Square Root Calculator
Advanced square root calculator — handles positive, negative, and decimal numbers. Get simplified radical form, step-by-step solution, and results in 10 different formats including fraction, scientific notation, and complex number form.
How to Use This Square Root Calculator
Type any number — integer, decimal, fraction, or negative — and get instant results in 10 formats with full step-by-step working.
Enter Your Number
Type any number into the input field. Supports integers (144), decimals (0.25), negative numbers (−25), fractions (3/4), and expressions like 2^10. The calculator handles all cases automatically.
Choose Root Type
Select square root (√), cube root (∛), 4th root, or any nth root up to the 10th root. The formula and step-by-step solution update automatically for each root type.
Set Precision
Choose between 2 and 15 decimal places for irrational results. All 10 output formats update instantly with your chosen precision — including scientific notation and engineering notation.
Copy Any Format
Click any of the 10 output format cards to copy that specific representation to your clipboard — simplified radical, decimal, fraction approximation, LaTeX, Unicode, and more.
Square Root Reference Table — 1 to 100
Common square roots with simplified radical form and decimal approximations. Perfect squares highlighted in blue.
| n | √n (Exact / Simplified) | √n (Decimal) | Perfect Square? | n² |
|---|
Understanding Square Roots
The square root of a number n is the value x such that x² = n. Every positive number has two square roots — one positive (the principal root) and one negative. For example, √25 = 5 because 5² = 25, but also (−5)² = 25, so −5 is also a square root of 25.
A perfect square is any integer whose square root is also an integer — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are the first ten perfect squares. For non-perfect-square integers, the square root is always an irrational number — it cannot be expressed as a fraction and its decimal expansion never terminates or repeats.
Simplifying radicals means expressing √n in the form a√b where b has no perfect square factors other than 1. For example, √72 = √(36 × 2) = 6√2, because 36 is the largest perfect square factor of 72.
- √(a × b) = √a × √b — the product rule for radicals
- √(a / b) = √a / √b — the quotient rule for radicals
- The square root of a negative number is an imaginary number: √(−n) = i√n
- Rationalising the denominator: 1/√2 = √2/2 (multiply top and bottom by √2)
- √(x²) = |x| — the absolute value, not just x
- For the nth root: ⁿ√(xⁿ) = x when n is odd; |x| when n is even
√ How to Simplify a Square Root Manually
To simplify √n: (1) Find the prime factorisation of n. (2) Group prime factors into pairs. (3) Each pair of identical primes comes out of the radical as one copy. (4) Any unpaired primes remain inside the radical. Example: √180 = √(2²×3²×5) = 2×3×√5 = 6√5. This calculator performs all steps automatically and shows the full prime factorisation in the step-by-step solution.
±i Square Root of Negative Numbers (Complex)
The square root of a negative number is not a real number — it is an imaginary number. By definition, i = √(−1), so √(−n) = i√n. For example, √(−16) = 4i. Complex numbers in the form a + bi arise when you take the square root of a negative number. This is fundamental in electrical engineering (impedance), quantum mechanics, and signal processing. This calculator returns the imaginary form automatically for negative inputs.
🔢 Babylonian Method — How Computers Calculate √n
The most common algorithm for computing square roots is the Babylonian (or Heron's) method: starting with an initial guess x₀, repeatedly compute x_{n+1} = (x_n + N/x_n) / 2 until convergence. This Newton-Raphson iteration doubles the number of correct decimal digits with each step — reaching 15-digit precision in about 5 iterations for most numbers. This calculator uses JavaScript's native Math.sqrt() which implements an optimised floating-point version of this algorithm internally.
📐 Square Roots in Geometry and Physics
Square roots appear throughout mathematics and science. In geometry: the diagonal of a square with side a is a√2; the hypotenuse of a right triangle is √(a²+b²) (Pythagoras). In physics: the speed of a pendulum involves √(L/g); relativistic effects use √(1−v²/c²); RMS voltage is √(mean of v²). In statistics: standard deviation is the square root of variance. Understanding square root properties is essential for any quantitative discipline.
Square Root Calculator FAQs
Common questions about calculating square roots, simplifying radicals, and how this calculator works.
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About This Square Root Calculator
This advanced square root calculator handles all real number inputs and returns results in 10 output formats. It uses JavaScript's native IEEE 754 double-precision floating-point arithmetic (Math.pow) for root calculations, achieving 15–17 significant digits of precision. Radical simplification uses prime factorisation up to 10,000 — larger numbers fall back to decimal form. Fraction approximation uses the Stern-Brocot / continued fraction algorithm to find the best rational approximation within the specified precision. Complex number output uses the standard mathematical convention i = √(−1). Step-by-step solutions are generated algorithmically from the computed result. Results are for educational and computational purposes — for critical engineering calculations, verify with a certified mathematical computing system.
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