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In arithmetic and algebra, the fifth power or sursolid [1] of a number n is the result of multiplying five instances of n together: n 5 = n × n × n × n × n. Fifth powers are also formed by multiplying a number by its fourth power, or the square of a number by its cube. The sequence of fifth powers of integers is:
Graphs of y = b x for various bases b: base 10, base e, base 2, base 1 / 2 . Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
For example, a just perfect fifth (for example C to G) is 3:2 (Play ⓘ), 1.5, and may be approximated by an equal tempered perfect fifth (Play ⓘ) which is 2 7/12 (about 1.498). If the A above middle C is 440 Hz , the perfect fifth above it would be E , at (440*1.5=) 660 Hz, while the equal tempered E5 is 659.255 Hz.
The term hyperpower [4] is a natural combination of hyper and power, which aptly describes tetration. The problem lies in the meaning of hyper with respect to the hyperoperation sequence. When considering hyperoperations, the term hyper refers to all ranks, and the term super refers to rank 4, or tetration.
Notably, an equal-tempered octave has twelve half steps (semitones) spaced equally in terms of the sound frequency ratio. The sound frequency doubles for corresponding notes from one octave to the next. The ratio is 3/2 = 1.5 for a perfect fifth, for example from C to G on a major scale, and 5/4 = 1.25 for a major third, for example from C to E.
However, the square of the distance (denoted d 2 or r 2), which has a paraboloid as its graph, is a smooth and analytic function. The dot product of a Euclidean vector with itself is equal to the square of its length: v⋅v = v 2. This is further generalised to quadratic forms in linear spaces via the inner product.
Redundant, but sometimes used for emphasizing that a number is positive, specially when other numbers in the context are or may be negative; for example, +2. 3. Sometimes used instead of for a disjoint union of sets. − 1. Denotes subtraction and is read as minus; for example, 3 – 2. 2.
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.