Nth Root Calculator
Find the square root, cube root, or any nth root — with radical simplification like √48 = 4√3 shown. Free, runs in your browser.
Square Root and Nth Root Calculator
How to Use the Nth Root Calculator
Enter a number
Type the number you want the root of — the radicand. It can be a whole number or a decimal.
Choose which root
Set the root index: 2 for a square root, 3 for a cube root, and so on up to any nth root you need.
Read the decimal answer
The principal real root appears instantly, to twelve significant figures.
See the simplified radical
When the number is a whole number, you also get the exact simplified surd — for example √48 = 4√3 — which most calculators never show.
Roots and Radicals
The Other Half of Powers
A root is the inverse of a power: the square root of 9 is 3 because 3² = 9, and the cube root of 27 is 3 because 3³ = 27. Where exponents build a number up by repeated multiplication, roots take it back down, asking "what number, raised to this power, gives me back the original?" The square root is by far the most common and has its own symbol, √, but the same idea extends to cube roots (∛), fourth roots, and any nth root at all. This calculator computes the principal real root — the positive answer where one exists — to high precision, for whatever root index you choose. It is the everyday workhorse behind the Pythagorean theorem (which needs a square root to find a distance), standard deviation, geometry, and any formula where a squared or cubed quantity has to be undone.
What sets this tool apart from a basic calculator is that it also simplifies the radical when the number is a whole number — and that is a genuinely useful, distinctly mathematical skill that most calculators skip entirely. The decimal value of √48 is about 6.9282, but its exact form is 4√3: you pull out the largest perfect square hiding inside 48 (which is 16 = 4²), leaving 3 behind under the root. Simplified surds like 4√3 are exact, they are the form expected in algebra and exam answers, and they make relationships between numbers visible that a decimal hides. The calculator finds this form by factorising the number into primes and pulling out every complete group — for a square root, every pair; for a cube root, every triple — so it works for any root index, turning ∛54 into 3∛2 just as readily as it turns √72 into 6√2.
"A decimal root is an approximation; a simplified surd like 4√3 is the exact truth. Pulling the perfect powers out from under the radical is where the real mathematics lives."
Real, and Not-Quite-Real
Roots have one famous boundary: you cannot take an even root of a negative number and stay among the real numbers, because no real number squared is negative. Asking for the square root of −4 has no real answer, and this calculator says so plainly rather than returning something misleading. (Such roots do exist as complex numbers, handled by the site's Complex Number Calculator.) Odd roots are friendlier — the cube root of −27 is a perfectly real −3, because a negative cubed stays negative — and the calculator computes those without complaint. Understanding which roots are real and which are not is part of the mathematics, not an inconvenience, and seeing the tool draw that line correctly helps build the intuition. Whether you are finding a side length, simplifying a surd for an exam, or just curious what the seventh root of something is, this calculator gives the precise decimal, the exact simplified form where one exists, and an honest answer at the edges — all instantly and privately in your browser.
The √ sign first appeared in print in 1525
A root is the inverse of a power: √9 = 3 because 3² = 9.
The square root (√) is the most common; cube root is ∛.
√48 simplifies to 4√3 by pulling out the perfect square 16.
A simplified surd like 4√3 is exact; its decimal is only an approximation.
You cannot take an even root of a negative and stay real.
But odd roots of negatives are fine: ∛(−27) = −3.
The √ sign first appeared in print in Christoff Rudolff's Die Coss (1525); the familiar story that it is a stylised letter r, for radix, was Euler's guess in 1775 and Cajori's A History of Mathematical Notations argues against it.
The Pythagorean theorem needs a square root to find a distance.
√2 is irrational — its decimal never ends or repeats.
An nth root equals raising to the power 1/n.
Frequently Asked Questions
- You pull out the largest perfect square that divides the number. For 48, that is 16 (= 4²), leaving 3 inside, so √48 = 4√3. This calculator does it automatically for any whole number by factorising into primes and pulling out complete pairs — and it works for cube roots and higher too.
- A square root undoes squaring (√9 = 3 because 3² = 9); a cube root undoes cubing (∛27 = 3 because 3³ = 27). Set the root index to 2 for square roots, 3 for cube roots, or any number for higher roots. They are all the same operation with a different index.
