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Power

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

x ^ y or Power[x, y] represents x to the power y.
Power is Listable, NumericFunction, and OneIdentity. Integer exponents
are reduced exactly (repeated squaring on GMP); Rational and Real
exponents evaluate numerically when the base is numeric; Power[0, 0]
stays Indeterminate; Power[x, 1/2] is canonicalised to Sqrt[x].

Examples

All examples below are verified against the current Mathilda build.

In[1]:= Sqrt[45]
Out[1]= 3 Sqrt[5]

In[2]:= (a * b)^2
Out[2]= a^2 b^2

In[3]:= (-1)^(3/2)
Out[3]= -I

In[4]:= (-1)^(7/4)
Out[4]= -(-1)^(3/4)

In[5]:= 18^(1/3)
Out[5]= 2^(1/3) 3^(2/3)

In[6]:= 12^(1/3)
Out[6]= 2^(2/3) 3^(1/3)

In[7]:= 60^(1/3)            (* 3 and 5 share eff 1/3 -> grouped *)
Out[7]= 2^(2/3) 15^(1/3)

In[8]:= 6^(1/3)             (* uniform exps -> stays *)
Out[8]= 6^(1/3)

Implementation notes

Algorithm. builtin_power evaluates Power[base, exp]. Power[x] is x; Power[b, e1, e2, ...] is right-associated into Power[b, Power[e1, e2, ...]] (matching Mathematica's right-grouping). The two-argument core handles, in order: infinity/Indeterminate algebra (0^Infinity -> 0, 1^Infinity -> Indeterminate with message, Infinity^n by sign of n, etc.); numeric exact folding (integer/rational/bigint powers via GMP, e.g. exact 2^10, (1/2)^3); inexact Real/MPFR exponentiation; partial radical simplification of integer^(p/q) (pulling out perfect-power factors so Sqrt[8] -> 2 Sqrt[2]); (b^m)^n -> b^(m·n) and product/zero/one identities; and Sqrt-style rational-exponent canonicalisation. Sqrt[x] is a thin wrapper (builtin_sqrt) that rewrites to Power[x, 1/2]. Symbolic cases that cannot be reduced return NULL, leaving the call unevaluated. Power is ONEIDENTITY | LISTABLE | NUMERICFUNCTION | PROTECTED (note: not Flat/Orderless — exponentiation is neither associative nor commutative).

Data structures. Expr* trees; exact integer/bigint exponentiation uses GMP mpz, rationals via make_rational, and MPFR for high-precision reals. Radical factor extraction works on integer factorisation of the base.

Complexity / limits. Integer powers are O(log exp) GMP multiplies; radical canonicalisation costs a factorisation of the integer base.

  • Listable.
  • Simplifies integer powers of integers.
  • Returns Overflow[] if the result exceeds 64-bit integer limits.
  • Reduces radicals (e.g., 8^(1/2) becomes 2*Sqrt[2]).
  • Supports complex results for negative bases (e.g., (-1)^(1/2) becomes I).

Attributes: Listable, NumericFunction, OneIdentity, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

  • Knuth, "The Art of Computer Programming, Vol. 2: Seminumerical Algorithms", on binary exponentiation.
  • Geddes, Czapor & Labahn, "Algorithms for Computer Algebra" (1992), on simplification of radical powers.
  • Source: src/power.c
  • Specification: docs/spec/builtins/arithmetic.md

Notes & additional examples

Worked examples

In[1]:= 2^200
Out[1]= 1606938044258990275541962092341162602522202993782792835301376

In[1]:= (1/2)^-5
Out[1]= 32

In[1]:= 27^(2/3)
Out[1]= 9

In[1]:= 0^0
Out[1]= Indeterminate

A Gaussian-integer base is raised exactly, keeping the real and imaginary parts as integers:

In[1]:= (3 + 4 I)^10
Out[1]= -9653287 + 1476984*I

A negative radicand is split into its real radical and the principal complex unit:

In[1]:= Sqrt[-12]
Out[1]= (2*I) Sqrt[3]

Irrational powers numericalise to arbitrary precision on request:

In[1]:= N[2^(1/2), 40]
Out[1]= 1.4142135623730950488016887242096980785697

Notes

Integer powers use binary exponentiation and promote to GMP bigints, so 2^200 is exact. A rational base with a negative integer exponent inverts and raises, giving (1/2)^-5 = 32. Rational exponents trigger perfect-power extraction: 27^(2/3) reduces to 9, while non-extractable cases such as 8^(1/3) of a non-cube stay symbolic. The indeterminate form 0^0 evaluates to Indeterminate rather than 1. Complex bases (Gaussian integers, negative radicands) are handled in closed form, and irrational powers of numeric bases evaluate to the requested precision under N[...].