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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 solution (g, h, k) is another solution to the original equation, but smaller (0 < g < d < x). Applying the same procedure to (g, h, k) would produce another solution, still smaller, and so on. But this is impossible, since natural numbers cannot be shrunk indefinitely. Therefore, the original solution (x, y, z) was impossible.
In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers) variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).
When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of their base. [2] Thus 3 + 5 2 = 28 and 3 × 5 2 = 75. These conventions exist to avoid notational ambiguity while allowing notation to remain brief. [4]
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
Over a field of characteristic 0, an algebra is power-associative if and only if it satisfies [,,] = and [,,] =, where [,,]:= () is the associator (Albert 1948). Over an infinite field of prime characteristic p > 0 {\displaystyle p>0} there is no finite set of identities that characterizes power-associativity, but there are infinite independent ...