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In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number n is denoted n 3, using a superscript 3, [a] for example 2 3 = 8. The cube operation can also be defined for any other mathematical expression, for example (x + 1) 3.
For nonnegative integers n and m, the value of n m is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet). Some examples for particular values of m and n are given in the ...
[20] [21] The acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction, [22] is common in the United States [23] and France. [24] Sometimes the letters are expanded into words of a mnemonic sentence such as "Please Excuse My Dear Aunt Sally". [ 25 ]
The third power of the trinomial a + b + c is given by (+ +) = + + + + + + + + +. This can be computed by hand using the distributive property of multiplication over addition and combining like terms, but it can also be done (perhaps more easily) with the multinomial theorem.
In the third expression, n is the height, but each of the bases is different. Care must be taken when referring to iterated exponentials, as it is common to call expressions of this form iterated exponentiation, which is ambiguous, as this can either mean iterated powers or iterated exponentials .
Fermat knew that a fourth power cannot be the sum of two other fourth powers (the n = 4 case of Fermat's Last Theorem; see Fermat's right triangle theorem). Euler conjectured that a fourth power cannot be written as the sum of three fourth powers, but 200 years later, in 1986, this was disproven by Elkies with: 20615673 4 = 18796760 4 ...
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In mathematics, a cube root of a number x is a number y that has the given number as its third power; that is =. The number of cube roots of a number depends on the number system that is considered. Every nonzero real number x has exactly one real cube root that is denoted x 3 {\textstyle {\sqrt[{3}]{x}}} and called the real cube root of x or ...