This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Power

 gives to the power .
• 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 and , Power gives the principal value of . »
• is automatically converted to only if is an integer.
• is automatically converted to only if 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 has a branch cut discontinuity for non-integer running from to 0 in the complex plane.
Enter as a superscript using Ctrl+^:
Explicit FullForm:
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Enter as a superscript using Ctrl+^:
 Out[1]=

Explicit FullForm:
 Out[1]=

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 Scope   (8)
Exact roots are found when possible:
Get a numerical approximation:
is immediately treated as an approximate number:
Power threads element-wise over lists and matrices:
Roots are factored out when possible:
Complex numbers are generated when necessary:
The principal root is always used:
Find powers of complex numbers:
Find limits at branch cuts:
Power can deal with real-valued intervals:
Vanishing and infinite arguments give symbolic results:
Depending on the real part of n, the result can be 0 or infinity:
Power is a numeric function:
Create a "power tower":
 Applications   (3)
5% compound interest:
Find the radius of a sphere of the same volume as a cube with side :
Contour plot of a complex inverse power:
Equivalent forms for square roots:
Whole powers of roots are automatically simplified:
Roots of powers cannot be automatically simplified:
Simplify with assumptions:
Use PowerExpand to do formal simplification:
Get results valid for all complex :
Use ExpToTrig to get trigonometric forms:
Reduce to single roots:
Use Solve or Root to find all roots:
Use Expand to expand out powers of polynomials:
Powers are automatically applied to series:
Equations involving powers can have infinitely many solutions:
Reciprocals, square roots, etc. are automatically converted to powers:
Exponentials are converted to powers:
Match powers of :
Include the case :
Branch cut structure for fractional powers in the complex plane:
Test whether powers are algebraic:
Integrals:
Integral transforms:
Sums:
Differential equations:
Power appears in special cases of many mathematical functions:
 Possible Issues   (13)
Power always computes principal roots:
Powers are not generically inverses of roots:
With approximate numbers, imaginary parts can be generated:
Use Chop to remove the small imaginary part:
The branch cut makes this function discontinuous:
Its derivative nevertheless generically simplifies to 0:
Machine-precision can give incorrect numerical results on the branch cut:
Machine-number inputs can give arbitrary-precision results:
Powers can be very large:
Some powers are too large for any computer:
Powers can give indeterminate expressions:
The precision of each result is determined by the precision of the zero:
Symbolic powers of 1 are only evaluated when the 1 is an exact or machine-precision number:
Numerical decision procedures with default settings cannot simplify this power: