Exponent and Root Calculator
Raise a number to any power and take its square, cube or nth root. A negative base needs a whole-number exponent
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Estimate only. Not engineering or safety advice. Check the result against the applicable standard and the equipment data, and have a competent engineer review it before it is relied on. Terms of use.
Open in the calculator with these values
About this calculator
This page calculates x to the power n, k-th root of x, x squared and 2 more results from number x, exponent n and root degree k. Change any value or its unit and the result updates. The steps below show the working for the default values, so you can check each line against your own method.
Limits of this calculator
Use this for square-law relations (power is current squared times resistance), for scaling laws such as the cube root in a transformer or fan relation, and for compound growth. The nth root of x is the same as x to the power 1/n. For the inverse of a power use the logarithm calculator: if xⁿ = y then n = log y / log x.
Cross-check: 31 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review.
Verification status: Engineering reference
Engineering reference. Formula and method documented and tested against separately coded reference calculations. Not yet Verified: the evidence listed on this page is still needed.
What this calculator checks
- The arithmetic of the published formula for the values you entered, in the units shown.
What remains unverified
- That the inputs suit your application. Take ratings, resistances and constants from the manufacturer data sheet or project specification.
Verification register entry
| Calculation author | Rahul Sharma (author) |
|---|---|
| Technical review | No independent technical review has been carried out |
| Individual professional approval | Not recorded |
| Formula reference | not recorded |
| Standard and edition | Not applicable: Pure mathematics or statistics; no electrical standard governs the method |
| Standard review state | Not applicable |
| Scope-screening note | not recorded |
| Cross-check test cases | 31 reference vectors (tests/reference_vectors.json) |
| Review date | not recorded |
| Next review date | not recorded |
| Status | Engineering reference |
Needed before this calculator can be marked Verified: reviewer name, qualification and review date; exact formula reference; formula source (textbook or standard); next review date.
Formula
Worked example
| Input | Value |
|---|---|
| Number x | 8 |
| Exponent n | 2 |
| Root degree k | 3 |
- x ^ n = 8 ^ 2 = 64
- k-th root: x ^ (1/3) = 2
- Check: (2) ^ 3 = 8
- x² = 64 x³ = 512 1 / x = 0.125
Assumptions
- Real numbers only: no complex result is returned
- A negative base is allowed only with a whole-number exponent, and zero cannot be raised to a negative power
- An odd root of a negative number is negative (the cube root of −8 is −2); an even root of a negative number is rejected
- Zero to the power zero is taken as 1
- The reciprocal is not defined for x = 0 and is shown as a dash
Use this for square-law relations (power is current squared times resistance), for scaling laws such as the cube root in a transformer or fan relation, and for compound growth. The nth root of x is the same as x to the power 1/n. For the inverse of a power use the logarithm calculator: if xⁿ = y then n = log y / log x.
Inputs and results
| Input | Unit | Allowed or shown | Note |
|---|---|---|---|
| Number x | (number) | any number | The base for the power and the number under the root. |
| Exponent n | (number) | any number | Any number, positive, negative or a fraction. A negative base needs a whole-number exponent. |
| Root degree k | (number) | 1 or more, up to 1,000 | A whole number from 1 to 1000: 2 for a square root, 3 for a cube root. Even roots need x of zero or more. |
| Result | Unit | Allowed or shown | Note |
|---|---|---|---|
| x to the power n (xⁿ) | (number) | Shown to 10 decimal places | |
| k-th root of x | (number) | Shown to 10 decimal places | |
| x squared | (number) | Shown to 10 decimal places | |
| x cubed | (number) | Shown to 10 decimal places | |
| Reciprocal 1 / x | (number) | Shown to 10 decimal places |
How it is calculated
Each input is converted to the base unit of its quantity, the formula above is applied in the order shown in the worked example, and each result is shown in the unit you choose. Results are held at full precision and rounded only for display, except where a step selects a standard size, as the worked example shows. A value outside its allowed range is refused with a message instead of giving a result.
How the result changes with number x
| Number x | x to the power n | k-th root of x | x squared |
|---|---|---|---|
| 0.8 | 0.64 | 0.9283177667 | 0.64 |
| 2 | 4 | 1.2599210499 | 4 |
| 4 | 16 | 1.587401052 | 16 |
| 8 | 64 | 2 | 64 |
| 16 | 256 | 2.5198420998 | 256 |
| 32 | 1,024 | 3.1748021039 | 1,024 |
| 80 | 6,400 | 4.3088693801 | 6,400 |
Other inputs are the same as in the worked example above.
Standards and further reading
Complex numbers, rectangular and polar form and angle conversion are standard mathematics, not the subject of a standard. Standards apply when the result feeds an electrical design, and they set the method there.
Frequently asked questions
What do I need to enter?
Enter number x, exponent n and root degree k. Change the unit of a value with its selector and the result stays the same.
What does this calculator assume?
Real numbers only: no complex result is returned. A negative base is allowed only with a whole-number exponent, and zero cannot be raised to a negative power. An odd root of a negative number is negative (the cube root of −8 is −2); an even root of a negative number is rejected. Zero to the power zero is taken as 1. The reciprocal is not defined for x = 0 and is shown as a dash.
Which calculators should I use next?
Common next steps are Logarithm Calculator, Quadratic Equation Solver, Trigonometry (Sin, Cos, Tan) and RMS, Peak and Peak-to-Peak. The related links at the bottom of this page open them.
How accurate are the results?
Every figure comes from the same tested engine as the Elecmetric calculators and is meant for estimating. 31 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review. Check design values against the standard and the equipment data you work to.
Formula guides and reference
Longer explanations with worked examples and tables, written for the same topic.
- Quadratic, Logarithm and Power Formulas: Formulas for the quadratic equation, logarithms and exponents with electrical examples: real and complex roots, decibels
- Trigonometry Formulas for Electrical Engineers: Sine, cosine and tangent formulas for engineers: the six trig functions, degrees and radians
- Complex Number Formulas: Polar and Rectangular: Complex number formulas for engineers: rectangular and polar form, conversion with atan2, addition
More Engineering Mathematics calculatorsAll engineering calculatorsExponent and Root Calculator in Elecmetric