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.

Published by · Reviewed against primary-source formulas

Square Root (√)

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Cube Root (∛)

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Perfect Square?

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Perfect Cube?

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Nearby Perfect Squares

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Nearby Perfect Cubes

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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

Methodology & Assumptions

This calculator implements standard formulas drawn from primary-source authorities. Values are point-in-time estimates; consult a licensed professional for high-stakes decisions. See the per-input definitions and source citations below.

How this works

Computations are deterministic and run client-side, no inputs leave your browser. Formulas are derived from standard published formulas for the calculator's domain (mortgage, taxes, energy, conversions, etc.). When the underlying agency publishes updated rates or thresholds we refresh defaults and update the page's lastmod timestamp.

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.

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This page identifies the inputs, method, and limitations behind its estimates. Calculator outputs are not professional advice and should be checked against the relevant primary source for a consequential decision. This calculator's formula and defaults are drawn from standard published sources for its domain, no figure is typed in by an editor. See our editorial standards & corrections policy, the methodology behind these numbers, or report a data error.

Inputs, defaults, and authoritative sources
Input Default Source / authority
All inputs Domain-typical defaults Editorial methodology, CalcMesh 2026