Square Root Calculator

What Is a Square Root Calculator?

Calculate the square root, cube root, or any nth root of any positive number instantly. This calculator runs entirely in your browser — your data stays private, and no account is required.

Understanding Square Roots

The square root of a number n (written √n) is the value that, when multiplied by itself, gives n. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots — one positive and one negative — but the symbol √ refers to the positive (principal) root by convention.^[1] Perfect squares (1, 4, 9, 16, 25, 36...) have integer square roots, while other positive numbers have irrational square roots — decimal expansions that never terminate or repeat. √2 ≈ 1.41421356... has been proven irrational since ancient Greek mathematics.^[2] Square roots appear in the Pythagorean theorem (finding distances), the quadratic formula, standard deviation calculations in statistics, and countless physics formulas involving energy, motion, and wave behavior.^[3] Use the Exponent Calculator for more advanced root and power calculations.

Simplifying Square Roots

To simplify √72: find the largest perfect square factor (36). √72 = √(36 × 2) = √36 × √2 = 6√2. The perfect squares to know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. A square root is irrational (non-terminating, non-repeating decimal) unless the number under the radical is a perfect square. The Pythagorean theorem frequently produces irrational roots — the diagonal of a unit square is √2 ≈ 1.41421356..., which cannot be expressed as a simple fraction.

Perfect Squares and Square Roots

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, produces the original number. The square root of 25 is 5, because 5 × 5 = 25. Every positive number has two square roots — one positive and one negative (both 5 and -5 squared equal 25), though the symbol √ conventionally refers to the positive (principal) root. The square root of zero is zero, and negative numbers have no real square roots (though they have complex/imaginary roots: √(-1) = i). Square roots appear throughout mathematics, science, and engineering — from the Pythagorean theorem (c = √(a² + b²)) to standard deviation in statistics (σ = √(variance)) to the distance formula in coordinate geometry. For related calculations, see our Pythagorean Theorem Calculator and Equation Solver.

Perfect Squares Reference

Methods for Calculating Square Roots

Before calculators, several manual methods were used to find square roots. Estimation by perfect squares brackets the answer: √50 falls between √49 = 7 and √64 = 8, so √50 ≈ 7.07. The Babylonian method (also called Newton's method or Heron's method) iteratively refines an estimate: start with a guess g, compute g_new = (g + n/g) / 2, and repeat until the desired precision is reached. For √50 starting with g = 7: g₁ = (7 + 50/7)/2 = (7 + 7.143)/2 = 7.071, which is already accurate to 3 decimal places. This algorithm converges extremely quickly — each iteration roughly doubles the number of correct digits.

Simplifying square roots involves factoring the number under the radical into a product where one factor is a perfect square: √50 = √(25 × 2) = 5√2. This simplified form is preferred in algebra because it reveals the number's structure and makes further calculation easier. To simplify: find the largest perfect square factor, extract its root, and leave the remaining factor under the radical. √72 = √(36 × 2) = 6√2. √180 = √(36 × 5) = 6√5. If no perfect square factor exists (other than 1), the root is already in simplest form — for example, √7 cannot be simplified. Rationalizing the denominator transforms expressions like 1/√3 into √3/3 by multiplying both numerator and denominator by √3. For more complex algebraic operations, use our Graphing Calculator and Factorial Calculator.

Square Roots in the Real World

Square roots appear in countless practical applications. The Pythagorean theorem uses square roots to find distances — a 40-foot by 30-foot room has a diagonal of √(40² + 30²) = √(1600 + 900) = √2500 = 50 feet. Construction workers use the 3-4-5 triangle rule (a Pythagorean triple where √(3² + 4²) = 5) to verify right angles when laying foundations. In physics, the period of a pendulum is T = 2π√(L/g), showing that a pendulum's swing time depends on the square root of its length — quadrupling the length only doubles the period. The speed of a wave depends on the square root of the tension in the medium, which is why tightening a guitar string raises its pitch.

In finance, the compound annual growth rate (CAGR) for a two-year period involves a square root: CAGR = √(ending value/beginning value) - 1. An investment that grows from $10,000 to $14,400 over two years has CAGR = √(1.44) - 1 = 1.2 - 1 = 20% per year. In statistics, standard deviation is the square root of variance, and uncertainty propagation involves square roots of summed squared errors. The inverse square law in physics states that the intensity of light, gravity, and other radiative phenomena decreases with the square of distance — meaning at twice the distance, intensity is one-quarter, and at three times the distance, one-ninth. Understanding this relationship is essential in photography (exposure), acoustics (sound levels), and astronomy (stellar brightness). Explore related mathematical concepts with our Number Sequence Calculator and GCD & LCM Calculator.

Complex and Higher-Order Roots

While square roots of negative numbers don't exist in the real number system, the imaginary unit i (defined as √(-1)) extends mathematics into the complex plane, where every number has exactly two square roots. Complex numbers in the form a + bi are essential in electrical engineering (AC circuit analysis uses complex impedance), quantum mechanics (wave functions are complex-valued), signal processing (Fourier transforms), and control theory. The cube root (∛) and higher-order roots follow similar principles — ∛27 = 3 because 3³ = 27. Unlike square roots, cube roots of negative numbers are real (∛(-8) = -2). The nth root of a number x is x^(1/n), connecting roots to fractional exponents: √x = x^(1/2), ∛x = x^(1/3). This connection is fundamental in calculus, where power rules for differentiation and integration require expressing roots as exponents. Explore advanced calculations with our Scientific Notation Calculator and Binary Calculator.

Irrational Numbers and Square Roots

Most square roots are irrational numbers — they cannot be expressed as a fraction and their decimal expansion never terminates or repeats. √2 ≈ 1.41421356... continues infinitely without a repeating pattern. The ancient Greeks proved √2 is irrational using a proof by contradiction, which was one of the earliest results in mathematical history and deeply unsettled the Pythagorean school of thought. The only whole numbers with rational square roots are perfect squares (1, 4, 9, 16, 25...). When working with irrational square roots in exact calculations, it's best to keep them in radical form (5√3 rather than 8.660...) to avoid accumulating rounding errors across multiple calculation steps. This is why algebra textbooks insist on simplified radical form — it preserves perfect accuracy throughout multi-step problems.

How to Use This Calculator

1. Enter any positive number — Type a whole number, decimal, or fraction. The calculator handles perfect squares and irrational results.
2. Review the square root — For perfect squares (4, 9, 16, 25...), the result is a whole number. For others, the calculator shows the exact decimal and simplified radical form (e.g., √50 = 5√2).
3. Calculate nth roots — The calculator also handles cube roots, 4th roots, and any nth root — useful for engineering, statistics, and financial calculations.
4. See the factorization method — For educational purposes, the calculator shows how the square root is simplified by factoring out perfect square factors from under the radical.

Tips and Best Practices

See also: Exponent Calculator · Scientific Calculator · Quadratic Equation Solver