√327 at a glance
- Exact value
- √327
- Decimal (10 places)
- 18.0831413200
- Rounded
- 18.1 · 18.08 · 18.083
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.083141
- Prime factorization
- 3 × 109
- Cube root
- 6.889419
How to simplify √327
The prime factorization of 327 is 3 × 109. Every prime appears only once, so there is no pair to bring outside the radical — √327 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 327, 3 and 109 appear an odd number of times, so √327 is irrational and 18.0831413200 is a rounded value.
Where √327 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √327 lies between 18 and 19. 327 is 3 above 324 and 34 below 361, so the root is closer to 18.
- Straight line between 324 and 361: 18.0811 (0.01% low)
- Tangent from 18, i.e. 18 + 3 ÷ 36: 18.0833 (0% high)
- Tangent from 19, i.e. 19 − 34 ÷ 38: 18.1053 (0.12% high)
For √327 the tangent at 18 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 327 is just 3 above 324.
Finding √327 with the Babylonian method
If a guess is too big, 327 ÷ 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√327) in one step.
Start from the nearest whole number, 18 (18² = 324):
| Step | Guess x | 327 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 18.0000000000 | 18.1666666667 | 18.0833333333 | 3 |
| 2 | 18.0833333333 | 18.0829493088 | 18.0831413210 | 8 |
| 3 | 18.0831413210 | 18.0831413190 | 18.0831413200 | 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 √327 = 18.0831413200 to every decimal shown.
√327 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √327 the pattern is [18; 12, 36] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √327 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 | 8.3 × 10⁻² |
| 217/12 | 18.0833333333 | 1.9 × 10⁻⁴ |
| 7,830/433 | 18.0831408776 | 4.4 × 10⁻⁷ |
| 94,177/5,208 | 18.0831413210 | 1.0 × 10⁻⁹ |
| 3,398,202/187,921 | 18.0831413200 | < 10⁻¹⁰ |
| 40,872,601/2,260,260 | 18.0831413200 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 327y² = 1. Its smallest solution in positive whole numbers is x = 217, y = 12.
√327 in geometry and everyday measurements
- A square patio or deck of 327 square feet is about 18.08 ft (18 ft 1 in) on each side, so edging all the way around takes 4 × √327 ≈ 72.3 ft.
- 327 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √327 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √327 as its space diagonal.
Square roots near √327 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √324 | 18 | 18.0000 | Yes |
| √325 | 5√13 | 18.0278 | No |
| √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 |
- The cube root of 327 is about 6.889419.
- Squaring undoes the root: (√327)² = 327, while 327² = 106,929 — the number whose square root is 327.
Frequently asked questions
What is the square root of 327?
The square root of 327 is √327, about 18.0831413200. The negative root, −18.083141, also squares to 327.
Is the square root of 327 rational or irrational?
Irrational. 327 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 √327 be simplified?
No. 327 = 3 × 109 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √327 rounded to two decimal places?
√327 ≈ 18.08 to two decimal places (18.1 to one, 18.083 to three). Check: 18.08² = 326.8864, close to 327.