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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] = ⏟.
The next line-width is not disturbed by large operators. The code for the math example reads: <math display= "inline" > \sum_{i=0}^\infty 2^{-i} </math> The quotation marks around inline are optional and display=inline is also valid. [2] Technically, the command \textstyle will be added to the user input before the TeX command is passed to the ...
The matrix exponential of another matrix (matrix-matrix exponential), [24] is defined as = = for any normal and non-singular n×n matrix X, and any complex n×n matrix Y. For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y , because the multiplication operator for matrix ...
Many algorithms for exponentiation do not provide defence against side-channel attacks. Namely, an attacker observing the sequence of squarings and multiplications can (partially) recover the exponent involved in the computation. This is a problem if the exponent should remain secret, as with many public-key cryptosystems.
In expressions such as , the notation for exponentiation is usually to write the exponent as a superscript to the base number .But many environments — such as programming languages and plain-text e-mail — do not support superscript typesetting.
The law of exponential growth can be written in different but mathematically equivalent forms, by using a different base, for which the number e is a common and convenient choice: = = /. Here, x 0 {\displaystyle x_{0}} denotes the initial value of the quantity x , k is the growth constant, and τ {\displaystyle \tau } is the time it takes the ...
The polynomials, exponential function e x, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b.
Use divide and conquer to compute the product of the primes whose exponents are odd; Divide all of the exponents by two (rounding down to an integer), recursively compute the product of the prime powers with these smaller exponents, and square the result; Multiply together the results of the two previous steps