^, _power

Raise an expression to a power

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For the power function in MATLAB®, see power.

Syntax

x ^ y
_power(x, y)

Description

x^y computes the y-th power of x.

x^y is equivalent to the function call _power(x, y).

The power operator ^ is left associative: x^y^z is parsed as (x^y)^z. Cf. Example 2.

If x is a polynomial of type DOM_POLY, then y must be a nonnegative integer smaller than 231.

_power is overloaded for matrix domains (matrix). In particular, x^(-1) returns the inverse of the matrix x.

Use powermod to compute modular powers. Cf. Example 3.

Mathematically, the call sqrt(x) is equivalent to x^(1/2). Note, however, that sqrt tries to simplify the result. Cf. Example 4.

If x or y is an element of a domain with a slot"_power", then this method is used to compute x^y. Many library domains overload the ^ operator by an appropriate "_power" slot. Powers are processed as follows:

x^y is searched for elements of library domains from left to right. Let z (either x or y) be the first term that is not of one of the basic types provided by the kernel (numbers, expressions, etc.). If the domain d = z::dom = domtype(z) has a slot"_power", it is called in the form d::_power(x, y). The result returned by d::_power is the result of x^y.

See Example 6 and Example 7.

For finite sets X, Y, the power X^Y is the set .

Examples

Example 1

Some powers are computed:

2^10, I^(-5), 0.3^(1/3), x^(1/2) + y^(-1/2), (x^(-10) + 1)^2

Use expand to “expand” powers of sums:

(x + y)^2 = expand((x + y)^2)

Note that identities such as (x*y)^z = x^z * y^z only hold in certain areas of the complex plane:

((-1)*(-1))^(1/2) <> (-1)^(1/2) * (-1)^(1/2)

Consequently, the following expand command does not expand its argument:

expand((x*y)^(1/2))

Example 2

The power operator ^ is left associative:

2^3^4 = (2^3)^4, x^y^z

Example 3

Modular powers can be computed directly using ^ and mod. However, powermod is more efficient:

123^12345 mod 17 = powermod(123, 12345, 17)

Example 4

The function sqrt produces simpler results than _power:

sqrt(4*x*y), (4*x*y)^(1/2)

Example 5

For finite sets, X^Y is the set :

{a, b, c}^2, {a, b, c}^{q, r, s}

Example 6

Various library domains such as matrix domains or residue class domains overload _power:

x := Dom::Matrix(Dom::IntegerMod(7))([[2, 3], [3, 4]]):
x^2, x^(-1), x^3 * x^(-3)

delete x:

Example 7

This example demonstrates the behavior of _power on user-defined domains. Without a "power"slot, powers of domain elements are handled like any other symbolic powers:

myDomain := newDomain("myDomain"): x := new(myDomain, 1): x^2

type(x^2), op(x^2, 0), op(x^2, 1), op(x^2, 2)

After the "_power" slot is defined, this method is used to compute powers of myDomain objects:

myDomain::_power := proc() begin "The result" end: x^2

delete myDomain, x:

Return Values

Arithmetical expression, a polynomial, a floating-point interval, or a set.

Overloaded By

x, y

See Also

MuPAD Functions