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1 is the number itself. The word “exponent” was coined in 1544 by Michael Stifel. The base 3 appears 5 times in the repeated multiplication, because the exponent is 5. 3 to the 5th” or “3 to the 5”. The identity above may be derived through a definition aimed at extending the range of exponents to negative integers. 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. 10 are also used to describe small or large quantities.

The first negative powers of 2 are commonly used, and have special names, e. If the exponentiated number varies while tending to 1 as the exponent tends to infinity, then the limit is not necessarily one of those above. This sign ambiguity needs to be taken care of when applying the power identities. The definition of exponentiation using logarithms is more common in the context of complex numbers, as discussed below. Powers of a positive real number are always positive real numbers.

2 is also a valid square root. If the definition of exponentiation of real numbers is extended to allow negative results then the result is no longer well-behaved. Before the invention of complex numbers, cosine and sine were defined geometrically. Using exponentiation with complex exponents may reduce problems in trigonometry to algebra. So the same method working for real exponents also works for complex exponents.

Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. 0, 1, 2, or 3 modulo 4. Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Neither of these options is entirely satisfactory. The rational power of a complex number must be the solution to an algebraic equation.

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