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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; Sqrt[x] is canonicalised to Power[x, 1/2].

Examples (15)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (8)

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

Floor reduces exponent into [0,1)

In[4]:= (-1)^(7/4) In[4b]:= (-1)^(-1/5) Out[4b]= -(-1)^(4/5)

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)

3 and 5 share eff 1/3 -> grouped

In[7]:= 60^(1/3)
Out[7]= 2^(2/3) 15^(1/3)

Uniform exps -> stays

In[8]:= 6^(1/3)
Out[8]= 6^(1/3)

Applications (7)

In[9]:= 2^200
Out[9]= 1606938044258990275541962092341162602522202993782792835301376

In[10]:= (1/2)^-5
Out[10]= 32

In[11]:= 27^(2/3)
Out[11]= 9

In[12]:= 0^0
Out[12]= Indeterminate

In[13]:= (3 + 4 I)^10
Out[13]= -9653287 + 1476984*I

In[14]:= Sqrt[-12]
Out[14]= (2*I) Sqrt[3]

In[15]:= N[2^(1/2), 40]
Out[15]= 1.4142135623730950488016887242096980785697

Performance

Against other systems, from the benchmark suite (same input, results cross-checked for agreement):

case Mathilda Wolfram Python
Fourier 1200000 (mixed radix) 12.7 s 9.68 s 8.53 s
ListConvolve 100000 x 2048 9.71 s 1.1 s 10.2 s
ListCorrelate 100000 x 2048 9.68 s 1.1 s 10.4 s
Fourier 262143 (awkward size) 5.91 s 5.13 s 4.42 s
Fourier 2^18 (262144) 4.46 s 2.7 s 2.35 s
InverseFourier 2^18 3.37 s 2.8 s 2.39 s

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, ...]] (right-associative 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). Higher-power cases for q == 2 now also reduce: (-1)^(3/2) → -I, (-1)^(5/2) → I, (-12)^(3/2) → -24 I Sqrt[3] (the principal-branch rule (-n)^(p/2) = I^p · |n|^(p/2)).
  • For a negative base, the residual (-1)^(b/q) exponent is reduced into [0, 1) by floor division of p/q, pulling out a (-1)^a = ±1 sign that merges into the coefficient (Mathematica canonical form). This covers negative exponents and |p| ≥ q alike, e.g. (-1)^(-1/5) → -(-1)^(4/5), (-1)^(-2/3) → -(-1)^(1/3), (-1)^(-7/5) → (-1)^(3/5), (-1)^(5/4) → -(-1)^(1/4), (-8)^(-1/3) → -(-1)^(2/3)/2. Positive bases keep truncation toward zero (residual exponent in (-1, 1)), unchanged. Negative bases other than -1 with even q ≥ 4 (e.g. (-16)^(1/4)) are still left unevaluated.
  • Distributes power over product if the exponent is an integer.
  • Nested rational powers compose for any base when |inner exponent| < 1. (B^r)^s → B^(r·s) holds on the principal branch for any complex B when r = p/q is a non-integer rational with |p| < q (then r·Arg(B) stays in (-π, π], so no branch cut is crossed). This works without a positivity assumption: Sqrt[a^(2/3)] → a^(1/3), Sqrt[Sqrt[a]] → a^(1/4). Inner exponents with |p| ≥ q (e.g. a^(3/2)) and symbolic inner exponents (e.g. 2^a) still stay unevaluated.
  • Positive numeric coefficient splits out of a mixed Times base under a rational power: (c·w)^(p/q) → c^(p/q) · w^(p/q) when c > 0 is a numeric rational/integer/real that fully reduces under q and w is the symbolic residual (valid for any w since Arg(c) = 0). Combined with the nested-power rule this gives Sqrt[(1/27/a)^(2/3)] → 1/3 (1/a)^(1/3). Gated to fire only when the coefficient genuinely reduces, so Sqrt[2 Pi], (4 Pi)^(2/3), Sqrt[2 Sqrt[3]] stay nested.
  • For Power[Integer, Rational] with positive base and positive p/q exponent, splits the base's prime factorisation into a product of distinct-prime powers grouped by reduced effective exponent (Mathematica canonical form). Triggers only when the resulting form is strictly more informative -- uniform-exponent inputs like 6^(1/3) and 30^(1/3) keep the compact form.

Attributes: Listable, NumericFunction, OneIdentity, Protected.

References

See also: I, Times

Notes & additional examples

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[...].