Square Root of 321

√321 at a glance

Exact value

√321

Decimal (10 places)

17.9164728672

Rounded

17.9 · 17.92 · 17.916

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.916473

Prime factorization

3 × 107

Cube root

6.847021

How to simplify √321

The prime factorization of 321 is 3 × 107. Every prime appears only once, so there is no pair to bring outside the radical — √321 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 321, 3 and 107 appear an odd number of times, so √321 is irrational and 17.9164728672 is a rounded value.

Where √321 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √321 lies between 17 and 18. 321 is 32 above 289 and 3 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.9143 (0.01% low)
• Tangent from 17, i.e. 17 + 32 ÷ 34: 17.9412 (0.14% high)
• Tangent from 18, i.e. 18 − 3 ÷ 36: 17.9167 (0% high)

For √321 the tangent at 18 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 321 is just 3 below 324.

Finding √321 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 321: following the tangent line down to zero simplifies to averaging x with 321 ÷ x.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √321 = 17.9164728672 to every decimal shown.

√321 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √321 the pattern is [17; 1, 10, 1, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √321 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 − 321y² = 1. Its smallest solution in positive whole numbers is x = 215, y = 12.

√321 in geometry and everyday measurements

• A square patio or deck of 321 square feet is about 17.92 ft (17 ft 11 in) on each side, so edging all the way around takes 4 × √321 ≈ 71.7 ft.
• 321 is not a sum of two whole-number squares — the prime factor 3 and 107 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √321 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 8 × 16 box, because 1² + 8² + 16² = 321.

Square roots near √321 and related roots

• The cube root of 321 is about 6.847021.
• Squaring undoes the root: (√321)² = 321, while 321² = 103,041 — the number whose square root is 321.

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