Square Root, Powers & Logarithm Calculator

The square root of a number x is the value that, when multiplied by itself, gives x. For example, the square root of 25 is 5 because 5 × 5 = 25. This calculator finds square roots, cube roots and nth roots for any positive number.

Perfect squares (4, 9, 16, 25, 36 ...) have exact integer roots. Non-perfect squares produce irrational numbers — the calculator shows results to up to 10 decimal places. It also handles cube roots of negative numbers (e.g. the cube root of −27 is −3).

What is a square root? The square root of X (written √X) is the number that, when multiplied by itself, equals X. √9 = 3 (because 3 × 3 = 9). √25 = 5. √100 = 10. Negative inputs have no real square root — but have an imaginary one (i = √-1). Most positive numbers have IRRATIONAL square roots — √2 = 1.41421356..., √3 = 1.7320508... — non-terminating decimals.

Common square roots to know. √1 = 1, √4 = 2, √9 = 3, √16 = 4, √25 = 5, √36 = 6, √49 = 7, √64 = 8, √81 = 9, √100 = 10, √121 = 11, √144 = 12, √169 = 13, √225 = 15, √400 = 20, √625 = 25, √1000 ≈ 31.62, √2025 = 45. Perfect squares (integer square roots): 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225...

Why square roots matter — Pythagoras' Theorem. Right-angled triangle: c² = a² + b², so c = √(a² + b²). Sample: right triangle with sides 3 and 4 has hypotenuse √(9+16) = √25 = 5. Used in: construction (verify square corners — 3:4:5 method), navigation (straight-line distances), surveying, computer graphics (pixel distances), and physics (vector magnitudes). UK GCSE Maths includes Pythagoras and surd manipulation.

How to calculate square roots by hand. Method 1 — estimation and refinement: guess, square it, adjust. √50: guess 7 (49), guess 7.1 (50.41) — close enough. Method 2 — long division-style algorithm taught pre-calculator era. Method 3 — prime factorisation for surds: √72 = √(36×2) = 6√2 = 6 × 1.4142 = 8.485. Most modern UK GCSE Maths emphasises surd manipulation: rationalise denominators (1/√2 = √2/2), simplify (√50 = 5√2).

Cube roots and higher roots. Cube root of X (∛X): the number that, cubed (multiplied by itself 3 times), equals X. ∛8 = 2, ∛27 = 3, ∛64 = 4, ∛125 = 5, ∛1000 = 10. Unlike square roots, cube roots of negative numbers ARE defined: ∛-8 = -2. Higher roots: ⁿ√X = the number which, raised to the nth power, equals X. ⁴√16 = 2 (because 2⁴ = 16). Used in: volume calculations, financial growth (CAGR), engineering (cross-sections, scaling). UK A-level Maths covers higher-order roots, complex roots, and roots of unity.