PowerExpand

PowerExpand[expr]

expands all powers of products and powers.

PowerExpand[expr,{x1,x2,}]

expands only with respect to the variables xi.

Details and Options

  • PowerExpand converts to , whatever the form of is.
  • PowerExpand also converts to , whatever the form of is.
  • The transformations made by PowerExpand are correct in general only if is an integer or and are positive real numbers.
  • PowerExpand converts Log[a^b] to bLog[a].
  • PowerExpand in general disregards all issues of branches of multivalued functions, so may not preserve the numerical values of expressions.
  • PowerExpand automatically threads over lists, as well as equations, inequalities and logic functions.
  • PowerExpand has the option Assumptions, specifying assumptions to use.
  • The default setting for the Assumptions option is Automatic, corresponding to a maximal set of assumptions.
  • You can specify default assumptions for PowerExpand using Assuming.

Examples

open allclose all

Basic Examples  (1)

Expand a square root, implicitly assuming positive real values:

Without PowerExpand, no expansion is done:

The expansion is only correct for positive real variables:

This gives a completely correct result:

This gives a result correct under the specified assumptions:

Scope  (10)

Expand a power of a product; the result may not be correct everywhere:

The general formula for expanding a power of a product:

Expand nested powers; the results may not be correct everywhere:

General formulas for expanding a nested power:

Expand the logarithm of a power; the result may not be correct everywhere:

The general formulas for expanding logarithms of powers:

Expand the logarithm of a product; the result may not be correct everywhere:

The general formula for expanding the logarithm of a product:

Expand compositions of inverse trigonometric and trigonometric functions:

This gives the universally correct formula:

Compute an expansion valid under the specified assumptions:

Expand the argument of a product:

Generalizations & Extensions  (1)

Expand only with respect to a and b:

Options  (3)

Assumptions  (3)

With the default setting Assumptions -> Automatic the expansions are not always correct:

When the assumptions are specified the result is correct under the given assumptions:

With Assumptions->True, PowerExpand gives a universally correct expansion formula:

Applications  (2)

Find universally correct expansion rules:

Expand under specified assumptions:

Properties & Relations  (5)

PowerExpand performs expansions valid under the given assumptions:

With Assumptions->True, PowerExpand gives general expansion formulas:

Refine and Simplify perform expansions valid under the given assumptions:

Use FunctionExpand to get a different representation of :

Use PiecewiseExpand to represent the result as a piecewise function:

Possible Issues  (1)

The result given by PowerExpand with Assumptions->Automatic may be incorrect:

Introduced in 1991
 (2.0)
 |
Updated in 1996
 (3.0)
2007
 (6.0)