√324 at a glance
- Exact value
- 18
- Decimal (10 places)
- 18
- Rounded
- 18
- Perfect square?
- Yes — 18²
- Rational?
- Rational (whole number)
- Both square roots
- ±18
- Prime factorization
- 2² × 3⁴
- Cube root
- 6.868285
How to simplify √324
Write 324 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√324 = 2 × 3² = 18
324 has 8 factor pairs (1 × 324, 2 × 162, 3 × 108, 4 × 81, 6 × 54, 9 × 36, 12 × 27, 18 × 18). The pair with two equal numbers, 18 × 18, is what makes it a perfect square — and why it has an odd number of factors.
Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 324 = 1 + 3 + 5 + 7 + … + 35, the first 18 odd numbers.
Where √324 sits between perfect squares
324 is 18². The perfect squares on either side are 289 (17²) and 361 (19²), so the gaps are 35 below and 37 above — consecutive squares always differ by consecutive odd numbers.
Finding √324 with the Babylonian method
Picture a rectangle with an area of 324 and one side x; the other side must be 324 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √324.
To show the method at work on a perfect square, start one above the answer, at 19:
| Step | Guess x | 324 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 17.0526315789 | 18.0263157895 | 1 |
| 2 | 18.0263157895 | 17.9737226277 | 18.0000192086 | 4 |
| 3 | 18.0000192086 | 17.9999807914 | 18.0000000000 | all 10 shown |
The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √324 = 18.0000000000 to every decimal shown.
√324 in geometry and everyday measurements
- A square patio or deck of 324 square feet is exactly 18 ft (18 ft) on each side, so edging all the way around takes 4 × √324 ≈ 72 ft.
- 324 is not a sum of two whole-number squares — no pair of squares adds up to it — so √324 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 8 × 16 box, because 2² + 8² + 16² = 324.
Square roots near √324 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √321 | √321 | 17.9165 | No |
| √322 | √322 | 17.9444 | No |
| √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 |
- The cube root of 324 is about 6.868285.
- Taking the square root twice gives the fourth root: √18 ≈ 4.242641.
- Because 324 = 4 × 81, the root is twice √81: 2 × 9 ≈ 18.
Frequently asked questions
What is the square root of 324?
The square root of 324 is 18, because 18 × 18 = 324. Strictly, 324 has two square roots, 18 and −18; the √ symbol means the positive one.
Is the square root of 324 rational or irrational?
Rational. 324 is a perfect square, so its square root is the whole number 18.
Can √324 be simplified?
Yes, all the way to a whole number: √324 = 18.
What is √324 rounded to two decimal places?
√324 is exactly 18, or 18.00 written to two decimal places.