Built-in Mathematica Symbol

PowerExpand

PowerExpand[expr]
expands all powers of products and powers.

PowerExpand[expr, {x₁, x₂, ...}]
expands only with respect to the variables x_i.

MORE INFORMATION

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 c is an integer or a and b 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:

Expand a square root, implicitly assuming positive real values:

In[1]:=

Out[1]=

Without PowerExpand, no expansion is done:

In[2]:=

Out[2]=

The expansion is only correct for positive real variables:

In[3]:=

Out[3]=

This gives a completely correct result:

In[4]:=

Out[4]=

This gives a result correct under the specified assumptions:

In[5]:=

Out[5]=

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)

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: