In
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
and
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, the fifth
power or sursolid of a
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ...
''n'' is the result of multiplying five instances of ''n'' together:
:.
Fifth powers are also formed by multiplying a number by its
fourth power
In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So:
:''n''4 = ''n'' × ''n'' × ''n'' × ''n''
Fourth powers are also formed by multiplying a number by its cube. Furthe ...
, or the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of a number by its
cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross.
The cube is the on ...
.
The sequence of fifth powers of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s is:
:0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 100000, 161051, 248832, 371293, 537824, 759375, 1048576, 1419857, 1889568, 2476099, 3200000, 4084101, 5153632, 6436343, 7962624, 9765625, ...
Properties
For any integer ''n'', the last
decimal digit of ''n''
5 is the same as the last (decimal) digit of ''n''. I.E:
:
By the
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means ...
, there is no general
algebraic formula (formula expressed in terms of
radical expression
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'':
:r^n = x,
where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A root ...
s) for the solution of
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equati ...
s containing a fifth power of the
unknown as their highest power. This is the lowest power for which this is true. See
quintic equation
In algebra, a quintic function is a function of the form
:g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\,
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
,
sextic equation, and
septic equation.
Along with the fourth power, the fifth power is one of two powers ''k'' that can be expressed as the sum of ''k'' − 1 other ''k''-th powers, providing counterexamples to
Euler's sum of powers conjecture
Euler's conjecture is a disproved conjecture in mathematics related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers and greater than 1, if the sum of many th powers of positive integers is ...
. Specifically,
: (Lander & Parkin, 1966)
See also
*
Eighth power
*
Seventh power
In arithmetic and algebra the seventh power of a number ''n'' is the result of multiplying seven instances of ''n'' together. So:
:.
Seventh powers are also formed by multiplying a number by its sixth power, the square of a number by its fifth ...
*
Sixth power
*
Fourth power
In arithmetic and algebra, the fourth power of a number ''n'' is the result of multiplying four instances of ''n'' together. So:
:''n''4 = ''n'' × ''n'' × ''n'' × ''n''
Fourth powers are also formed by multiplying a number by its cube. Furthe ...
*
Cube (algebra)
In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together.
The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or ...
*
Square (algebra)
In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square ...
*
Perfect power
In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n' ...
Footnotes
References
*
*
*
*
*
*
Integers
Number theory
Elementary arithmetic
Integer sequences
Unary operations
Figurate numbers
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