Square Root Calculator
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Calculate the square root (√), cube root (∛), and square (n²) of any number. Shows the formula and exact result.
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📐 Formula
√n = n^(1/2). ∛n = n^(1/3). n² = n × n
How to Use the Square Root Calculator
Enter the number
Input any positive number. The calculator computes the principal square root (the positive root). For negative inputs, it returns the imaginary result in i notation.
Switch to cube or nth root
Toggle between square root (√), cube root (∛), and nth root (ⁿ√) using the mode selector. Enter any integer n for a custom root.
View the step-by-step method
Expand the working section to see the long-division or estimation method — useful for understanding manual calculation or checking maths homework.
Use the perfect squares reference
The sidebar shows the first 20 perfect squares. If your number appears here, the square root is an exact whole number.
Understanding Square Roots
The square root of a number n is the value x such that x × x = n. Every positive number has two square roots: a positive one (the principal root) and a negative one. √25 = ±5, because 5² = 25 and (-5)² = 25. In most practical contexts, the positive root is used. Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100...) produce whole-number square roots. All other positive integers produce irrational numbers — they cannot be expressed as exact fractions and have non-repeating, non-terminating decimal expansions. √2 = 1.41421356...; √3 = 1.73205080...
How to Calculate Square Root by Hand
For a rough estimate: identify the two perfect squares your number falls between, then interpolate. √50 falls between √49 = 7 and √64 = 8. 50 is 1/15 of the way from 49 to 64, so √50 ≈ 7 + (1/15) = 7.07. The actual value is 7.0711. For more precision, use the Babylonian method (Newton's method): start with an initial guess g, then iterate: g_new = (g + n/g) ÷ 2. Starting with g=7 for √50: (7 + 50/7) ÷ 2 = (7 + 7.143) ÷ 2 = 7.071. Highly accurate after just 2–3 iterations.
Square Roots in Real-World Applications
Square roots appear throughout practical mathematics: in the Pythagorean theorem (c = √(a² + b²) for right triangles); in the standard deviation formula in statistics (√ of variance); in physics for calculating velocity, displacement, and wave frequency; in finance for volatility calculations; and in geometry for finding side lengths from areas. The distance between two points (x₁,y₁) and (x₂,y₂) is √((x₂-x₁)² + (y₂-y₁)²) — the core formula behind GPS positioning, computer graphics, and map applications.
How to Calculate a Square Root by Hand: Worked Example (Newton's Method)
Estimate √2 without a calculator, using Newton's method (also called the Babylonian method): repeatedly apply xₙ₊₁ = 0.5 × (xₙ + 2/xₙ), starting from a rough guess.
Starting guess: x₀ = 1.5 (a reasonable estimate, since 1.5² = 2.25, close to 2).
Iteration 1: x₁ = 0.5 × (1.5 + 2/1.5) = 0.5 × (1.5 + 1.3333) = 1.41667.
Iteration 2: x₂ = 0.5 × (1.41667 + 2/1.41667) = 1.414216.
Iteration 3: x₃ = 0.5 × (1.414216 + 2/1.414216) = 1.4142136.
After just three iterations, the estimate matches the true value of √2 ≈ 1.41421356 to seven decimal places. This is why Newton's method is the algorithm most calculators and computers actually use internally — it converges extremely fast, roughly doubling the number of correct digits with each iteration.
How do you estimate a square root without any iteration at all?
Find the two nearest perfect squares. For √50: 49 = 7² and 64 = 8², so √50 falls between 7 and 8, closer to 7 since 50 is much nearer to 49. A linear estimate: 7 + (50−49)/(64−49) = 7 + 1/15 ≈ 7.067 — reasonably close to the true value of 7.071, good enough for a quick sanity check without a calculator.
Where Do Square Roots Show Up in Everyday Calculations?
How is the Pythagorean theorem a square-root problem?
Finding the diagonal of a rectangle (or the hypotenuse of a right triangle) with legs of 6 and 8 uses c = √(a² + b²) = √(36 + 64) = √100 = 10 — a clean example, though most real measurements produce a non-perfect-square result requiring the calculator (or Newton's method above) to resolve precisely.
Why does standard deviation in statistics involve a square root?
Standard deviation is defined as the square root of variance specifically to return the spread of data to the same units as the original data — variance (an average of squared deviations) is in squared units, which aren't directly interpretable. Taking the square root converts a statistic like "variance = 25 square inches" back into a directly meaningful "standard deviation = 5 inches."
Is there a difference between "square root" and "principal square root"?
Every positive number technically has two square roots — a positive and a negative one, since both (5)² and (−5)² equal 25. The principal square root (what a calculator returns by default) is always the non-negative one. Physics and engineering problems (like solving for a distance or time, which can't be negative) typically only need the principal root, while pure algebra problems solving x² = 25 correctly require both x = 5 and x = −5 as solutions.
Frequently Asked Questions
Sources & Methodology
Calculations are based on the most current publicly available data from authoritative government and industry sources: