Square Root Calculator — Simplify Radicals

Calculate the square root of any number, simplify radicals (e.g., √72 = 6√2), and find cube roots. See step-by-step simplification, perfect square check, and nearest perfect squares. 100% private — runs in your browser.

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How to Simplify Square Roots (Radicals)

Simplifying a square root means removing perfect square factors from under the radical sign. The key idea is that √(a × b) = √a × √b. To simplify √n, find the largest perfect square that divides n, take its square root outside the radical, and leave the remainder inside. For example, √72 = √(36 × 2) = 6√2.

This process is essential in algebra, geometry, trigonometry, and physics. Simplified radicals are the standard form for exact answers in most math courses, as opposed to decimal approximations which lose precision.

Radical Simplification Rules

√(a × b) = √a × √b

√(a / b) = √a / √b

√(a²) = |a|

ⁿ√x = x1/n

Steps to simplify √n:

  • 1. Find the prime factorization of n
  • 2. Group factors into pairs
  • 3. Move each pair outside the radical as a single factor
  • 4. Multiply the factors outside and inside separately

Example: Simplify √180

  • 180 = 2 × 2 × 3 × 3 × 5 = 2² × 3² × 5
  • Take 2 and 3 outside: 2 × 3 = 6
  • Leave 5 inside: √5
  • √180 = 6√5 ≈ 13.4164

Perfect Squares

A perfect square is a number whose square root is a whole number. The first 25 perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625. Recognizing perfect squares quickly helps you simplify radicals faster and estimate square roots mentally.

Cube Roots and Higher Roots

The cube root of a number n, written ³√n, is the value that when cubed gives n. Unlike square roots, cube roots can be negative: ³√(-8) = -2 because (-2)³ = -8. Higher roots (4th root, 5th root, etc.) follow the same pattern: ⁿ√x = x^(1/n). This calculator shows both the square root and cube root of your input.

Square Roots in Geometry

Square roots appear constantly in geometry. The Pythagorean theorem (c = √(a² + b²)) uses square roots to find the hypotenuse of a right triangle. The distance formula between two points (d = √((x&sub2;-x&sub1;)² + (y&sub2;-y&sub1;)²)) is built on square roots. The diagonal of a square with side s is s√2, and the height of an equilateral triangle with side s is (s√3)/2.

Mental Estimation of Square Roots

To estimate a square root mentally, find the nearest perfect squares above and below. For √50: 7² = 49 and 8² = 64, so √50 is slightly more than 7 (actually 7.071). For more precision, use linear interpolation: √50 ≈ 7 + (50-49)/(64-49) = 7.067. This calculator shows the nearest perfect squares to help you build this intuition.