Square Root Calculator › Square root of 333
Square Root of 333
√333 ≈ 18.248288
333 is not a perfect square, so √333 is an irrational number that is approximately 18.248288. In simplified radical form, √333 = 3√37.
√333 lies between √324 = 18 and √361 = 19.
| Number | 333 |
| Square root (decimal) | 18.248288 |
| Simplified radical | 3√37 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 333
333 = 3^2 × 37. This is what determines the simplified radical: for √333 you pull out one copy of any prime that appears twice.
Here the paired factor 3 comes out from under the root, leaving √333 = 3√37.
Estimating √333 by hand
√333 falls between √324 = 18 and √361 = 19, so it starts at 18.something. The Babylonian method refines that guess by averaging it with 333 ÷ the guess:
| Step | Estimate |
| Start | 18 |
| Step 1 | 18.25 |
| Step 2 | 18.248288 |
| Step 3 | 18.248288 |
After 3 steps the estimate is 18.248288, matching √333 ≈ 18.248288.
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Frequently asked questions
Is 333 a perfect square?
No, 333 is not a perfect square, so its square root is irrational and approximately 18.248288.
What is the square root of 333 in simplest radical form?
√333 = 3√37 in simplest radical form.