Square Root of 322

√322 at a glance

Exact value

√322

Decimal (10 places)

17.9443584449

Rounded

17.9 · 17.94 · 17.944

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.944358

Prime factorization

2 × 7 × 23

Cube root

6.854124

How to simplify √322

The prime factorization of 322 is 2 × 7 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √322 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 322, 2, 7 and 23 appear an odd number of times, so √322 is irrational and 17.9443584449 is a rounded value.

Where √322 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √322 lies between 17 and 18. 322 is 33 above 289 and 2 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.9429 (0.01% low)
• Tangent from 17, i.e. 17 + 33 ÷ 34: 17.9706 (0.15% high)
• Tangent from 18, i.e. 18 − 2 ÷ 36: 17.9444 (0% high)

For √322 the tangent at 18 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 322 is just 2 below 324.

Finding √322 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, 18 (18² = 324):

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √322 = 17.9443584449 to every decimal shown.

√322 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √322 the pattern is [17; 1, 16, 1, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √322 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 − 322y² = 1. Its smallest solution in positive whole numbers is x = 323, y = 18.

√322 in geometry and everyday measurements

• A square patio or deck of 322 square feet is about 17.94 ft (17 ft 11 in) on each side, so edging all the way around takes 4 × √322 ≈ 71.8 ft.
• 322 is not a sum of two whole-number squares — the prime factor 7 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √322 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 12 × 13 box, because 3² + 12² + 13² = 322.

Square roots near √322 and related roots

• The cube root of 322 is about 6.854124.
• Squaring undoes the root: (√322)² = 322, while 322² = 103,684 — the number whose square root is 322.

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