# 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

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## 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 the expansions are not always correct:

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

With , 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 , 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 may be incorrect:

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