√329 at a glance
- Exact value
- √329
- Decimal (10 places)
- 18.1383571472
- Rounded
- 18.1 · 18.14 · 18.138
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.138357
- Prime factorization
- 7 × 47
- Cube root
- 6.903436
How to simplify √329
The prime factorization of 329 is 7 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √329 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 329, 7 and 47 appear an odd number of times, so √329 is irrational and 18.1383571472 is a rounded value.
Where √329 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √329 lies between 18 and 19. 329 is 5 above 324 and 32 below 361, so the root is closer to 18.
- Straight line between 324 and 361: 18.1351 (0.02% low)
- Tangent from 18, i.e. 18 + 5 ÷ 36: 18.1389 (0% high)
- Tangent from 19, i.e. 19 − 32 ÷ 38: 18.1579 (0.11% high)
For √329 the tangent at 18 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 329 is just 5 above 324.
Finding √329 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 329: following the tangent line down to zero simplifies to averaging x with 329 ÷ x.
Start from the nearest whole number, 18 (18² = 324):
| Step | Guess x | 329 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 18.0000000000 | 18.2777777778 | 18.1388888889 | 3 |
| 2 | 18.1388888889 | 18.1378254211 | 18.1383571550 | 8 |
| 3 | 18.1383571550 | 18.1383571394 | 18.1383571472 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √329 = 18.1383571472 to every decimal shown.
√329 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √329 the pattern is [18; 7, 4, 2, 1, 1, 4, 1, 1, 2, 4, 7, 36] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √329 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 18/1 | 18.0000000000 | 1.4 × 10⁻¹ |
| 127/7 | 18.1428571429 | 4.5 × 10⁻³ |
| 526/29 | 18.1379310345 | 4.3 × 10⁻⁴ |
| 1,179/65 | 18.1384615385 | 1.0 × 10⁻⁴ |
| 1,705/94 | 18.1382978723 | 5.9 × 10⁻⁵ |
| 2,884/159 | 18.1383647799 | 7.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 329y² = 1. Its smallest solution in positive whole numbers is x = 2,376,415, y = 131,016.
√329 in geometry and everyday measurements
- A square patio or deck of 329 square feet is about 18.14 ft (18 ft 2 in) on each side, so edging all the way around takes 4 × √329 ≈ 72.6 ft.
- 329 is not a sum of two whole-number squares — the prime factor 7 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √329 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 18 box, because 1² + 2² + 18² = 329.
Square roots near √329 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √326 | √326 | 18.0555 | No |
| √327 | √327 | 18.0831 | No |
| √328 | 2√82 | 18.1108 | No |
| √329 | √329 | 18.1384 | No |
| √330 | √330 | 18.1659 | No |
| √331 | √331 | 18.1934 | No |
| √332 | 2√83 | 18.2209 | No |
- The cube root of 329 is about 6.903436.
- Squaring undoes the root: (√329)² = 329, while 329² = 108,241 — the number whose square root is 329.
Frequently asked questions
What is the square root of 329?
The square root of 329 is √329, about 18.1383571472. The negative root, −18.138357, also squares to 329.
Is the square root of 329 rational or irrational?
Irrational. 329 is not a perfect square — it falls between 324 and 361 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √329 be simplified?
No. 329 = 7 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √329 rounded to two decimal places?
√329 ≈ 18.14 to two decimal places (18.1 to one, 18.138 to three). Check: 18.14² = 329.0596, close to 329.