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

By CalcMesh Editorial · Reviewed against primary-source formulas

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

Square Roots in Applied Math

Square roots appear in standard deviation (σ = √variance), which appears in every empirical study published. NIST's 2023 Engineering Statistics Handbook notes the standard deviation of the sample mean scales as σ/√n — doubling sample size only reduces measurement error by roughly 29%, and a 10x reduction requires 100x the sample. This √n rule drives sample-size decisions in clinical trials, A/B testing, and polling.

The Pythagorean theorem (c = √(a² + b²)) remains the most widely applied square-root identity. GPS positioning, CAD/CAM manufacturing, and video game physics engines execute trillions of √ operations daily. Intel x86 processors implement the SQRTPD instruction directly in silicon — a modern CPU computes a 64-bit double-precision square root in 10-20 clock cycles, roughly 5 nanoseconds.

Newton's method (xₙ₊₁ = ½(xₙ + S/xₙ)) converges quadratically — the number of correct digits roughly doubles with each iteration, reaching double-precision accuracy in 4-6 iterations from a reasonable starting estimate. The famous fast inverse square root trick used in Quake III Arena (1999) exploited IEEE 754 bit-level manipulation to approximate 1/√x in just 4 instructions, roughly 4x faster than the x87 FPU could at the time — a historic demonstration of how much performance math trickery enables.

Sources: NIST Engineering Statistics Handbook, Intel Architecture Software Developer Manual

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.