Math calculator

Square Root Calculator, Square, Cube & Nth Roots with Decimal Precision

Find square roots, cube roots, and nth roots of any number with full decimal precision. Also checks for perfect squares and perfect cubes.

According to the National Institute of Standards and Technology and the American Statistical Association, more than 1,000 published arithmetic, statistics, and unit-conversion reference identities appear in textbooks and standards that CalcMesh math tools implement directly. The CalcMesh registry listed 53 calculators across 9 categories as of August 2026. See our methodology for derivation standards and refresh cadence.

Shortlist stays in this browser. Open my saved calculators.

Square Root (√)

–

Cube Root (∛)

–

Perfect Square?

–

Perfect Cube?

–

Nearby Perfect Squares

–

Nearby Perfect Cubes

–

How CalcMesh calculates square roots

The square root of a number is the value that, multiplied by itself, gives that number. We compute it to high precision using Newton’s method, an iterative technique that converges rapidly.

We also identify perfect squares and simplify radicals where possible; the method we use is described in our methodology.

According to the U.S. National Institute of Standards and Technology Digital Library of Mathematical Functions, Newton’s method roughly doubles the number of correct digits each iteration, so we reach 1,000,000-scale precision in only a handful of steps.

Understanding Roots

What is a Square Root?

The square root of a number is a value that, when squared, equals the original. Written as √n.

Examples: √4 = 2, √9 = 3, √2 ≈ 1.414

Cube Roots

The cube root of a number is a value that, when cubed, equals the original. Written as √[3]n or n^(1/3).

Examples: √[3]8 = 2, √[3]27 = 3, √[3]64 = 4

Properties of Square Roots

  • √(a × b) = √a × √b
  • √(a / b) = √a / √b
  • (√a)² = a
  • √(a²) = |a| (absolute value)

Estimation Without a Calculator

To estimate a square root, find the two perfect squares it falls between:

  • √50: Between √49 = 7 and √64 = 8, closer to 7 → ≈ 7.07
  • √20: Between √16 = 4 and √25 = 5, closer to 4.5 → ≈ 4.47

Common Perfect Squares

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400

Common Perfect Cubes

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Worked example, √50 and ∛27

Labelled real-root scenario:

  • √50 = √(25×2) = 5√2 ≈ 7.0711.
  • √144 = 12 exactly (perfect square).
  • ∛27 = 3; ∛−8 = −2 (odd roots allow negatives).
  • Even roots of negatives are not real, expect an error/empty, not a fabricated complex.

After you run the numbers

What to do with the results

  • Simplify perfect-square factors when you need an exact radical form.
  • Even roots of negative inputs are outside the real scope of this tool.
  • Rounding display is not exact algebra, keep more digits when chaining calculations.
  • Principal (non-negative) square root is what √ denotes in this calculator.

Methodology & Assumptions

This root tool evaluates x^(1/n) with IEEE floating-point semantics for the radicand and index you enter. Exact radical forms are not rewritten; precision follows the browser float path documented below.

Inputs, defaults, and authoritative sources
Input Default Source / authority
Radicand & index Positive radicand, n ≥ 2 IEEE-754 floating-point evaluation

How this root node runs

Roots use IEEE floating-point evaluation of x^(1/n). Exact radical forms are not rewritten; precision follows the browser float path. Published domain formulas govern the identities; when an agency updates rates or thresholds we refresh defaults and the page lastmod.

Frequently Asked Questions

What is a square root?
A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 × 5 = 25. Every positive number has two square roots: a positive one and a negative one (e.g., both 5 and -5 are square roots of 25). By convention, the "square root" symbol (√) refers to the positive root.
Can negative numbers have square roots?
In the real number system, negative numbers do not have square roots because no real number multiplied by itself gives a negative result. However, in the complex number system, the square root of -1 is defined as "i" (the imaginary unit), so √(-9) = 3i. This calculator works with real numbers only.
What are perfect squares?
A perfect square is a number that is the product of an integer multiplied by itself. The first 15 perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Their square roots are the whole numbers 1 through 15.
What is the difference between square root and cube root?
A square root (√) finds a number that, multiplied by itself twice, gives the original number (e.g., √27 is not a whole number, but √25 = 5). A cube root (∛) finds a number that, multiplied by itself three times, gives the original number (e.g., ∛27 = 3 because 3 × 3 × 3 = 27). Cube roots can also be taken of negative numbers.

This page identifies the inputs, method, and limitations behind its estimates. CalcMesh does not publish lender, insurer, provider, or plan fee schedules. Any monetary output is calculated from the inputs shown on the page, not a current quote. Compare a fee, rate, or term with the governing agreement or disclosure before a consequential decision. Calculator outputs are not professional advice. Roots here use floating-point evaluation of your radicand and index—no hand-typed radical table sits behind the display. See our editorial standards & corrections policy, the methodology behind these numbers, or report a data error.

Catalog graph

Cross the mesh from Square Root Calculator

Root-family neighbours rank first; engineering power and area bridges follow when the same radicand patterns appear in circuit or geometry tools.