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When an exponent is a positive integer, that exponent indicates how many copies of the base are multiplied together. For example, 3 5 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power.
In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation.
In the first two expressions a is the base, and the number of times a appears is the height (add one for x). In the third expression, n is the height , but each of the bases is different. Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as ...
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.
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. One such notation is (,).
See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm.
It is a binary operation defined with two numbers a and b, where a is tetrated to itself b − 1 times. The type of hyperoperation is typically denoted by a number in brackets, []. For instance, using hyperoperation notation for pentation and tetration, 2 [ 5 ] 3 {\displaystyle 2[5]3} means tetrating 2 to itself 2 times, or 2 [ 4 ] ( 2 [ 4 ] 2 ...
See Big O notation for a comparison of the rate of growth of various functions. The inverse of the double exponential function is the double logarithm log(log( x )). The complex double exponential function is entire , due to the fact that it is the composition of two entire functions f ( x ) = a x = e x ln a {\displaystyle f(x)=a^{x}=e^{x ...