Power

x^y

gives x to the power y.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Exact rational number results are given when possible for roots of the form .
  • For complex numbers x and y, Power gives the principal value of . »
  • (a b)^c is automatically converted to a^c b^c only if c is an integer.
  • (a^b)^c is automatically converted to a^(b c) only if c is an integer.
  • For certain special arguments, Power automatically evaluates to exact values.
  • Power can be evaluated to arbitrary numerical precision.
  • Power automatically threads over lists.
  • Power[x,y] has a branch cut discontinuity for noninteger y running from to 0 in the complex x plane.
  • Power[x,y,z,] is taken to be Power[x,Power[y,z,]].

Background & Context

  • Power is a mathematical function that raises an expression to a given power. The expression Power[x,y] is commonly represented using the shorthand syntax x^y or written in 2D typeset form as xy. A number to the first power is equal to itself (), and 1 to any complex power is equal to 1 (). The inverse of a power function is given by Log, so solving the equation for gives a principal solution of .
  • The operation of taking an expression to the second power is known as squaring and the operation of taking an expression to the third power is known as cubing. The rules for combining quantities containing powers are called the exponent laws, and the process of raising a base to a given power is known as exponentiation. Many expressions involving Power, Exp, Log, and related functions are automatically simplified or else may be simplified using Simplify or FullSimplify. PowerExpand can be used to do formal expansion and associated simplification, and ExpToTrig can be used to get trigonometric forms of Power expressions.
  • The function Sqrt[x] is represented using Power[x,1/2]. Exponentiation using the base of the natural logarithm E can be input as Exp[x] but is represented using Power[E,x].
  • Power[x,y] has a branch cut discontinuity for y running from to 0 in the complex x plane for noninteger y. Because of this branch cut, Power[x,1/n] returns a complex root by default instead of the real one for negative real x and odd positive n. To obtain a real-valued n^(th) root, Surd[x,n] can be used. The special case CubeRoot[x] corresponds to Surd[x,3].

Examples

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Basic Examples  (4)

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Enter as a superscript using :

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Explicit FullForm:

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Power threads element-wise over lists:

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Scope  (8)

Generalizations & Extensions  (5)

Applications  (3)

Properties & Relations  (25)

Possible Issues  (13)

Neat Examples  (3)

See Also

Sqrt  Exp  CubeRoot  Surd  PowerExpand  PowerMod  Log  Log10  Log2  LogLogPlot  FactorialPower  PowerRange  PolygonalNumber

Tutorials

Introduced in 1988
(1.0)