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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:
Power threads element-wise over lists:
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Enter as a superscript using Ctrl+^:
In[1]:=
Click for copyable input
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Explicit FullForm:
In[1]:=
Click for copyable input
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Power threads element-wise over lists:
In[1]:=
Click for copyable input
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Click for copyable input
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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 threads over sparse arrays:
Power is a numeric function:
Create a "power tower":
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:
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:
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:
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