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The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication. [ nb 2 ] The definition of x 0 requires further the existence of a multiplicative identity .
To compare numbers in scientific notation, say 5×10 4 and 2×10 5, compare the exponents first, in this case 5 > 4, so 2×10 5 > 5×10 4. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×10 4 > 2×10 4 because 5 > 2.
If n is an integer, the functional equation of the logarithm implies = () = (). Since the right-most expression is defined if n is any real number, this allows defining b x {\displaystyle b^{x}} for every positive real number b and every real number x : b x = exp ( x ln b ) . {\displaystyle b^{x}=\exp(x\ln b).}
Because superscript exponents like 10 7 can be inconvenient to display or type, the letter "E" or "e" (for "exponent") is often used to represent "times ten raised to the power of", so that the notation m E n for a decimal significand m and integer exponent n means the same as m × 10 n.
An integer is the number zero , a positive natural number (1, 2, 3, . . .), ... (since the result can be a fraction when the exponent is negative).
Two to the power of n, written as 2 n, is the number of values in which the bits in a binary word of length n can be set, where each bit is either of two values. A word, interpreted as representing an integer in a range starting at zero, referred to as an "unsigned integer", can represent values from 0 (000...000 2) to 2 n − 1 (111...111 2) inclusively.
When evaluating polynomials, it is convenient to define 0 0 as 1. A (real) polynomial is an expression of the form a 0 x 0 + ⋅⋅⋅ + a n x n, where x is an indeterminate, and the coefficients a i are real numbers. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents
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 . As for definition (5), the additive property together with the complex derivative f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} are sufficient to guarantee f ( x ) = e x ...