the slope of the tangent to the graph at each point is equal to its height (its y -coordinate) at that point.
1.000 2 ×2 0 + (1.000 2 ×2 0 + 1.000 2 ×2 4) = 1.000 2 ×2 0 + 1.000 2 ×2 4 = 1.00 0 2 ×2 4 Even though most computers compute with 24 or 53 bits of significand, [ 8 ] this is still an important source of rounding error, and approaches such as the Kahan summation algorithm are ways to minimise the errors.
This set is ordered lexicographically with the least significant position first: we write f < g if and only if there exists x ∈ β with f(x) < g(x) and f(y) = g(y) for all y ∈ β with x < y. This is a well-ordering and hence gives an ordinal number. The definition of exponentiation can also be given by transfinite recursion on the exponent β.
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.