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.

Results

Square Root √n12
Cube Root ∛n—
Squared n²20,736

📐 Formula

√n = n^(1/2). ∛n = n^(1/3). n² = n × n

How to Use the Square Root Calculator

1

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.

2

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.

3

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.

4

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.

Sources & Methodology

Calculations are based on the most current publicly available data from authoritative government and industry sources:

Frequently Asked Questions

A perfect square is a number that is the product of an integer multiplied by itself. Examples: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. The square root of a perfect square is always a whole number.
In real mathematics, no — the square root of a negative number is undefined. In complex mathematics, √−1 = i (the imaginary unit). This calculator works with real numbers only.
√n (square root) asks "what number multiplied by itself equals n?" while n² (squared) multiplies n by itself. They are inverse operations: √(n²) = n and (√n)² = n.
The perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. These are the squares of integers 1 through 10. Perfect squares are useful in geometry (area calculations), the Pythagorean theorem, and simplifying radical expressions in algebra.
Find the two perfect squares it falls between, then interpolate. Example: √50 — 49 = 7² and 64 = 8², so √50 is between 7 and 8, closer to 7. More precisely: 50 − 49 = 1, 64 − 49 = 15, so √50 ≈ 7 + 1/15 ≈ 7.07. The actual value is 7.071 — this method is accurate to one decimal place.