How to Use the IMSQRT and IMPOWER Functions in Microsoft Excel for Complex Radicals and Exponents

In electrical engineering, quantum mechanics, acoustic modelling, and fluid dynamics, formulas routinely demand raising complex numbers to powers (such as squaring an AC impedance or taking cube roots of transfer polynomials). Standard spreadsheet operators and functions like ^ or SQRT only operate on the real number line; evaluating =SQRT(-16) produces an immediate #NUM! error. Microsoft Excel provides two dedicated engineering functions designed specifically for complex arithmetic: IMSQRT (for square roots) and IMPOWER (for arbitrary integer and fractional exponentiation).

Mathematical Definitions of Complex Roots and Powers

Both functions utilise polar forms and de Moivre’s theorem to compute solutions:

• IMSQRT: Evaluates the principal square root of a complex number \(z = x + yi\): \[ \sqrt{z} = \sqrt{|z|} \left( \cos\frac{\theta}{2} + i\sin\frac{\theta}{2} \right) \] where \(|z| = \sqrt{x^2 + y^2}\) is the modulus and \(\theta = \text{atan2}(x, y)\) is the principal phase angle \((-\pi < \theta \le \pi)\).
• IMPOWER: Raises a complex number \(z\) to any real numerical power \(p\): \[ z^p = |z|^p \left( \cos(p\theta) + i\sin(p\theta) \right) \] This formulation supports positive integers, negative integers (reciprocals), and decimal fractions (such as \(p = 0.5\) for square roots or \(p = 0.3333\) for cube roots).

Function Syntax and Arguments

1. IMSQRT Syntax

=IMSQRT(inumber)

• inumber: A complex number formatted as text in the form "x+yi" or "x+yj", or a reference to a cell containing such a value. Plain numbers entered as digits are automatically converted to complex numbers with a zero imaginary part.

2. IMPOWER Syntax

=IMPOWER(inumber, number)

• inumber: The base complex number formatted as text ("x+yi").
• number: The real numerical exponent to which the complex base is raised.

Practical Excel Examples

1. Extracting Square Roots with IMSQRT

To take the square root of a negative real number, such as \(-16\):

=IMSQRT("-16")

Excel maps \(-16\) onto the complex plane and outputs:

4i

For a complete two-dimensional coordinate like "3+4i" in cell A2:

=IMSQRT(A2)

Because \(|3+4i| = 5\) and \(\theta \approx 0.9273\) radians, Excel returns the exact integer coordinate:

2 + i

2. Arbitrary Exponentiation with IMPOWER

To cube the complex number "1+2i":

=IMPOWER("1+2i", 3)

Using binomial expansion \((1+2i)^3 = 1 + 3(2i) + 3(2i)^2 + (2i)^3 = 1 + 6i – 12 – 8i = -11 – 2i\), Excel returns:

-11 - 2i

3. Calculating Negative Powers (Complex Inversion)

To compute the inverse square (\(z^{-2}\)) of "2+3i":

=IMPOWER("2+3i", -2)

Excel calculates the reciprocal magnitude and phase inversion, returning:

-0.029586 - 0.071006i

Companion Formulas for Engineering Analysis

Once roots or powers are evaluated, engineers frequently need to extract real values, imaginary parts, magnitudes, and phase degrees:

Desired Metric	Formula Pattern	Output Description
Real Resistance Component	=IMREAL(IMSQRT(A2))	Real scalar numeric value
Imaginary Reactance	=IMAGINARY(IMSQRT(A2))	Numeric imaginary coefficient
Magnitude / Amplitude	=IMABS(IMPOWER(A2, 3))	Distance from origin \(|z|^3\)
Phase Angle (Degrees)	=DEGREES(IMARGUMENT(IMPOWER(A2, 3)))	Angular rotation in degrees

Troubleshooting Common Errors

• #VALUE! Error: Occurs when the complex number string is malformed (e.g. contains stray spaces like "2 + 3i" instead of "2+3i"), or when the exponent in IMPOWER is not a valid real number.
• #NUM! Error: Occurs if inumber cannot be parsed as a complex coordinate or if zero is raised to a negative exponent (division by zero).