- Because no real number squared is negative — a positive squared is positive, and a negative squared is also positive. So an even root of a negative has no real value, and the calculator says so. Odd roots are different: the cube root of −27 is a real −3. Complex (imaginary) roots are handled by the Complex Number Calculator.
- It is the exact form of a root with all perfect powers pulled out, like 4√3 instead of √48. Unlike the decimal (6.9282…), the surd is exact and is the form expected in algebra and exams. The calculator shows it whenever the number is a whole number with a non-trivial factor to extract.
- Taking the nth root of x is exactly raising x to the power 1/n. So √x is x^(1/2) and ∛x is x^(1/3). This is why the Exponent calculator and this Nth Root calculator are two views of the same maths — use whichever notation fits your problem.
- Yes — the square root of 2 cannot be written as a fraction, and its decimal expansion (1.41421356…) never ends or repeats. The calculator shows a high-precision approximation, and where a whole number simplifies neatly it shows the exact surd instead.
- The root is computed as x^(1/n) in IEEE 754 double precision, which carries 15–17 significant decimal digits, and is displayed rounded to twelve — so every digit you see is reliable. For an irrational root that is still an approximation, as on any calculator; for a perfect power the simplified-radical line gives the exact integer, and for a number like 48 the exact answer is the surd 4√3.
- Yes — set the root index to any value up to 20. The calculator computes the principal real root for any index and, when the number is a whole number, pulls out complete groups of that size to simplify the radical (for a 4th root it extracts every group of four equal prime factors, and so on).
- Yes. The calculation runs in your browser — nothing is uploaded, stored, or logged — and it works offline once loaded.
- The simplifier factorises the number into primes by trial division up to its square root, then pulls out every complete group of n equal factors. That is instant for numbers up to roughly 10¹⁴, and exact for any whole number below 2⁵³ (about 9 × 10¹⁵), the largest integer double-precision arithmetic represents exactly. Decimals are not simplified — only whole numbers have a meaningful surd form — but their decimal root is still computed.
Method & sources
How it computes
Principal real nth root evaluated as x^(1/n) in IEEE 754 binary64 arithmetic and displayed to twelve significant figures: for x ≥ 0 the non-negative root, for x < 0 the real root only when n is odd (an even root of a negative is refused as having no real value). For a whole-number radicand the exact simplified surd is given as well, by factorising x into primes and extracting every complete group of n equal factors — √48 = 4√3, ∛54 = 3∛2, and a perfect power is reported exactly.
What this tool implements
- Principal-root convention: one real root is returned, never the ± pair and never a complex root; an even root of a negative number is refused with a message rather than approximated, and the page points the reader at the complex-number tool instead.
- Radical simplification by complete prime factorisation — coefficient = Π p^⌊e/n⌋, radicand = Π p^(e mod n) — with trial division by 2, 3 and then 6k ± 1 up to √x (Knuth, TAOCP Vol. 2, §4.5.4). Only integer radicands are simplified; a decimal gets the decimal root alone.
- IEEE 754-2019 binary64 throughout: 15-17 significant decimal digits carried, twelve displayed. x^(1/n) is not a correctly-rounded root, so a perfect power can land one unit in the last place low; the twelve-figure rounding absorbs it and the simplified-radical line states the exact integer.
- Root index entered as an integer; an index below 2 falls back to a square root.
Sources
- Cajori F. A History of Mathematical Notations, Volume I: Notations in Elementary Mathematics. Chicago: Open Court Publishing; 1928. Signs for roots — the first printed appearance of √ in Christoff Rudolff's Die Coss (1525), and the argument against deriving the sign from the letter r for radix.
- Knuth DE. The Art of Computer Programming, Volume 2: Seminumerical Algorithms. 3rd ed. Reading, MA: Addison-Wesley; 1997. §4.5.4 Factoring into primes (trial division by 2, 3 and then 6k ± 1).
- IEEE. IEEE Std 754-2019, IEEE Standard for Floating-Point Arithmetic. New York: IEEE; 2019. The binary64 format and its 15-17 significant decimal digits.
What can make this go out of date
- None at runtime — pure arithmetic on the number entered. No tables, rates or published series are consulted, so nothing here goes stale on a schedule.
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