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2. Exponentiation - Wikipedia

   en.wikipedia.org/wiki/Base_(exponentiation)

   This definition of exponentiation with negative exponents is the only one that allows extending the identity + = to negative exponents (consider the case =). The same definition applies to invertible elements in a multiplicative monoid , that is, an algebraic structure , with an associative multiplication and a multiplicative identity denoted 1 ...

3. Sums of powers - Wikipedia

   en.wikipedia.org/wiki/Sums_of_powers

   In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.

4. Sixth power - Wikipedia

   en.wikipedia.org/wiki/Sixth_power

   In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So: n 6 = n × n × n × n × n × n. Sixth powers can be formed by multiplying a number by its fifth power, multiplying the square of a number by its fourth power, by cubing a square, or by squaring a cube. The sequence of sixth ...

5. Power of two - Wikipedia

   en.wikipedia.org/wiki/Power_of_two

   By comparison, powers of two with negative exponents are fractions: for positive integer n, 2 −n is one half multiplied by itself n times. Thus the first few negative powers of 2 are ⁠ 1 / 2 ⁠, ⁠ 1 / 4 ⁠, ⁠ 1 / 8 ⁠, ⁠ 1 / 16 ⁠, etc. Sometimes these are called inverse powers of two because each is the multiplicative inverse of ...

6. Sums of three cubes - Wikipedia

   en.wikipedia.org/wiki/Sums_of_three_cubes

   Semi-log plot of solutions of + + = for integer , , and , and .Green bands denote values of proven not to have a solution.. In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum.

7. Tetration - Wikipedia

   en.wikipedia.org/wiki/Tetration

   It is not known whether n q is rational for any positive integer n and positive non-integer rational q. [21] For example, it is not known whether the positive root of the equation 4 x = 2 is a rational number. [citation needed] It is not known whether e π or π e (defined using Kneser's extension) are rationals or not.