Square Root of 82

√82 at a glance

Exact value

√82

Decimal (10 places)

9.0553851381

Rounded

9.1 · 9.06 · 9.055

Perfect square?

No — between 9² and 10²

Rational?

Irrational

Both square roots

±9.055385

Prime factorization

2 × 41

Cube root

4.344481

How to simplify √82

The prime factorization of 82 is 2 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √82 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 82, 2 and 41 appear an odd number of times, so √82 is irrational and 9.0553851381 is a rounded value.

Where √82 sits between perfect squares

81 = 9² and 100 = 10² are the nearest perfect squares, so √82 lies between 9 and 10. 82 is 1 above 81 and 18 below 100, so the root is closer to 9.

• Straight line between 81 and 100: 9.0526 (0.03% low)
• Tangent from 9, i.e. 9 + 1 ÷ 18: 9.0556 (0% high)
• Tangent from 10, i.e. 10 − 18 ÷ 20: 9.1000 (0.49% high)

For √82 the tangent at 9 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 82 is just 1 above 81.

Finding √82 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √82 = 9.0553851381 to every decimal shown.

√82 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √82 the pattern is [9; 18] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 82 is one more than a perfect square (9² + 1). A pattern that never ends is one more proof that √82 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 82y² = 1. Its smallest solution in positive whole numbers is x = 163, y = 18. Because the period is odd, the equation with −1 on the right also has a solution: 9² − 82 × 1² = −1.

√82 in geometry and everyday measurements

• A square room or garden bed covering 82 square feet measures about 9.06 ft (9 ft 1 in) along each wall.
• 82 = 1² + 9², so by the Pythagorean theorem √82 is the diagonal of a 1 × 9 rectangle — and the distance between the points (0, 0) and (1, 9) on a grid.

Square roots near √82 and related roots

• The cube root of 82 is about 4.344481.
• Four times the radicand doubles the root: √328 = 2 × √82 ≈ 18.11077.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.