√326 at a glance
- Exact value
- √326
- Decimal (10 places)
- 18.0554700853
- Rounded
- 18.1 · 18.06 · 18.055
- Perfect square?
- No — between 18² and 19²
- Rational?
- Irrational
- Both square roots
- ±18.055470
- Prime factorization
- 2 × 163
- Cube root
- 6.882389
How to simplify √326
The prime factorization of 326 is 2 × 163. Every prime appears only once, so there is no pair to bring outside the radical — √326 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 326, 2 and 163 appear an odd number of times, so √326 is irrational and 18.0554700853 is a rounded value.
Where √326 sits between perfect squares
324 = 18² and 361 = 19² are the nearest perfect squares, so √326 lies between 18 and 19. 326 is 2 above 324 and 35 below 361, so the root is closer to 18.
- Straight line between 324 and 361: 18.0541 (0.01% low)
- Tangent from 18, i.e. 18 + 2 ÷ 36: 18.0556 (0% high)
- Tangent from 19, i.e. 19 − 35 ÷ 38: 18.0789 (0.13% high)
For √326 the tangent at 18 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 326 is just 2 above 324.
Finding √326 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):
| Step | Guess x | 326 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 18.0000000000 | 18.1111111111 | 18.0555555556 | 4 |
| 2 | 18.0555555556 | 18.0553846154 | 18.0554700855 | 9 |
| 3 | 18.0554700855 | 18.0554700851 | 18.0554700853 | all 10 shown |
The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √326 = 18.0554700853 to every decimal shown.
√326 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √326 the pattern is [18; 18, 36] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √326 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 | 5.5 × 10⁻² |
| 325/18 | 18.0555555556 | 8.5 × 10⁻⁵ |
| 11,718/649 | 18.0554699538 | 1.3 × 10⁻⁷ |
| 211,249/11,700 | 18.0554700855 | 2.0 × 10⁻¹⁰ |
| 7,616,682/421,849 | 18.0554700853 | < 10⁻¹⁰ |
| 137,311,525/7,604,982 | 18.0554700853 | < 10⁻¹⁰ |
The same fractions solve Pell’s equation, x² − 326y² = 1. Its smallest solution in positive whole numbers is x = 325, y = 18.
√326 in geometry and everyday measurements
- A square patio or deck of 326 square feet is about 18.06 ft (18 ft 1 in) on each side, so edging all the way around takes 4 × √326 ≈ 72.2 ft.
- 326 is not a sum of two whole-number squares — the prime factor 163 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √326 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 18 box, because 1² + 1² + 18² = 326.
Square roots near √326 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √323 | √323 | 17.9722 | No |
| √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 |
- The cube root of 326 is about 6.882389.
- Squaring undoes the root: (√326)² = 326, while 326² = 106,276 — the number whose square root is 326.
Frequently asked questions
What is the square root of 326?
The square root of 326 is √326, about 18.0554700853. The negative root, −18.055470, also squares to 326.
Is the square root of 326 rational or irrational?
Irrational. 326 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 √326 be simplified?
No. 326 = 2 × 163 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √326 rounded to two decimal places?
√326 ≈ 18.06 to two decimal places (18.1 to one, 18.055 to three). Check: 18.06² = 326.1636, close to 326.