Zoho Sheet IMPOWER: Raise Complex Numbers to Real Powers

• Introduced: The original release and date are not established by the checked official sources.
• Availability: Currently documented in Zoho Sheet’s rolling web-service help; there is no verified final supported-version endpoint.
• Scope: The English cloud reference does not establish identical behavior in every mobile client or imported workbook.
• Checked: October 14, 2026, against the official IMPOWER reference. Documentation and independent arithmetic checks; no native Zoho Sheet test.

=IMPOWER(complex_number; number)

All arguments are required; no optional parameter or default is documented in the official IMPOWER reference.

Use a fresh worksheet for each function so the results in column C do not overwrite another exercise. Enter the following in A1:B10, or import the two-column practice CSV at A1. Keep column A as text and column B as numbers. The literal text does not include enclosing quotation marks in cells.

Rows 7, 8 and 10 are optional boundary inputs, not part of the worked result checks. In particular, do not fill IMLN through the zero row and assume it has a logarithm.

For an integer exponent n, (a+bi)^n follows ordinary complex multiplication. Expanding (a+bi)^2 gives (a²−b²)+2abi. For positive integer powers, repeated multiplication is a convenient independent validation, but a real or fractional exponent requires a complex logarithm/branch convention. A general mathematical expression for nonzero z and real t is z^t = exp(t·Log(z)), where Log denotes a chosen branch of the complex logarithm. We do not assume Zoho’s branch rules for arbitrary negative or fractional arguments merely from its examples.

Reproduce the official examples

Source suffix inconsistency. The source shows "2+3j" producing -5+12i and "5j" producing 0-0.2i. Those numeric coefficients agree with mathematics; the string suffix differs. An output-suffix rule is not established by these examples. Consult IMPOWER’s original examples when comparing results.

Use the practice data above in A1:B10. A2 is 2+3j, A3 is 1-4i, A4 is 5j, A5 is 3+4j, and A6 is 2-1j. Put exponents B2=2, B3=3, B4=-1, B5=2, and B6=3. Then enter the following formulas one per row in column C (do not literally type the display text in the result cells):

The first three correspond to vendor examples; the last two are independent predictions. Compare real and imaginary coefficients rather than exact formatting. The exponent is not optional. No default exponent is documented by Zoho.

Squaring 3+4j: (3+4j)(3+4j)=9+12j+12j+16j². Since j²=−1, combine terms to obtain −7+24j. Confirm a matching coefficient pair from the row with A5, rather than demanding a specific imaginary-unit suffix or number of decimals.

Cubing 2−j: First (2−j)²=3−4j; multiply by (2−j) again: (3−4j)(2−j)=6−3j−8j+4j²=2−11j.

Power as repeated multiplication audit

For positive integer exponents, compare a square computed two ways: =IMPOWER(A5;2) and =IMPRODUCT(A5;A5). Both have expected coefficients -7 and 24; do not require identical result strings. Check the separately documented IMPRODUCT argument contract before using it with large ranges.

Reciprocal when exponent is negative

=IMPOWER("5j";-1) is officially documented. It is mathematically equivalent to 1/(5j). A comparison formula is =IMDIV("1+0j";"5j"), but exact parsing of "1+0j" and the results’ displayed suffixes must be tested. If you need a reciprocal in an operational sheet, test a nonzero known divisor first; negative exponent does not magically make division by zero meaningful.

Magnitude forecasting

For a complex value z, |z^n|=|z|^n for ordinary integer n, and mathematically also for real n for nonzero z with standard complex power conventions. For A5=3+4j, |z|=5; its square has magnitude 25 because sqrt((-7)²+24²)=25. A worksheet comparison could use =IMABS(IMPOWER(A5;2)) against =IMABS(A5)^2. IMABS documentation.

Powers on row-wise signal phasors

In a signal-processing or control-theory exercise, place one complex phasor in each A-row and an exponent in its B-row, then fill C2 with =IMPOWER(A2;B2) and copy down. A phase change from squaring a phasor can be checked mathematically through polar form: (r·e^{iθ})²=r²·e^{i2θ}. This is an interpretation of complex arithmetic, not an assurance about a real-world device’s calibration or measurement errors.

Fractional powers need branch awareness

The mathematical principal square root of 3+4j is 2+j, but there is also -2-j as another square root. If testing =IMPOWER(A5;0.5), compare it with =IMSQRT(A5) but do not assume equality in every branch-cut case without testing. For ambiguous inputs on negative real axes, explicitly document which branch your model expects. IMSQRT documentation.

• Zero raised to zero: Mathematically indeterminate in many contexts, with software conventions varying. Zoho’s IMPOWER page does not establish a 0^0 result; do not publish a predicted error code.
• Zero with a negative exponent: Mathematically involves division by zero. The vendor page does not identify a particular returned error code or parsing behaviour for this case.
• Fractional and negative-real bases: Complex roots involve branches. The documentation provides integer and negative integer exponents but does not specify comprehensive branch conventions.
• Non-numeric exponents: The documented second parameter is a number; any text coercion beyond documented behaviour requires testing.
• Large magnitudes/exponents: Overflow thresholds, maximum exponent, and precision loss are not established on this help page. Guard extreme inputs when production accuracy matters.
• Ranges/arrays: The official syntax specifies a single complex-number text input and a number. Do not assume an array formula of many bases or exponents will work in one call.
• Case-sensitive suffixes: COMPLEX explicitly documents lowercase i/j; IMPOWER’s own documentation does not establish arbitrary case acceptance.
• Blank input cells: Neither implicit empty-to-zero conversion nor resulting error is established here. Validate required inputs before filling formulas down.

Can IMPOWER replace POWER? POWER is a separately documented mathematical function. IMPOWER accepts text-form complex numbers, so use the function designed for the data type rather than assuming the two are interchangeable. POWER.

Should a j input produce a j output? Mathematically the imaginary-unit name is interchangeable. The official IMPOWER page unexpectedly shows i in results for two j inputs. Its real/imaginary coefficients still match arithmetic. Exact engine behaviour remains a test item.

Does IMPOWER take a complex exponent? The documented second argument is number, and the help does not establish a complex exponent. Do not silently promote it to one.

What about imported Excel files? The name suggests a similar mathematical concept, not a certified cross-application formula contract. Re-test in the target product.

Quote complex literals, such as "2+3j", but do not quote a reference such as A2. Multi-argument formulas here use the semicolons printed by Zoho’s English help; follow your workbook’s formula editor if its syntax differs.

The official page lists broad error families: inspect spelling and quotation marks for #NAME!, input types for #VALUE!, and cell references for #REF!. Its generic list does not establish the exact error returned for each malformed complex input. Avoid applying unverified blank-to-zero or uppercase-suffix conversions.

• COMPLEX constructs a complex representation.
• IMABS extracts magnitude.
• IMREAL and IMAGINARY let you inspect the two coefficients.

Compare independently expected coefficients within a suitable numerical tolerance. Printed precision and simplified zero components can vary; the examples do not establish a universal string-serialization rule.