Square Root of 323

√323 at a glance

Exact value

√323

Decimal (10 places)

17.9722007556

Rounded

18.0 · 17.97 · 17.972

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.972201

Prime factorization

17 × 19

Cube root

6.861212

How to simplify √323

The prime factorization of 323 is 17 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √323 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 323, 17 and 19 appear an odd number of times, so √323 is irrational and 17.9722007556 is a rounded value.

Where √323 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √323 lies between 17 and 18. 323 is 34 above 289 and 1 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.9714 (0% low)
• Tangent from 17, i.e. 17 + 34 ÷ 34: 18.0000 (0.15% high)
• Tangent from 18, i.e. 18 − 1 ÷ 36: 17.9722 (0% high)

For √323 the tangent at 18 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 323 is just 1 below 324.

Finding √323 with the Babylonian method

If a guess is too big, 323 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√323) in one step.

Start from the nearest whole number, 18 (18² = 324):

Because the starting guess was already close, two steps are enough to match √323 = 17.9722007556 to every decimal shown.

√323 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √323 the pattern is [17; 1, 34] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √323 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 − 323y² = 1. Its smallest solution in positive whole numbers is x = 18, y = 1.

√323 in geometry and everyday measurements

• A square patio or deck of 323 square feet is about 17.97 ft (18 ft) on each side, so edging all the way around takes 4 × √323 ≈ 71.9 ft.
• 323 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √323 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 17 box, because 3² + 5² + 17² = 323.

Square roots near √323 and related roots

• The cube root of 323 is about 6.861212.
• Squaring undoes the root: (√323)² = 323, while 323² = 104,329 — the number whose square root is 323.

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