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, and handles powers and logarithms alongside them.

Perfect squares such as 4, 9, 16, 25 and 36 have exact integer roots, and the sequence continues 49, 64, 81, 100, 121, 144, 169, 196 and 225. Everything else produces an irrational number that never terminates, which is why √2 = 1.41421356... and √3 = 1.7320508... run on indefinitely. Results here are shown to up to 10 decimal places. Negative numbers have no real square root, only an imaginary one built on i = √-1, though cube roots of negatives are perfectly ordinary, since the cube root of −27 is −3.

A dozen or so roots are worth knowing by heart. The perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100 give the whole numbers 1 through 10 in order, and 121, 144 and 169 give 11, 12 and 13. Further up, 225 gives 15, 400 gives 20 and 625 gives 25. Two useful outliers are 1000, whose root is approximately 31.62, and 2025, whose root is exactly 45. Recognising these turns a good deal of GCSE arithmetic into recall rather than calculation.

Square roots earn their place in geometry through Pythagoras' Theorem, where c² = a² + b² and therefore c = √(a² + b²). A right-angled triangle with sides 3 and 4 has a hypotenuse of √(9+16) = √25 = 5, which is the basis of the 3:4:5 method builders use to check that a corner is square. The same operation underlies straight-line distances in surveying and route-finding, pixel distances in computer graphics and vector magnitudes in physics. UK GCSE Maths covers both Pythagoras and surd manipulation.

Working a root out by hand is still a useful skill. Estimation and refinement is the quickest route, so to find √50 you guess 7, square it to get 49, try 7.1, square it to get 50.41 and stop there. A long division-style algorithm existed before calculators and still appears in older textbooks. Prime factorisation handles surds neatly, so √72 becomes √(36×2) = 6√2 = 6 × 1.4142 = 8.485. Modern GCSE work leans on that third approach, asking you to rationalise denominators, turning 1/√2 into √2/2, and to simplify, turning √50 into 5√2.

Cube roots and higher roots follow the same logic with a different exponent. The cube root of a number, written ∛X, is what you multiply by itself three times to reach X, so the cube roots of 8, 27, 64, 125 and 1000 come out as 2, 3, 4, 5 and 10. Unlike square roots, cube roots of negatives are defined, and ∛-8 = -2. An nth root generalises the idea, so ⁴√16 = 2 because 2⁴ = 16. Applied to 144, the square root is 12 because 12 × 12 = 144, the cube root is 5.2415 and the fourth root is 3.4641. Higher roots turn up in volume calculations, financial growth rates, engineering cross-sections and, at A-level, complex roots and roots of unity.