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Square Root Calculator

Calculate square and nth roots of any number, with exact simplified radicals and decimal approximations.

Updated

What is the Square Root Calculator?

Calculate square roots, cube roots, or any n‑th root instantly — and get the exact simplified radical form too. This free online square root calculator combines a fast decimal approximator with a built‑in radical simplifier, so you never have to choose between a precise number and an exact expression.

Why you'll love it

  • Two modes — square root (default) or custom root degree for cube roots, 4th roots, etc.
  • Exact radicals — simplifies integer radicands into clean forms like 2√3 or 3∛5.
  • Decimal precision — high‑accuracy floating‑point result always visible.
  • Copy in one click — copy the simplified radical and its decimal approximation together.
  • Error‑proof — real‑time validation catches impossible inputs (e.g., even root of a negative number).

Whether you're solving homework, checking your work, or exploring mathematics, this calculator gives you both the numeric answer and the underlying structure.

How it works

Decimal approximation

The calculator uses JavaScript's Math.pow(radicand, 1/rootDegree) to compute the principal real root. For example, the square root of 25 is Math.pow(25, 0.5), which yields 5. This method handles both integers and decimals with very high accuracy.

Radical simplification

When the radicand is a whole number (integer), the calculator attempts to express the root in the simplest radical form a√b (or a∛b, etc.). It does this by:

  1. Prime factorising the absolute value of the radicand.
  2. For each prime factor, grouping the exponents into full n‑th powers and a remainder.
  3. Moving all full powers outside the radical, leaving the remainder inside.

Example: simplify the cube root of 54.

  • Radicand = 54, n = 3.
  • Prime factorisation: 54 = 2¹ × 3³.
  • Exponents: 2 has 1 (remainder after dividing by 3 → 1); 3 has 3 (full group → 3/3 = 1 full power).
  • Outside the radical: 3^(3/3) = 3¹ = 3.
  • Inside the radical: 2¹ = 2.
  • Result: 3∛2 (the decimal approximation is also shown, ≈ 3.7798).

For large integers (>10¹⁰) the simplification is skipped to keep the calculator instant.

Edge cases

  • Radicand = 0 → always 0.
  • Negative radicand with even root degree → no real root; an error is shown.
  • Negative radicand with odd root degree → the negative sign is preserved outside the radical (e.g., -∛8 simplifies to -2).
  • Non‑integer radicands → only the decimal approximation is provided.

Examples

Square root of a perfect square

Find the square root of 144.

Radicand: 144, Root: 2
Decimal: 12, Simplified: 12

Simplify a radical

Simplify the cube root of 54.

Radicand: 54, Root: 3
Decimal: ~3.7798, Simplified: 3∛2

Nth root of a decimal

Calculate the 5th root of 2.5.

Radicand: 2.5, Root: 5
Decimal: ~1.2011244

Frequently asked questions

What is a square root?

A square root of a number x is a value that, when multiplied by itself, gives x. For example, √16 = 4 because 4 × 4 = 16. Every positive number has two square roots (positive and negative), but this calculator shows the principal (non‑negative) root.

How do I calculate an nth root?

In the “Nth Root” mode, enter the radicand (the number under the radical) and the root degree (n). The calculator computes radicand^(1/n) for the decimal result. If the radicand is a whole number, it also gives the simplified radical form when possible.

Why can’t I take the even root of a negative number?

Even‑index roots (square root, 4th root, 6th root, …) of a negative number are not real numbers. In the real number system, no number multiplied by itself an even number of times yields a negative product. For example, there is no real number x such that x² = -4. Use odd roots (cube root, 5th root, etc.) for negative radicands.

How accurate is the decimal approximation?

The calculator uses JavaScript’s native 64‑bit floating‑point arithmetic, which provides about 15–17 significant decimal digits of precision. This is far more than enough for everyday calculations, homework, and most engineering tasks.

What does “simplified radical form” mean?

It’s the standard way to write a root expression with the smallest possible number under the radical sign. For example, √12 simplifies to 2√3, and ∛54 simplifies to 3∛2. Our calculator shows this form whenever the radicand is a small‑ to medium‑sized integer.

Can I simplify radicals that contain variables or fractions?

No — this tool only simplifies integer radicands. It does not handle algebraic expressions, variables, or fractional radicands. For those, a symbolic algebra system is needed.

Does the calculator work offline?

Yes. All calculations run directly in your browser using client‑side JavaScript. No internet connection is required after the page has loaded.

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