Calculate square roots, simplify radicals, find square roots of fractions, imaginary square roots, and nth roots — all with step-by-step solutions. No signup needed.
Five modes for every square root need. Pick the right mode, enter your number, and get a complete step-by-step solution instantly.
Select Basic √x for decimal results, Simplify for radical form, Fraction for √(a/b), Negative for imaginary, or Nth Root for cube roots and beyond.
Type the number you want the square root of. For fractions, enter numerator and denominator separately. Negative mode handles imaginary results.
Press Calculate Square Root and see your result instantly — shown as a decimal, simplified radical form, and full step-by-step working.
Every result includes complete working — prime factorization, pairing, simplification — so students and teachers always understand the method.
The most commonly needed square roots — from 1 to 30. Perfect squares have whole number square roots; non-perfect squares show their simplified radical form.
| Number (n) | n² (Square) | √n (Decimal) | Simplified Radical | Type |
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Square roots appear in geometry, physics, finance, engineering, and everyday problem-solving. Here's where our free calculator helps the most.
Solve algebra problems, simplify radical expressions, and verify homework answers with complete step-by-step working shown.
Find the hypotenuse using Pythagoras theorem (c = √(a²+b²)), calculate triangle sides, and find areas of squares from diagonal measurements.
Calculate RMS values, velocity from kinetic energy, resonance frequency, and impedance in electrical engineering circuits.
Calculate standard deviation (takes square root of variance), find geometric means, and solve compound interest problems.
Find the side length of a square when area is known (side = √area). Calculate room dimensions and material quantities.
Use the simplified radical form for exam preparation and teaching. The step-by-step solution helps explain the method to students clearly.
A square root calculator is a free online tool that instantly finds the square root of any number — and does much more than just give you a decimal answer. Our online square root calculator also simplifies radicals to their simplest form, handles square roots of fractions, calculates imaginary square roots of negative numbers, and computes any nth root (cube root, 4th root, and beyond).
Whether you need a quick decimal answer or a fully simplified radical expression for algebra homework, this square root calculator with steps shows every step of the working — so you always understand how the answer was found, not just what it is.
Knowing how to find a square root manually is an important skill — especially for exams. Here are the three main methods:
144 = 2 × 2 × 2 × 2 × 3 × 3
= (2×2) × (2×2) × (3×3)
= 2² × 2² × 3²
√144 = 2 × 2 × 3 = 12
The long division method finds square roots of any number, including non-perfect squares, to as many decimal places as needed. This is the method used in schools for finding √2, √3, etc. without a calculator.
For a quick approximation of √N, use Newton's iterative formula:
x₁ = (x₀ + N/x₀) / 2
Start with any reasonable guess x₀ and iterate until it converges. For example, to find √10: start with x₀ = 3, then x₁ = (3 + 10/3)/2 = 3.167, x₂ = 3.1623... which converges to √10 = 3.16228.
Pro Tip: For exams where you cannot use a calculator, memorize the square roots of perfect squares from 1 to 20. Most exam problems are built around these values. Our Perfect Squares tab lists them all for quick reference.
Our simplify square root calculator uses prime factorization to reduce any radical to its simplest form. Here is the process:
72 = 2 × 2 × 2 × 3 × 3
Pairs: (2,2) and (3,3) — unpaired: 2
Outside: 2 × 3 = 6 — Inside: 2
√72 = 6√2 ≈ 8.485
When you take the square root of a negative number, the result is an imaginary number. This is because no real number multiplied by itself gives a negative result. The unit of imaginary numbers is i = √(-1).
Imaginary numbers are used in electrical engineering (AC circuits), quantum physics, signal processing, and complex number mathematics. Use our negative square root calculator mode to calculate √(-x) with the imaginary unit shown clearly.
To find the square root of a fraction, take the square root of the numerator and denominator separately:
√(a/b) = √a / √b
√75 = √(25×3) = 5√3
√16 = 4
√(75/16) = 5√3 / 4 ≈ 2.165
The nth root of a number x is the value y such that yⁿ = x. Square root is the 2nd root; cube root is the 3rd root:
Use our Nth Root mode to calculate any root instantly. The formula used is: ⁿ√x = x^(1/n)
Use the prime factorization method for perfect squares: break the number into prime factors, group them into pairs, and take one from each pair outside the radical. For non-perfect squares, use the long division method or Newton's approximation (start with a guess, average it with N/guess, repeat). For example: √144 = √(2²×2²×3²) = 2×2×3 = 12.
To simplify √n: find all perfect square factors of n, extract the largest one from under the radical. For example: √48 = √(16×3) = 4√3. Or use prime factorization: 48 = 2⁴×3, so √48 = 2²×√3 = 4√3. Use our Simplify Radical mode for automatic simplification with steps.
The square root of a negative number is an imaginary number. It uses the imaginary unit i, where i = √(-1). For example: √(-9) = √9 × √(-1) = 3i. √(-50) = √50 × i = 5√2 · i. Imaginary numbers are used in electrical engineering, quantum physics, and complex number mathematics.
Apply the rule: √(a/b) = √a / √b. Take the square root of numerator and denominator separately, then simplify. Example: √(9/16) = √9/√16 = 3/4 = 0.75. For √(3/4) = √3/2 ≈ 0.866. Use our Fraction mode for instant results with full working.
The square root (√x) finds a number that when multiplied by itself gives x (e.g. √9 = 3 because 3×3 = 9). The cube root (∛x) finds a number that when multiplied three times gives x (e.g. ∛27 = 3 because 3×3×3 = 27). Use our Nth Root mode to calculate cube roots and any other root.
A perfect square is a number that is the square of a whole number, giving a whole number square root. 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 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15. Click the Perfect Squares tab in our calculator to see them all.