CalcaTools

Root Calculator | Compute Square, Cube, Or Any Nth Root Precisely

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Find the square root, cube root or any nth root of a number to high precision, with the exponent form shown. Handles perfect roots exactly and irrational roots to many decimal places — ideal for geometry, physics and algebra work.

Last updated: July 2026 · Free · No sign-up required

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Results

Enter values above and click Calculate to see your result instantly.

Quick reference

ExpressionValue
√255
∛644
√0.040.2
4th root of 813

functions Shows the working, not just the answer

For students and teachers, the final number is only half the value. Every math tool here exposes the formula it applied, the intermediate steps, and the rounding rule, so you can follow along, check your homework, or use the answer in a proof or report with confidence.

calculate Accurate to the spec

Calculations use 64-bit floating point with sensible rounding for the domain (currency to 2 decimals, percentages to 4 decimals, algebra to 6 significant figures). Where exact rational arithmetic matters — fractions, factorials, simplification — we use a dedicated BigNumber path so 1/3 + 1/6 returns ½, not 0.49999.

school Free for classroom use

Educators are welcome to link to any math calculator on CalcaTools from a class site, Google Classroom, or worksheet. The pages are mobile-friendly, free, ad-supported (so we can keep them free) and have no sign-up wall — students just click and use them in class or at home.

tips_and_updates Pair with the spoke articles

Below the calculator we link a small set of plain-English explainer pages — "What is a percentage?", "Why does PEMDAS matter?", and so on. They cover the underlying concept in 4–6 short paragraphs. Read those before the calculator if the topic is new, or after if you want the extra context.

Interpretation guide

RootMeaning
Square rootNumber that multiplies by itself once
Cube rootNumber cubed equals the input
nth rootInverse of exponent n

lightbulb Worked example: cube size from volume

A shipping crate must hold exactly 2.744 m³. How long is each edge? Edge = ∛2.744 = 2.744^(1/3). Since 1.4³ = 2.744, the cube root is exactly 1.4 — a perfect root, so no rounding is involved.

Result: Each edge of the crate is exactly 1.4 m long.

What this means: Because 2.744 is a perfect cube, the answer is exact. For non-perfect radicands the calculator returns the full-precision decimal instead — ∛30 ≈ 3.107 — so you can tell at a glance whether a root is exact or approximate.

Formula & methodology

Formula: nth root of x = x^(1/n)

How roots are computed

An nth root is a fractional exponent:

ⁿ√x = x^(1/n)

  • Square root: n = 2; cube root: n = 3; any positive index works.
  • Perfect roots are detected and returned exactly: ∛27 = 3, √144 = 12.
  • Irrational results show full double precision (~15–16 digits): √2 ≈ 1.4142135623730951.
  • Even roots of negative numbers have no real answer; odd roots do (∛(−27) = −3).

Reading the result

A 2.744 m³ crate has edges ∛2.744 = 1.4 m, since 1.4³ = 2.744. Results include the simplified radical form where useful — √72 = 6√2 — alongside the decimal value.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do I use the root calculator?
Type the number you want the root of and choose the index: 2 for a square root, 3 for a cube root, or any other n for an nth root. The tool computes x^(1/n) to many decimal places. For example, the square root of 625 is 25 because 25 × 25 = 625, and the cube root of 27 is 3 because 3 × 3 × 3 = 27.
What size cube holds 2.744 cubic meters?
Take the cube root: ∛(2.744) = 1.4 m per edge, since 1.4³ = 2.744. In exponent form, the nth root of x is x^(1/n) — so ∛(2.744) = 2.744^(1/3). The calculator evaluates fractional exponents directly and flags perfect roots.
How do you simplify √72 by hand?
Factor out the largest perfect square: 72 = 36 × 2, so √72 = 6√2 ≈ 8.4853. The simplified radical form 6√2 is exact, while 8.4853 is a decimal approximation. The calculator shows both the simplified surd and the decimal value.
Why do my results differ from another calculator?
The Root Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. This root calculator handles square roots, cube roots, and any nth root you specify. Type a number and an index and the tool returns the exact root to many decimal places, distinguishes perfect roots from irrational results, and shows the calculation method There is no limit on usage, and it works on any device.
How do I simplify a square root like √72?
Pull out the largest perfect-square factor: 72 = 36 × 2, so √72 = 6√2. If the largest factor is not obvious, factor stepwise — √72 = √4 × √18 = 2√18 = 2 × 3√2 = 6√2 lands in the same place. A root is fully simplified when no perfect square remains inside, which is the form math and physics problems expect.
What is the difference between a square root and a cube root?
The square root asks which number times itself gives x; the cube root asks which number cubed gives x. √64 = 8 (and −8), while ∛64 = 4 only — odd roots of positive numbers have a single real answer, even roots have two. The calculator handles any index n you specify.
Can you take an even root of a negative number?
Not over the real numbers — no real value squared gives −9, so √(−9) has no real answer (it is 3i in complex numbers). Odd roots are fine: ∛(−27) = −3 because (−3)³ = −27. The calculator returns real results and reports even roots of negatives as undefined.
How many digits do the results carry?
Roots are computed in double precision — about 15–16 significant digits — so √2 shows as 1.4142135623730951 rather than a rounded 1.41. Perfect roots like ∛27 = 3 are detected and returned exactly. Irrational results are shown to full precision with no artificial truncation.

Explore the full math toolkit

Every CalcaTools math calculator — algebra, geometry, trigonometry, statistics, sequences, and unit conversions — with step-by-step workings on every result.