IMPOWER in Google Sheets: Verify Complex Powers and Correct the Polar Formula
Raise complex values to numeric powers in Google Sheets, audit integer and reciprocal results and correct the source’s missing angle multiplier explicitly.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: IMPOWER(complex_base,exponent) raises a complex value to a numeric power. Both arguments are required. Repeated multiplication provides a transparent independent check for positive integer powers.
Availability and version history
- Introduced: first introduction date is not established by the reviewed official history; no desktop release number is inferred.
- Availability: documented in the rolling Google Sheets service; these examples use the web worksheet workflow.
- Checked: October 14, 2026; official IMPOWER reference. Documentation and independent models, with no native Sheets execution or wider client certification.
Syntax and the source equation problem
=IMPOWER(complex_base,exponent)
Google describes complex bases in i or j notation and says the exponent must be a number. Its examples include positive integers, negative integers and a negative fractional exponent.
The displayed polar equation omits multiplication of the angle by the exponent, even though its worked outputs require that multiplication. The correct mathematical model uses magnitude raised to n and angle n times theta.
For example, its published cube of 5+2i is 65+142i, confirmed by direct multiplication. Keeping the original angle unchanged would not produce that result. This guide uses independently checked powers and discloses the source defect rather than copying it.
Example 1: a power table for 3+i
Enter the exact complex text 3+i in A2. Put exponents 1, 2, 3, −1 and −2 in B2:B6. Enter =IMPOWER($A$2,B2) in C2 and fill down.
| B, exponent | Expected real coefficient | Expected imaginary coefficient |
|---|---|---|
| 1 | 3 | 1 |
| 2 | 8 | 6 |
| 3 | 18 | 26 |
| −1 | 0.3 | −0.1 |
| −2 | 0.08 | −0.06 |
The absolute A2 reference keeps the base fixed while the exponent changes. Extract coefficients into D and E with IMREAL and IMAGINARY if you want visible numeric checks.
These values are exact mathematical targets. A calculated complex text result may include small rounding differences, especially when evaluated through a polar representation.
Example 2: inspect repeated multiplication
The square of 3+i has real coefficient 9−1=8 and imaginary coefficient 2×3×1=6. Multiplying 8+6i by 3+i gives real coefficient 24−6=18 and imaginary coefficient 8+18=26.
Compare the cubic result with:
=IMPRODUCT("3+i","3+i","3+i")
The independent expected quantity is 18+26i. IMPRODUCT uses complete complex multiplication; squaring or cubing the separate numeric coefficients is a different operation.
For another source-supported control, =IMPOWER("5+2i",3) has expected quantity 65+142i. That published output independently exposes the missing angle multiplier in the source equation.
Advanced example: negative powers use a reciprocal
The reciprocal of 3+i is (3−i)/10, so the coefficients are 0.3 and −0.1. Squaring that reciprocal gives 0.08−0.06i, matching exponent −2.
Compare exponent −1 with:
=IMDIV(COMPLEX(1,0),"3+i")
Multiplying the reciprocal by 3+i recovers the mathematical value one. Its real coefficient is one and imaginary coefficient zero, regardless of the simplified display form chosen for that quantity.
Google separately documents =IMPOWER("-1-j",-1) returning −0.5+0.5j. This exact negative-integer control does not require choosing among fractional roots.
Advanced example: a bounded square-root control
For =IMPOWER("5+12i",0.5), an independent positive-real-root model gives 3+2i. Squaring that result recovers 5+12i.
Google’s IMSQRT reference explicitly equates its function with IMPOWER using exponent 0.5 and provides positive-real-component worked roots. That supports the bounded root workflow without defining a global branch interval.
The other mathematical root, −3−2i, also squares to 5+12i. A single returned root does not enumerate all possible roots or preserve the sign of an earlier squared quantity.
Do not infer Microsoft’s principal-angle interval as a Google Sheets rule. The reviewed Google IMPOWER page does not specify that complete interval or negative-axis endpoint behavior.
Advanced example: compare magnitudes and combined exponents
The base 3+i has magnitude SQRT(10). Its square has magnitude ten, and its cube has magnitude 10×SQRT(10), approximately 31.622776602.
Multiplying the square 8+6i by the cube 18+26i gives the fifth power, with coefficients −12 and 316. Its squared magnitude is 100000, equal to ten to the fifth power.
Check =IMPRODUCT(IMPOWER(A2,2),IMPOWER(A2,3)) against =IMPOWER(A2,5). This integer exponent-addition model is different from assuming unrestricted fractional-power identities across branch boundaries.
Errors and precision limits
The exponent must be numeric, but the dedicated source does not classify the exact error for every invalid exponent or base. Zero to zero, zero to negative powers, Boolean/blank coercion and competing errors remain unverified.
The source also prints a negative fractional example for 0.732−5.349i. Its numerical output can be independently corroborated, but one example does not establish every branch choice. New worked fractional outputs stay within the positive-real-root scope above.
Use both coordinate differences with an explicit tolerance for approximate results. No native Sheets execution was performed.
References and verification
Examples use fictional data, English function names and comma separators. Expected outputs are independent mathematical calculations or explicitly documented rules, not native Sheets captures. Check the spreadsheet locale when reproducing punctuation-sensitive formulas.