Square Root Calculator › Square root of 321
Square Root of 321
√321 ≈ 17.916473
321 is not a perfect square, so √321 is an irrational number that is approximately 17.916473. In simplified radical form, √321 = √321.
√321 lies between √289 = 17 and √324 = 18.
| Number | 321 |
| Square root (decimal) | 17.916473 |
| Simplified radical | √321 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 321
321 = 3 × 107. This is what determines the simplified radical: for √321 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √321 is already in simplest radical form as √321.
Estimating √321 by hand
√321 falls between √289 = 17 and √324 = 18, so it starts at 17.something. The Babylonian method refines that guess by averaging it with 321 ÷ the guess:
| Step | Estimate |
| Start | 17 |
| Step 1 | 17.941176 |
| Step 2 | 17.91649 |
| Step 3 | 17.916473 |
After 3 steps the estimate is 17.916473, matching √321 ≈ 17.916473.
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Frequently asked questions
Is 321 a perfect square?
No, 321 is not a perfect square, so its square root is irrational and approximately 17.916473.
What is the square root of 321 in simplest radical form?
√321 = √321 in simplest radical form.