Root Calculator
The n-th root with the parity doctrine — odd index keeps the sign, even index refuses negatives, and the index itself must earn its place.
Root Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Real roots only — complex roots not printed
- Integer indexes 2 and up — no radical simplification
In short: The 4th root of 81 is 3 (3⁴ = 81). The 5th root of −32 is −2 — odd index, sign preserved, since (−2)⁵ = −32. The 4th root of −16 is refused: an even index cannot reach a negative on the real line. And an index of 1 is refused on principle — the 1st root of x is x, a page would be a lie.
Formula
ⁿ√x = sign-law root for odd n
ⁿ√x = principal root for even n, x ≥ 0
the index decides the domain — parity is the whole law.
Worked Example
- Check the index. a whole number, 2 or more — anything else is not a root, it is identity or nonsense.
- Apply parity. odd n: one real root for every real x, sign kept. even n: x must be non-negative and the root is the positive one.
- Verify. raise the answer to the n-th power — the check is printed, not assumed.
The exponent bridge: the n-th root of x is x^(1/n) — typed on the Exponent page, a negative x with odd n returns not-a-number on binary floats for the same 1/3-not-representable reason; this page’s sign law is the honest route.
Strengths & Limits Of This Model
Where this engine is strong
- Parity doctrine printed beside the answer
- Power-back verification on every result
Where it stops
- No symbolic radical simplification
- No indexed-radical input (⁵√ typed as x and n)
Practical Use Cases
Finance
the n-th root of a growth factor is the per-period rate (CAGR)
Engineering
RMS values are 2nd roots; higher roots appear in fatigue laws
Geometry
an area-to-side back-solve with n = 2, volume-to-edge with n = 3
Methodology & Editorial Standards
Refuse an index that is not an integer of at least 2. Even index: refuse a negative radicand (no real root); return the principal (positive) root. Odd index: apply the sign law — sign(x) × |x|^(1/n). Perfect powers are detected by rounding and verified by raising back; the power-back check prints on every run.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Root Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
What does the index mean?
How many equal factors you are un-multiplying. Index 2 asks for a square, 3 for a cube, 5 for a fifth. The index is the root’s whole personality — it decides the domain and the sign behaviour.
Why do odd roots of negatives exist but not even roots?
Sign bookkeeping. An odd number of negative factors multiplies to a negative — so (−2)⁵ = −32 and the 5th root of −32 is −2. An even number of factors is always positive — so nothing real raised to an even power lands on −16, and the even root of a negative is refused on the reals.
Why refuse an index of 1?
The 1st root of x is x itself — a rule, but not a rooting. Indexes below 1, or between integers, are refused: the index is a COUNT of factors, and counts are whole and at least 2 before the question means anything.
How is this the same as a fractional exponent?
Exactly the same operation: the n-th root of x is x^(1/n). The Exponent page prints that bridge. The reason to keep a dedicated page is precision honesty: x^(1/n) evaluated as a power goes not-a-number for negative x with odd n (1/n is never exactly storable in binary), while the sign law — root the magnitude, keep the sign — never misses.
Which root does the symbol name for even indexes?
The positive one — the principal root, same doctrine as the Square Root page. The equation x⁴ = 81 has the pair ±3 as solutions; ⁴√81 names 3. Odd indexes need no convention: the root is unique.
What is a CAGR, and is it a root?
Compound annual growth rate — and yes. A factor of 2.07 over 6 years is a 2.07^(1/6) ≈ 1.129 per-year factor, about 12.9% a year. Roots are how compound growth is read backwards, which is most of what finance calls analysis.
Root page or Exponent page?
The exponent page owns the LAWS — products add, powers multiply — and this page owns the INVERSE question with the parity doctrine. They share the fractional-exponent bridge and disagree, honestly, on negative bases: the exponent page refuses what this page accepts by the sign law, and both pages say so.
Can the root be irrational?
Almost always. ⁴√81 = 3 is the rarity; ⁵√100 is an endless non-repeating decimal. The page prints significant digits and the power-back check — the check is the honesty, the digits are the portrait.