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.

What should I know about Simplifying Surds (Radical Expressions)?

A surd is an irrational root, like √12, that's left in root form rather than converted to a decimal, and simplifying one means factoring out the largest perfect-square factor possible. For √12, since 12 = 4 × 3 and 4 is a perfect square, √12 = √4 × √3 = 2√3. Surds combine like algebraic terms only when they share the same root after simplification: 2√3 + 5√3 = 7√3, but 2√3 + 5√2 cannot be combined further, much like unlike algebraic terms. Multiplying surds follows √a × √b = √(ab), which is how √3 × √12 simplifies directly to √36 = 6 without ever converting to decimals. This exact-value approach is preferred in many math contexts over decimal approximations because it avoids rounding error entirely and keeps answers in their precise mathematical form.

Square Root Reference Table

Memorizing perfect squares up to 15² makes mental estimation of nearby square roots much faster:

Since 50 sits roughly a quarter of the way from 49 (7²) to 64 (8²), a quick mental estimate would put √50 close to 7.1-7.2 — and the true value, 7.0711, confirms that linear-interpolation intuition is a reasonable first approximation, though it systematically underestimates slightly because the square root function curves (its graph is concave, not straight).

Worked Example: Approximating a Square Root by Hand (Babylonian Method)

Approximate √40 to four decimal places without a calculator, using the Babylonian (Newton's) method: xₙ₊₁ = (xₙ + N/xₙ) / 2. Start with a rough guess, x₀ = 6 (since 6² = 36 is close to 40). Iteration 1: x₁ = (6 + 40/6) / 2 = (6 + 6.6667) / 2 = 12.6667/2 = 6.3333. Iteration 2: x₂ = (6.3333 + 40/6.3333) / 2 = (6.3333 + 6.3158) / 2 = 12.6491/2 = 6.3246. The true value of √40 is 6.32456 — the method already matches to four decimal places after just two iterations, which is typical: the Babylonian method roughly doubles the number of correct digits with each iteration once it gets close. This is the same algorithm that pocket calculators and computer processors used for decades before dedicated hardware square-root instructions became standard, because it converges so quickly and needs only basic arithmetic.

Square Roots of Negative Numbers and Imaginary Numbers

Ordinary square roots are only defined for non-negative numbers within the real number system, because no real number multiplied by itself can produce a negative result (a negative times a negative is positive, and a positive times a positive is positive). To handle square roots of negative numbers, mathematics defines the imaginary unit i = √(−1). Using this definition, √(−16) = √(16 × −1) = √16 × √(−1) = 4i. Similarly, √(−50) = √(50 × −1) = √50 × i = 5√2 × i ≈ 7.071i. These "imaginary" results aren't just abstract curiosities — they're essential in electrical engineering (analyzing AC circuits), signal processing, and solving polynomial equations that have no real roots (like x² + 9 = 0, whose solutions are x = ±3i). A standard calculator will show an error for √(−16) specifically because it's restricted to real-number output; a scientific or complex-number calculator will correctly return 4i instead. Complex numbers like 4i also follow their own arithmetic rules — for instance, (4i)² = 16 × i² = 16 × (−1) = −16, which correctly "undoes" the square root and confirms 4i really is a valid square root of −16, just not a real one; every negative number in fact has two square roots in the complex system, ±4i in this case, mirroring the ± pair that every positive real number has. This extension from real to complex numbers is exactly why the fundamental theorem of algebra guarantees every polynomial equation has a full set of roots once complex numbers are allowed, even when no real solution exists.