Related Function:

The IMPOWER function returns a complex number in x + yi or x + yj text format raised to a power.

Syntax

=IMPOWER(inumber,number)

Arguments

Argument Description
inumber A complex number you want to raise to a power
number The power to which you want to raise the complex number

Note: Use the COMPLEX function to convert real and imaginary coefficients into a complex number.

Examples

  A B C D
1 Data Formula Result Notes
2 3+4i =IMPOWER(A2,3) -117+44i 3+4i raised to the power of 3
3   =IMPOWER(“3+4i”,3) -117+44i  
4   =IMPOWER(“2-i”,3) 2-11i  
5   =IMPOWER(COMPLEX(4,1)) 52+47i  

Common Function Error(s)

Problem What went wrong
#NUM! Occurs if the supplied inumber argument is not recognized as a complex number
#NAME? Occurs when Analysis ToolPak add-in is not enabled

A complex number raised to a power is calculated as follows:  

    \[   (x+yi)^n = r^ne^{in\theta} = r^n\ \text{cos}n\theta + ir^n\ \text{sin}n\theta  \]

    \begin{align*} \text{where: } & r = \sqrt{x^2+y^2}  \\\text{and: } & \theta = \tan^{-1} \left(\frac{y}{x}\right) \\\text{and: } & \theta\epsilon(-\pi;\pi]  \\\end{align*}

See Wikipedia for more information on complex numbers.