Square Root Calculator › Square root of 357
Square Root of 357
√357 ≈ 18.894444
357 is not a perfect square, so √357 is an irrational number that is approximately 18.894444. In simplified radical form, √357 = √357.
√357 lies between √324 = 18 and √361 = 19.
| Number | 357 |
| Square root (decimal) | 18.894444 |
| Simplified radical | √357 |
| Perfect square? | No |
| Rational? | No |
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Prime factorization of 357
357 = 3 × 7 × 17. This is what determines the simplified radical: for √357 you pull out one copy of any prime that appears twice.
No prime appears twice, so nothing can be taken out — √357 is already in simplest radical form as √357.
Estimating √357 by hand
√357 falls between √324 = 18 and √361 = 19, so it starts at 18.something. The Babylonian method refines that guess by averaging it with 357 ÷ the guess:
| Step | Estimate |
| Start | 18 |
| Step 1 | 18.916667 |
| Step 2 | 18.894457 |
| Step 3 | 18.894444 |
After 3 steps the estimate is 18.894444, matching √357 ≈ 18.894444.
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Frequently asked questions
Is 357 a perfect square?
No, 357 is not a perfect square, so its square root is irrational and approximately 18.894444.
What is the square root of 357 in simplest radical form?
√357 = √357 in simplest radical form.