Square Root & Cube Root Calculator

Square Roots Guide

What Is a Square Root?

The square root of a number n is the value x such that x² = n. In other words, it's the number you multiply by itself to get n. The square root of 25 is 5 because 5 × 5 = 25; the square root of 144 is 12 because 12 × 12 = 144. The symbol √ is called a radical sign. Every positive number has two square roots — a positive one (called the principal root) and a negative one — because both 5 and −5 squared equal 25. By convention, √25 refers to the positive root (5), unless stated otherwise. Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100...) have whole-number square roots. Numbers that are not perfect squares have irrational square roots — they go on forever without repeating. √2 = 1.41421356... and √3 = 1.73205080... are the most famous examples. The square root of a negative number is not a real number — it produces an imaginary number (written with i), which is part of complex number mathematics. This calculator handles real numbers only.

What's the key thing to understand about Estimating Square Roots by Hand?

Finding square roots by hand is a useful skill for estimation and checking calculator results. For a quick estimate, find the two consecutive perfect squares your number falls between. √50 lies between √49 = 7 and √64 = 8, so it's somewhere between 7 and 8. Since 50 is close to 49, a good first estimate is 7.1. For a more precise method, use Newton's Method (also called the Babylonian method): start with a guess g, then repeatedly apply g_new = (g + n/g) / 2 until the answer stabilises. For √50 with g = 7: (7 + 50/7) / 2 = (7 + 7.143) / 2 = 7.071. One more iteration: (7.071 + 50/7.071) / 2 = (7.071 + 7.072) / 2 = 7.071. That's accurate to three decimal places after just two steps. For GCSE and A-level maths, recognising that √50 = √(25 × 2) = 5√2 is a simplification technique — always check if the number contains a perfect square factor before calculating.

What should I know about Cube Roots and nth Roots?

A cube root ∛n is the value x where x³ = n. The cube root of 8 is 2 (because 2³ = 8); the cube root of 27 is 3; the cube root of 125 is 5. Unlike square roots, cube roots of negative numbers are real: ∛(−8) = −2 because (−2)³ = −8. The nth root of a number generalises this: the nth root of n is x where xⁿ = n, written as ⁿ√n or equivalently n^(1/n) in exponent notation. So √n = n^(1/2), ∛n = n^(1/3), and the 4th root = n^(1/4). This exponent notation is why calculators and spreadsheets can compute any root: to find the 5th root of 243, calculate 243^0.2 = 3. In the formula x = n^(1/k), k is the root and n is the number. Excel/Sheets: =POWER(243,1/5). Python: 243**(1/5). For engineering and science, cube roots appear in volume calculations (side length = ∛volume), and nth roots appear in compound interest formulas where you need to find a periodic rate from a total growth factor.

What should I know about Applications?

Square roots appear throughout mathematics, science, and engineering. In geometry, the Pythagorean theorem requires square roots: if a right triangle has sides a and b, the hypotenuse c = √(a² + b²). Finding the diagonal of a square with side 10m: √(10² + 10²) = √200 = 10√2 ≈ 14.14m. In statistics, the standard deviation is the square root of the variance — a key measure of how spread out data is. In physics, the period of a simple pendulum is T = 2π√(L/g), where L is length and g is gravitational acceleration. In finance, volatility is often expressed as an annualised standard deviation, computed from daily returns using square roots of time. Speed is proportional to √(kinetic energy) in mechanics. The quadratic formula x = (−b ± √(b² − 4ac)) / 2a always involves a square root. Area calculations work in reverse: if you know a square's area is 200 m², the side length is √200 ≈ 14.14m — useful for working out room dimensions, garden plots, and tiling projects.