BUILT-IN MATHEMATICA SYMBOL

Power

gives

to the power

.

MORE INFORMATION

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.

Power is taken to be Power.

EXAMPLES

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Basic Examples (4)

Enter as a superscript using Ctrl+^:

Explicit FullForm:

Power threads element-wise over lists:

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Out[1]=

Enter as a superscript using Ctrl+^:

In[1]:=

Out[1]=

Explicit FullForm:

In[1]:=

Out[1]=

Power threads element-wise over lists:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

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:

Generalizations & Extensions (5)

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 threads over sparse arrays:

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:

Properties & Relations (20)

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:

Machine-precision numerical evaluation is inadequate:

Higher internal precision resolves the result:

Nonrational powers are not absorbed into series:

Power applies element-wise to matrices:

Use MatrixPower for matrix powers:

Neat Examples (3)

Plot successive powers:

Generate successive power towers:

Contour plot of the argument of such a tower:

Magnitudes of power towers of

:

Find the limit:

Solve for the limit: