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8 Μαρ 2014 · Exponentiation is a correspondence between addition and multiplication. Think of a number line, with $0$ in the "middle", and tick marks at each integer. Moving a certain distance to the right corresponds to adding a positive number, and adding the same number moves you the same distance, no matter where you are on the line.
10 Οκτ 2016 · If I know the outcome of a value being raised to a certain power, is it possible to know what the original base was? Example: x ^ 0.25 = 2.5045 What's the proper way to calculate x?
$\begingroup$ I disagree, you can actually write any decimal as a fraction. 0.14 is fourteen hundredths, or simplified 7/50. then add in 2/1 by raising to lowest common denominator, you end up with 2.14=107/50.
Thanks for pointing this out, Asaf. To avoid any anomalies, you may indeed want to distinguish set exponentiation from, say, exponentiation on the natural numbers when it is defined only as repeated multiplication. $\endgroup$ –
10 Σεπ 2013 · Your teachers aren’t hiding what comes next after exponentiation, it’s just that such a “fourth operation” doesn’t seem to have any practical application. The expression x x, useful in ballistics, can be thought of as x “operated on” by 2 (using the fourth operation) but that’s rather stretching to find an example.
5 Δεκ 2020 · Exponentiation is similar to functions in general in that they can both be represented as ordered pairs. In lambda calculus, the standard algorithmic definition of exponentiation is $\lambda be.eb$ , which is literally just defining it as a function and is by far the simplest mathematical operation in Church encoding.
13 Μαρ 2012 · This is known as Exponentiation by repeated squaring (see also Modular exponentiation) It deserves to be better known that this arises simply from writing the exponent in binary radix in Horner polynomial form, i.e. $\rm\ d_0 + 2\, (d_1 + 2\, (d_2\ +\:\cdots)).\, $ Below is an example of computing $\ x^{25}\ $ by repeated squaring.
You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the in...
This is impossible, so the discoverer of exponentiation (whoever that may have been) came up with a compromise: 1 would be a null on the left (1^x = 1) but an identity on the right (x^1 = x). This implies that, in general, 1^x is different from x^1, and thus exponentiation is not commutative.
5 Σεπ 2012 · Over the reals, the concept that "exponentiation = repeated multiplication" breaks down when you have non-integer exponents, so you have to start defining exponentiation using suprema of sets, which exploits the ordered field nature of the reals. The complex field is not an ordered field, so the equivalent notion of a supremum doesn't exist.
