Calculate square roots, cube roots and nth roots of any number — with decimal precision control, verification and simplified radical form.
Enter your figures and click Calculate to see your results.
Enter the number, the root degree (2 for square root, 3 for cube root) and how many decimal places you want.
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The square root of n is a value x where x² = n. Every positive number has two square roots: +√n and −√n. The principal root is the positive one. √16 = 4 because 4² = 16. Negative numbers have no real square roots — they require complex numbers involving i (the imaginary unit where i² = −1).
The nth root of x is written ⁿ√x = x^(1/n) — a value that gives x when raised to the nth power. Square root = 2nd root. Cube root = 3rd root. ⁴√81 = 3 (since 3⁴ = 81). Even roots of negative numbers are not real; odd roots of negatives are real: ∛(−8) = −2, since (−2)³ = −8.
A root is irrational when the result cannot be written as p/q with integers p and q. √2 ≈ 1.41421356... is non-repeating and non-terminating. Only perfect nth powers have rational nth roots: √4=2, √9=3, ∛8=2, ⁴√16=2. The irrationality of √2 was a shocking discovery in ancient Greece that revolutionised mathematics.
Find the largest perfect square that divides the radicand. √72 = √(36×2) = 6√2 (since √36 = 6). For cube roots: ∛54 = ∛(27×2) = 3∛2. The simplified form ⁿ√(aⁿ×b) = a×ⁿ√b is used throughout algebra and is the standard form for presenting root answers.
Newton's method finds ⁿ√x iteratively: start with a guess g, then update: g = ((n−1)g + x/g^(n−1)) / n, repeating until convergence. For square roots: g = (g + x/g) / 2. It converges extremely rapidly — doubling the number of correct digits with each iteration — and is how computers calculate roots internally.
